Limited broadcast domination

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by

Feiran Yang

B.Sc., Sichuan University, 2011

B.Sc., University of Prince Edward Island, 2012 M.Sc., University of Victoria, 2015

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of Mathematics and Statistics

c

Feiran Yang, 2019 University of Victoria

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Limited Broadcast Domination

by

Feiran Yang

B.Sc., Sichuan University, 2011

B.Sc., University of Prince Edward Island, 2012 M.Sc., University of Victoria, 2015

Supervisory Committee

Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics)

Dr. Michael A. Henning, Co-supervisor

(Department of Mathematics, University of Johannesburg)

Dr. Jing Huang, Academic Unit Member (Department of Mathematics and Statistics)

Dr. Kieka Mynhardt, Academic Unit Member (Department of Mathematics and Statistics)

Dr. Frank Ruskey, Non-Unit Member (Department of Computer Science)

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Supervisory Committee

Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics)

Dr. Michael A. Henning, Co-supervisor

(Department of Mathematics, University of Johannesburg)

Dr. Jing Huang, Academic Unit Member (Department of Mathematics and Statistics)

Dr. Kieka Mynhardt, Academic Unit Member (Department of Mathematics and Statistics)

Dr. Frank Ruskey, Non-Unit Member (Department of Computer Science)

ABSTRACT

Let G = (V, E) be a graph and f be a function such that f : V → {0, 1, 2, . . . , k}. Let V_f+ = {v : f (v) > 0}. If for every vertex v 6∈ V_f+ there exists a vertex w ∈ V_f+ such that d(v, w) ≤ f (w) then f is called a k-limited dominating broadcast of G. The quantity P

v∈V

f (v) is called the cost of the broadcast. The minimum cost of a dominating broadcast is called the k-limited broadcast domination number of G, and is denoted by γb,k(G). This parameter γb,k(G) is a variation of the well-studied

broadcast domination number. The value γb,k(G) can also be defined as a solution to

an integer linear programming problem. The solution to the dual problem is defined as the k-limited multipacking number.

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We begin with a survey of known results and background related to these broadcast domination related parameters. In Chapter 3, we give a proof of NP-completeness for the problem of determining the k-limited broadcast domination number for a fixed graph G, as well as a proof of NP-completeness for its dual problem of determining the k-limited multipacking number. Chapter 4 focuses on cubic and subcubic graphs. Here we give an upper bound for the 2-limited broadcast domination number of (C4, C6)-free cubic graphs. In Chapter 5, we describe algorithms which determine the

k-limited broadcast domination number for strongly chordal graphs, interval graphs, circular arc graphs and proper interval bigraphs in polynomial time. In Chapter 6, we show that the k-limited broadcast domination number for trees can be determined in linear time. We specifically give a linear time algorithm which determines the 2-limited broadcast domination number of trees.

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List of Figures

Figure 2.1 A graph G with γb(G) = 4, mp(G) = 2. . . 7

Figure 2.2 A graph H with γb(H) = 4, mp(H) = 3. . . 7

Figure 2.3 The tree T9. . . 11

Figure 2.4 The tree Tk of order 3k +3 that is tight for the bound of k-limited broadcast domination number in Theorem 2.18. . . 12

Figure 2.5 Tree T . . . 14

Figure 2.6 First step to construct ST. . . 14

Figure 2.7 Shadow tree ST. . . 14

Figure 2.8 A graph G with an induced subgraph having larger broadcast domination number. . . 15

Figure 2.9 The 3-trampoline. . . 18

Figure 3.1 A replacement of edge uv when k = 3. . . 24

Figure 3.2 A replacement of edge uv when k = 4. . . 24

Figure 3.3 One of the three special vertex structures in Hv when k = 4. . . 26

Figure 3.4 The vertex gadget associated with v in G0 when k = 4. . . 27

Figure 5.1 An interval graph and its interval representation. . . 57

Figure 5.2 An interval bigraph and its interval representation. . . 61

Figure 5.3 Graph G used to illustrate Algorithm 1. . . 63

Figure 6.1 A tree and its shadow tree. . . 69

Figure 6.2 A tree and its parse tree. . . 72

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Introduction

The original concept of broadcast domination was introduced by Erwin in 2001 [28]. For a graph G, each vertex v in G is assigned a strength f (v) ∈ {0, 1, 2, ..., diam(G)}. Let V_f+ denote the set of vertices with positive strengths. For vertex u ∈ V (G), if there exists a vertex v ∈ V_f+ such that d(u, v) ≤ f (v), then we say u hears the broadcast from v and f (v) is the called the weight or cost of v. If every vertex of G hears the broadcast from at least one vertex, then f is called a dominating broadcast. The cost of f is the quantityP

v∈V f (v). The broadcast domination number of G is the

minimum cost of a dominating broadcast of G, and is denoted by γb(G). A broadcast

is called optimal if it has a cost of γb(G). Notice that if we restrict f (v) ∈ {0, 1} for

all v ∈ V (G), then f is the characteristic function of a dominating set. It follows immediately then that γb(G) is at most the graph’s domination number, γ(G). In

this dissertation, we will refer to the broadcast domination problem as the task of finding the broadcast domination number and also an optimal dominating broadcast. Unless specified otherwise, all broadcasts are considered to be dominating in this dissertation.

In the original paper of Erwin [29], another version of broadcast was mentioned but not studied. In this scenario, each vertex v in G can be assigned with strength f (v) ∈ {0, 1, 2, ..., k} for some fixed k. To distinguish this problem from the standard broadcast domination problem, we call this the k-limited broadcast domination prob-lem. Again, we want to find the minimum cost of k-limited dominating broadcast and also the function f corresponding to the optimal result. We denote the k-limited broadcast domination number as γb,k(G). One can easily see that the 1-limited

broad-cast domination number γb,1(G) is the domination number γ(G). By definition, we

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This dissertation is organized as follows. We introduce basic notation and ter-minology and survey all previous results related to broadcast domination in Chapter 2. In Chapter 3, we will give proofs to the NP-completeness of finding the k-limited broadcast domination number and the k-limited multipacking number (which will be defined in Chapter 2) for any fixed graph G. In Chapter 4, we give an upper bound for the 2-limited broadcast domination number on (C4, C6)-free subcubic graphs and

also on (C4, C6)-free cubic graphs. The proof technique is similar to the method

used by Henning, L¨owenstein and Rautenbach [38]. In Chapter 5, we give algorith-mic results which find optimal k-limited dominating broadcasts for strongly chordal graphs, interval graphs and proper interval bigraphs. The k-limited broadcast domi-nation problem in these graph classes can be solved in O(n3), O(n2) and O(n3) time, respectively. In Chapter 6, we focus on the k-limited broadcast domination problem for trees. We first show that two different, established methods give linear algorithms to find the k-limited broadcast domination number of a given tree. Then we focus on 2-limited dominating broadcasts and give a detailed linear time algorithm for find-ing the 2-limited broadcast domination number and an optimal 2-limited dominatfind-ing broadcast for trees. This algorithm is different from the algorithm given in [15]. In Chapter 7, we give a summary of open problems and conjectures that are related to the broadcast domination problem.

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Chapter 2 Background

The theory surrounding the broadcast domination problem has developed during the past years since its introduction in 2001. In this chapter we will have a deep look at the problem and related topics. Our notation is standard and follows [19]. Only terminology which is not standard in graph theory is explicitly defined in the the-sis. In particular, the closed neighbourhood of v is the set of vertices N (v) ∪ {v}, and is denoted as N [v]. The k-closed neighbourhood of v is the set of vertices Nk[v] = {u ∈ V (G) : d(u, v) ≤ k}. The eccentricity of a vertex v in graph G is the

maximum value of d(u, v) for any possible vertex u. The radius of graph G is the minimum eccentricity of the graph, which is denoted as rad(G).

2.1 General Broadcast Domination Theory

Broadcast domination was first introduced in Erwin’s PhD thesis [28] in 2001 as a model of communication networks, and was further investigated in the resulting publication [29] in 2002. From this work comes the following introductory results which give bounds on the broadcast domination number.

Proposition 2.1. [29] For every connected graph G, γb(G) ≤ min{rad(G), γ(G)}.

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Theorem 2.2. [29] If G is a connected graph, then γb(G) ≥ diam(G) + 1 3 .

Since the path on n vertices, Pn, has γ(Pn) =

l

diam(Pn)+1

3

m

= dn₃e, these bounds give the exact value of the broadcast domination number for Pn where n ≥ 2.

Corollary 2.3. [29] For every integer n ≥ 2, γb(Pn) = γ(Pn) =

ln 3 m

. In fact we will see later in this chapter that_n

3 is an upper bound for the broadcast

domination number of any graph on n vertices. Erwin also gave a sufficient condition for a graph G to have γb(G) ≤ 3.

Proposition 2.4. [29] Let G be a connected graph and k = min{rad(G), γ(G)}. If 1 ≤ k ≤ 3, then γb(G) = k.

If a graph G has rad(G) = γb(G), then we call G radial. This is an important

class of graphs with respect to broadcast domination and various papers exist which classify radial graphs.

Later in 2006, in the paper by Dunbar et al. [27], broadcast domination was inves-tigated further. In this work, they defined a key concept called efficient broadcast. A dominating broadcast is efficient if no vertex hears the broadcast from two different vertices.

Theorem 2.5. [27] Every graph G has an optimal efficient dominating broadcast. The idea given in the proof of this result is that for an inefficient dominating broadcast f , there exists a vertex v such that d(v, x) ≤ f (x) and d(v, y) ≤ f (y) where x, y are vertices that are broadcasting. Then we can replace f (x) and f (y) by assigning a vertex w that is within distance f (y) from x and also within distance f (x) from y with weight f (x) + f (y), and reassigning f (x) = f (y) = 0. The cost of the new broadcast is equal to the cost of the original broadcast and this process can be repeated until an efficient broadcast is found.

Given a graph G, the problem of determining the domination number, γ(G) is NP-complete [33]. Since a dominating broadcast is a generalized dominating set, it

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could be hypothesized that the problem of determining γb(G) is also NP-complete.

However, in 2006, Heggernes and Lokshtanov [36] used the fact that every graph has an optimal efficient dominating broadcast to give a polynomial time algorithm to find the broadcast domination number of any graph.

Theorem 2.6. [36] The broadcast domination number of a graph G can be found in O(n6_).

To find an optimal dominating broadcast, Heggernes and Lokshtanov first consid-ered a ball graph of the original graph. The ball graph of a dominating broadcast is a graph whose vertices are the broadcast neighbourhoods of the original graph where two vertices of the ball graph are adjacent if the two broadcast neighbourhoods con-tain a pair of adjacent vertices in the original graph. Since there exists an optimal efficient broadcast, there exists a ball graph which is a path or a cycle for an opti-mal dominating broadcast. The idea is to assume that for each vertex v ∈ V (G), the broadcast neighbourhood of v is an end-point of a ball graph which is a path. This finds all possible optimal dominating broadcasts which are paths. Next the case when the ball graph is a cycle is considered. A broadcast neighbourhood from the original graph is first removed, giving a path ball graph for the remaining subgraph. The running time of this process when the ball graph is a path is O(n4), and when the ball graph is a cycle the running time is O(n6). Heggernes and Sæther [37] later conjectured that the broadcast domination problem can be solved in O(n5_{) time in}

general.

2.2 Other Types of Broadcasts

The domination number of a graph and the packing number of a graph are related in that both values provide an optimal solution to two integer linear programming problems which are duals. In this sense, a concept that comes alongside a broadcast is that of a multipacking. The term multipacking was first introduced in the Master’s thesis of Teshima [60] in 2012. Here broadcast domination was considered as a linear programming problem and the linear programming dual was used to give the definition of a multipacking.

The definition of γb(G) leads to a 0-1 integer program, B(G), which we now

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variable giving the truth value of the statement “the strength of the broadcast at vertex i equals k”. The formulation of B(G) is:

Minimize rad (G) P k=1 n P i=1 kxik, subject to: P d(i,j)≤k

xik ≥ 1 for each vertex j,

xik ∈ {0, 1} for each vertex i and integer k ∈ {1, 2, . . . , rad (G)}.

We use Bf(G) to denote the linear programming relaxation of B(G). The linear

programming dual of Bf(G) is M Pf(G): Maximize n P j=1 yj, subject to: P d(i,j)≤k

yj ≤ k for each vertex i and integer k ∈ {1, 2, . . . , rad (G)},

yj ≥ 0 for each j.

Let M P (G) denote the 0-1 integer program whose linear programming relaxation is M Pf(G). For each feasible solution of M P (G), the vector y = (y1, y2, . . . , yn) is

the characteristic vector of a multipacking, a subset S ⊆ V (G) such that for every v ∈ V and for every 1 ≤ k ≤ rad(G), |Nk[v] ∩ S| ≤ k. The multipacking number of G

is the maximum cardinality of a multipacking of G, and is denoted by mp(G). Multipacking was studied further by Hartnell and Mynhardt [35] in 2014. Here an extension of Erwin’s inequality chain was given with the addition of the multipacking number.

Theorem 2.7. [35] For any connected graph G, diam(G) + 1

3

≤ mp(G) ≤ γb(G) ≤ min{rad(G), γ(G)}.

Proof : For a graph G, consider a diametrical path v0, v1, ..., vdiam(G)with diam(G) + 1

vertices. The set {v0, v3, ...} is a multipacking. Therefore mp(G) ≥

l

diam(G)+1 3

m . The inequality mp(G) ≤ γb(G) comes directly from the duality theorem of linear

programming and γb(G) ≤ min{rad(G), γ(G)} is from Proposition 2.1.

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Theorem 2.8. [35] For any connected graph G, γb(G)/mp(G) < 3.

Proof : Since ldiam(G)+1₃ m≤ mp(G), we have

3mp(G) ≥ diam(G) + 1 > rad(G) ≥ γb(G).

So γb(G)/mp(G) < 3.

Interestingly, no graph G has been found such that γb(G)/mp(G) > 2. The

small example shown in Figure 2.1 (which was provided in [35]) has γb(G) = 4 and

mp(G) = 2.

Figure 2.1: A graph G with γb(G) = 4, mp(G) = 2.

Below is an example (as seen in [35]) which shows that for any given integer i, there is a graph G that has |V (G)| ≥ i and γb(G)/mp(G) = 4/3. To obtain G, make i

copies of the graph H in Figure 2.2 and connect them together in series by adding an edge between vertex rj in Hj and vertex lj+1 in Hj+1. The resulting graph will have

γb(G)/mp(G) = 4/3.

lj rj

Figure 2.2: A graph H with γb(H) = 4, mp(H) = 3.

For the graph G as described, we can see that γb(H) = 4 and mp(H) = 3, thus

γb(G) ≤ 4i and mp(G) ≥ 3i. Hartnell and Mynhardt proved that these are equalities.

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Theorem 2.9. [4] For any connected graph G, γb(G) ≤ 2mp(G) + 3.

Here it was further conjectured that γb(G) ≤ 2mp(G) for any connected graph G

and it was showed in the paper that this inequality holds when mp(G) ≤ 4. Further examples of a graph G with γb(G) = 4 and mp(G) = 2 were also given.

For any integer programming problem, a natural variation of the problem can be obtained by considering the LP relaxation. Since broadcast domination and multi-packing can be regarded as integer programming problems, Brewster, Mynhardt and Teshima [14] used this idea to study fractional broadcast domination and fractional multipacking. Here, the broadcast strength of a vertex can be a fraction, and a vertex can be considered to be fractionally in a multipacking. For example, we can assign 1/3 strength to all vertices in C4, for a total cost of 4/3, resulting in a dominating

broadcast where each vertex hears a total strength at least one. On the other hand we can pack 1/3 for each vertex in C4 and it will give a multipacking of size 4/3.

We denote the fractional broadcast domination number as γb,f(G) and the fractional

multipacking number as mpf(G). The duality theorem of linear programming [20]

gives the result below.

Proposition 2.10. [14] For a connected graph G,

mp(g) ≤ mpf(G) = γb,f(G) ≤ γb(G).

The difference mpf(G) − mp(G) can be arbitrarily large. The graph shown in

Figure 2.2 has fractional multipacking number 4 since we can pack 1/3 on the degree 2 and 4 vertices with the exception of lj and rj, which are not packed. Using k copies

of the graph joined in series as before will give mpf(G) − mp(G) = k.

A set X ⊆ V (G) is irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X (note that X may not necessarily be a dominating set). The concept of broadcast irredundance was first introduced by Ahmadi et al. [1]. A broadcast f is irredundant if reducing the strength of any broadcast vertices in f strictly decreases the number of vertices that hear a broadcast. The broadcast irredundance number, irb(G), is defined to equal the minimum cost of

a maximal irredundant dominating broadcast. The upper broadcast irredundance number IRb(G) is defined to be the maximum cost of an irredundant dominating

broadcast. Note that if the strength of any vertex in an optimal dominating broadcast is reduced, this will decrease the number of vertices that hear the broadcast (as

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all vertices in G can no longer hear the broadcast), so every optimal dominating broadcast is maximal irredundant. As a result, we have that irb(G) ≤ γb(G). Ahmadi

et al. gave some inequalities between broadcasts and other graph parameters.

The upper domination number of a graph G is the maximum cardinality of a minimal dominating set of G. Mynhardt and Roux [50] considered a similar variation for broadcasts. A dominating broadcast is minimal if reducing the strength of any vertex that is broadcasting results in a broadcast that is no longer dominating. The maximum cost of a minimal dominating broadcast is defined to be the upper broadcast number of G, denoted as Γb(G). The inequality chain of parameters was extended to

include the upper domination number and the upper broadcast number. Theorem 2.11. [50] For any graph G,

irb(G) ≤ γb(G) ≤ γ(G) ≤ Γ(G) ≤ Γb(G) ≤ IRb(G).

The upper broadcast number was further studied in the paper by Bouchemakh and Fergani [10]. In this paper they gave an upper bound for the upper broadcast domination number.

Theorem 2.12. [10] For any graph G, Γb(G) ≤ n − δ(G) and the bound is sharp on

paths, stars and complete graphs.

Bouchemakh and Fergani also studied the upper broadcast number on grids. The Cartesian product of graphs G and H is denoted by GH.

Theorem 2.13. [10] For m ≥ n ≥ 2, Γb(PmPn) = m(n − 1).

Another variation of broadcast domination is the k-broadcast domination, which was first studied by Henning, MacGillivray and Yang [39] (also see [61]). In a k-broadcast, instead of requiring every vertex to hear at least one k-broadcast, this time we restrict every vertex to hear at least k broadcasts. Additionally, as was seen for multipacking, if the dual is considered in terms of the linear programming problem for a k-broadcast, we can define k-multipacking. The k-multipacking is a pair (c,r), where c : V → N and r : V → N such that for every vertex v ∈ V we have:

X

d(v,u)≤`

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for each ` ∈ [r]. The value of a k-multipacking (c, r) is the quantity X

v∈V

{k · c(v) − r(v)}.

The minimum cost of a k-dominating broadcast is the k-broadcast domination number, denoted by γbk(G), and the largest value of a k-multipacking of G is the

k-multipacking number, denoted by mpk(G).

In the paper they specifically studied 2-broadcast domination and showed that there exists an arbitrarily large constant difference between γb2(G) and 2 · γb(G).

Theorem 2.14. [39] For any integer t, there exists a connected graph G with γb2(G) ≤

2 · γb(G) − t and there exists a graph H with γb2(H) ≥ 2 · γb(H) − t.

Henning, MacGillivray and Yang also gave a bound of the 2-broadcast domination number:

Theorem 2.15. [39] For any connected graph G on n vertices, γb2(G) ≤

_4n

5 .

The bound of Theorem 2.15 was first shown for trees. The proof is by induction on the number of vertices when the tree is split into two subtrees and various cases are considered in terms of different properties of the trees. Since the bound holds for any spanning tree of a specific graph G, it also holds for the graph itself. It was conjectured that the bound can be reduced to 2n+4₃ .

Another variation of broadcast domination is broadcast domination with a cost function. Instead of assigning cost k to a vertex to broadcast with strength k, in this variation a vertex will have c(k) as the cost to broadcast with strength k. This variation was studied by Paul [31] but the basic theory is still lacking.

Limited broadcast domination is another variation to the standard broadcast dom-ination. In this variation, vertices are no longer allowed to broadcast with strength {0, 1, 2, ..., rad(G)}, but only with strength {0, 1, 2, ..., k}. The first paper for limited broadcast domination was by Rad and Khosvravi [54]. In the paper few results were given and the reader is urged to be cautious with them. The first major results for limited broadcast domination were given by C´aceres et al. [15].

Theorem 2.16. [15] The 2-limited broadcast domination problem is NP-complete. In the proof, C´aceres et al. reduced the 2-limited broadcast domination problem to 3-SAT. Another contribution in this paper is an upper bound of the 2-limited broadcast domination number.

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Theorem 2.17. [15] If T is a tree with n vertices, then the 2-limited broadcast dom-ination number of T is at most d4n/9e.

Since every connected graph has a spanning tree, this bound also holds for all connected graphs. The proof is taken by induction on the number of vertices when the tree is split into subtrees. It was shown that the trees that are tight for the bound are F ∪ {P1, P2, P4}. The minimum graph in family F is T9 in the figure below and

all the other trees in the family are formed by adding an edge between the vertices named l and r in different copies of T9.

l r

Figure 2.3: The tree T9.

In this paper, they also proved that there exists a linear algorithm to find the 2-limited broadcast domination number and an optimal dominating 2-2-limited broadcast for trees. In Chapter 6, we will give a simpler linear algorithm to find the 2-limited broadcast domination number and an optimal dominating 2-limited broadcast for trees. We will also extend their result to show that for any fixed k, there exists a linear algorithm to find the k-limited broadcast domination number and an optimal dominating k-limited broadcast for trees.

C´aceres et al. further studied k-limited broadcast domination in a related work [16] from the same year. In this paper they gave an upper bound for the k-limited broadcast domination number.

Theorem 2.18. [16] If T is a tree with n vertices, then the k-limited broadcast dom-ination number of T is at most k+2_k+1 · n

3.

This tree result can be extended to any connected graph. A similar induction technique as in the proof for the 2-limited bound in Theorem 2.17 was used in this proof. Moreover, it was shown that the bound is tight for any k and a construction of a tree Tk that is tight for the bound on k-limited broadcast domination number

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v1 v2 v3 v4 v5 v6 v7 v8 v2k−2v2k−1 v2k v2k+1

v₁0 v0₂ v0₄ v₆0 v₈0 v_2k−20 v0_2k v_2k+10

Figure 2.4: The tree Tk of order 3k + 3 that is tight for the bound of k-limited

broadcast domination number in Theorem 2.18.

In the same paper it was also proved that the problem of finding an optimal k-limited broadcast domination solution is NP-complete.

Theorem 2.19. [16] For each fixed k ≥ 1, the k-limited broadcast domination problem is NP-complete.

Here the problem was reduced to 3-SAT. In Chapter 3 of this dissertation, we will use a completely different technique to prove the same result, and also give a complexity result for the k-limited multipacking problem, which is the dual of the k-limited broadcast domination problem. The definition of k-limited broadcast domination leads to the following 0-1 integer program:

Minimize k P `=1 n P i=1 `xi`, subject to: P d(vi,vj)≤`

xi`≥ 1 for each vertex vj,

xi`∈ {0, 1} for each vertex vi and integer ` ∈ {0, 1, . . . , k}.

The solution of this problem is the k-limited broadcast domination number, γb,k(G).

The dual of the LP relaxation of the previous integer program leads to the following 0-1 integer program: Maximize n P j=1 yj, subject to: P d(vi,vj)≤`

yj ≤ ` for each vertex vi and integer ` ∈ {0, 1, . . . , k},

yj ∈ {0, 1} for each j.

The solution of this problem is the k-limited multipacking number. We therefore define a k-limited multipacking to be a subset S ⊆ V such that for every u ∈ V and

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every integer ` such that 1 ≤ ` ≤ k, |N`[v] ∩ S| ≤ `. The maximum size of such a set

is the k-limited multipacking number.

Broadcast independence is another problem related to broadcast domination. A broadcast f on G is an independent broadcast if every vertex v and u that have f (v) > 0 and f (u) > 0 have d(u, v) > max{f (u), f (v)}. The maximum of the total cost a independent broadcast for a graph G is the broadcast independence number of G, which is denoted as αb(G). This variation was first studied by Bouchmakh

and Zemir [11]. In this paper, they studied broadcast independence on grids and gave bounds for the broadcast independence number on 2 × n, 3 × n and 4 × n grids. Later, Ahmane, Bouchmakh and Sopena studied the broadcast independence number for caterpillars [2]. Bessy and Rautenbach [55] studied the relationship between the broadcast independence number and the independence number and gave the following results relating these two graph parameters.

Theorem 2.20. For any graph G, α(G) ≤ αb(G) ≤ 4α(G).

Note that α(G) denotes the independence number of G. They also further bounded the broadcast independence number for graphs with specific girth and minimum de-gree.

Theorem 2.21. For graph G of girth at least 6 and minimum degree at least 3 or of girth at least 4 and minimum degree at least 5, αb(G) < 2α(G).

2.3 Broadcast Domination and Multipacking on

Trees

Broadcasts in trees have a special structure, which was exploited in the thesis by Herke [40] in 2007 and in the paper by Herke and Mynhardt [35] in 2009 as well as in the paper by Cockayne, Herke and Mynhardt [22] in 2011.

An important definition is that of a shadow tree. Suppose P : v0v1v2...vd is a

diametrical path of the tree T . The shadow tree is constructed in two steps. First, consider the forest F = T − {v0v1, v1v2, v2v3, ..., vd−1vd}. For each vertex vk of P , let

Qk be a longest path in F that starts at vk and ends at bk (possibly vk = bk). Reduce

tree T to the subtree induced by the vertices belonging to V (P ) ∪ (

d

S

k=1

V (Qk−1)). For

the second step, if there exists some k such that d(vk, bk) ≥ d(vk, b0k), remove Q 0 k\{v

0 k}

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from the tree. This is the shadow tree of T , which is denoted by ST. The shadow of

vertex bk is the set of vertices {v : d(vk, bk) ≥ d(vk, v)}.

For example, consider the tree T in Figure 2.5.

v0 v1 v2 v3 v4 v5 v6 v7

b1 b4

b5

Figure 2.5: Tree T .

One diametrical P is v0v1...v7 and for each vertex vi we only keep the longest branch

starting from it. After the first step we reduce T to the tree in Figure 2.6:

v0 v1 v2 v3 v4 v5 v6 v7

b1 b4

b5

Figure 2.6: First step to construct ST.

We have d(b4, v5) ≤ d(b5, v5). According to the second step of our construction, we

should remove b4 from the tree. So, finally, ST is the tree in Figure 2.7:

v0 v1 v2 v3 v4 v5 v6 v7

b1

b5

Figure 2.7: Shadow tree ST.

Shadow trees are extremely helpful in finding the broadcast domination number for trees.

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This theorem allows attention to be restricted to the simplified tree knowing it has the same broadcast domination number as the original tree T .

Although the shadow tree is an induced subgraph of the original tree, the analog of Theorem 2.22 does not hold for all graphs.

v

Figure 2.8: A graph G with an induced subgraph having larger broadcast domination number.

Consider the graph shown in Figure 2.8. Since v is a dominating vertex of G, γb(G) = 1. However if we remove v, the remaining graph is P4 and γb(P4) = 2.

Herke and Mynhardt also introduced the important definitions of split-sets and split-edges. Let T be a tree with diametrical path P . A split-set is a set of edges on P whose removal splits T into components such that for each component Ti, Ti has

even positive diameter and Ti∩ P is a diametrical path of Ti. A split-edge is an edge

that is contained in some split-set. For example, in Figure 2.7, v2v3 is a split-edge.

On the other hand, the edge v3v4 is not a split edge since its removal creates a subtree

with diametrical path from b5 to v7. In general, all the edges that have two ends in

some shadow (visually in Figure 2.7, the only edge that is not in some shadow is v2v3)

cannot be a split-edge.

Herke and Mynhardt showed that the broadcast domination number is a function of the largest size of a split-set.

Theorem 2.23. [35] If M is split-set with maximum cardinality m of a tree T , then

γb(T ) =

diam(T ) − m 2

.

Theorem 2.24. [35] A tree T is radial if and only if it has no nonempty split-set. They also gave an upper bound for the broadcast domination number of a tree in terms of its order.

Theorem 2.25. [35] For any tree T of order n, γb(T ) ≤

_n

3.

The proof is by induction on the number of vertices. It was shown that the tree can be partitioned into two subtrees where one subtree has exactly k vertices, where

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k ≡ 0 (mod 3). In this subtree, the bound holds without the ceiling. Since for a graph G, a dominating broadcast of a spanning tree is also a dominating broadcast of G, Theorem 2.25 gives an upper bound for γb(G).

Corollary 2.26. For any connected graph G of order n, γb(G) ≤

_n

3.

This bound is tight for paths and cycles.

In the literature, several algorithms have been given to find the broadcast number as well as the multipacking number for trees. In 2009, Dabney, Dean and Hedetniemi [26] (also see [25]) gave a linear algorithm to find an optimal dominating broadcast for trees. The linearity of the algorithm is based on a complex data structure. Later Brewster, MacGillivray and Yang [13] (also see [61]) gave a simpler greedy algorithm which makes use of shadow trees, split-edges and split-sets.

Theorem 2.27. [25, 26] The broadcast domination number for trees can be found in O(n) time.

Mynhardt and Teshima [51] (also in [60]) showed that the multipacking number of trees can be found in linear time. Brewster, MacGillivray and Yang [13] (also see [61]) gave a simpler algorithm for finding a optimal multipacking on trees.

Theorem 2.28. [51, 60] The multipacking number for trees can be found in O(n) time.

In her Master’s thesis, Teshima [60] proved the nice result shown below. This theorem can be seen as a generalization of the classic theorem of Meir and Moon [49] that the domination number equals the 2-packing number for trees.

Theorem 2.29. [60] For any tree T , γb(T ) = mp(T ).

Some work in the literature has been done towards classifying trees into different categories. Graphs G with γb(G) = γ(G) are called 1-cap graphs and there are several

papers about 1-cap trees. Seager [56] first classified all 1-cap caterpillars. Lunney and Mynhardt [48] studied trees with γb(T ) = γ(T ). In this work, a characterization

of 1-cap trees was given. Mynhardt and Wodlinger [52] further studied 1-cap trees. The other extremal case is when γb(T ) = rad(T ). This case was first studied by

Herke and Mynhardt [41] and later by Mynhardt and Wodlinger [53]. In the second paper, using a complex case analysis, the authors characterized the trees for which the equality γb(T ) = rad(T ) holds for a unique broadcast.

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The final category considered is that of the diametrical tree, where the upper broadcast domination number equals the diameter. Gemmrich and Mynhardt [34] characterized diametrical caterpillars in the paper.

2.4 Broadcast Domination on Different Graph

Classes

While the broadcast domination problem has been studied much for general graphs, there have also been various approaches towards studying specific graph classes.

Blair, Heggernes, Horton and Manne [8] focused on studying interval graphs, series parallel graphs and trees. Here, a dynamic programming method was given to find an optimal broadcast for interval graphs and for series-parallel graphs.

Theorem 2.30. [8] The broadcast domination problem restricted to interval graphs can be solved in O(n3_{) time.}

Theorem 2.31. [8] The broadcast domination problem restricted to series-parallel graphs can be solved in O(nr4) time, where r is the radius.

Note the second solution is an improvement since in general complexity O(n6) is needed to calculate the broadcast domination number for any given graph G and r = O(n). Blair, Heggernes, Horton and Manne also used the same method to find optimal solution for trees. The algorithm runs in O(nr) time, which is worse than the algorithm discussed in Theorem 2.27.

The interval graph solution was later improved by Chang and Peng [18]. The authors used a better data structure than Blair et al. and a similar method to improve the running time. Although the title of their paper indicates the algorithm is linear, we remark that it is linear in terms of the number of edges of the graph instead of the number of vertices.

Theorem 2.32. [18] The broadcast domination problem for interval graphs can be solved in O(n + m) time, where m is the number of edges.

In general m = O(n2_{) so this algorithm is in fact quadratic.}

Another class of graphs that is well studied is the class of block graphs. A block graph is a graph where every vertex is contained in a clique and if two cliques intersect,

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they intersect on exactly one vertex. With regard of the structure, it is sometimes also called as clique tree. Heggernes and Sæther [37] found an O(n + m) algorithm to find an optimal solution for broadcast domination on block graphs. Due to the tree-like structure of a block graph, the algorithm is efficient and elegant.

Theorem 2.33. [37] The broadcast domination problem for block graphs can be solved in O(n + m) time, where m is the number of edges.

In this paper, Heggernes and Sæther also mentioned that the original O(n6) al-gorithm can be implemented for chordal graphs in just O(n4) time. They further conjectured that the broadcast domination problem can be solved on chordal graphs in O(n2) time.

Although a full proof providing evidence that the broadcast domination problem for chordal graphs can be solved in O(n2) has yet to be found, there has been pleasing progress for the important subclass of strongly chordal graphs, which we now define. For k ≥ 3, a graph G is called a k-trampoline if it contains a k-clique with vertex set {v1, v2, . . . , vk} and, for each pair {vi, vi+1} such that 1 ≤ i ≤ k and addition

modulo k, there is a vertex wi such that N (wi) ∩ {v1, v2, . . . , vk} = {vi, vi+1}. A

graph is a strongly chordal graph if it is chordal and does not contain a k-trampoline as a subgraph, for any k.

Figure 2.9: The 3-trampoline.

Brewster, MacGillivray and Yang [13] (also see [61]) showed that for strongly chordal graphs, the broadcast domination problem can be solved in O(n3_{) time. This}

algorithm is different from the other algorithms presented so far since it uses integer programming. Recall that in the previous section, the broadcast domination problem was defined as a linear programming problem, so it is natural to use linear program-ming to solve the problem. However, if the solution to the linear program is fractional, it is not a good solution towards the broadcast domination problem. In general, al-though the class of chordal graphs do not necessarily have an integer solution to the linear program, but the subclass, strongly chordal graphs, always have integer solu-tions. A matrix is totally balanced if it does not contain any cycle of length at least 3.

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It was proved in [13] that the constraint matrix of a strongly chordal graph is totally balanced. A matrix is called Γ-free if it does not contain Γ =

" 1 1 1 0 #

as a subma-trix. Lubiw [46, 47] proved that a totally balanced matrix can have a Γ-free ordering and Farber [32] showed that the linear programming problem associated with Γ-free matrices always has an integer solution and it can be solved greedily. Combining all the results above, an efficient algorithm for the class of strongly chordal graphs was given.

Theorem 2.34. [13, 61] The broadcast domination problem for strongly chordal graphs can be solved in O(n3) time.

It is noted that for any strongly chordal graph G, we have γb(G) = mp(G) by the

duality theorem of the linear programming.

Corollary 2.35. [13, 61] If G is strongly chordal, then γb(G) = mp(G).

Another graph class that has been studied is the products. The first paper on the topic is by Braser and Spacaman [12]. In this paper they studied the strong product (), the direct product (×) and the Cartesian product () of graphs. In each case of the graph products of G1 and G2, the product of graphs has vertex set

{(x1, x2)|x1 ∈ V (G1) and x2 ∈ V (G2)}. Two vertices (v1, v2) and (u1, u2) are adjacent

in the product of G1 and G2 if

• either v1 = u1 and v2u2 ∈ E(G2), or v2 = u2and v1u1 ∈ E(G1), or v1u1 ∈ E(G1)

and v2u2 ∈ E(G2) when considering strong product.

• v1u1 ∈ E(G1) and v2u2 ∈ E(G2) when considering direct product.

• either v1 = u1and v2u2 ∈ E(G2), or v2 = u2and v1u1 ∈ E(G1) when considering

Cartesian product.

Theorem 2.36. [12] For graphs G and H, the broadcast domination number of the Cartesian product of the two graphs has γb(GH) ≤ 3₂(γb(G) + γb(H)).

Theorem 2.37. [12] For graphs G and H, the broadcast domination number of the strong product of the two graphs has γb(G H) ≤ 3₂max{γb(G), γb(H)}.

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Theorem 2.38. [12] For graphs G and H, the broadcast domination number of the direct product of the two graphs has

γb(G × H) ≤

  

3max{γb(G), γb(H)} if rad(G) 6= rad(H),

3min{γb(G), γb(H)} + 1 otherwise.

Braser and Spacaman further studied the Cartesian products of paths and cycles. Theorem 2.39. [12] The broadcast domination number of the Cartesian product of Pm and Pn has γb(PmPn) = rad(PmPn) = bm₂c + bn₂c.

Theorem 2.40. [12] The broadcast domination number of the Cartesian product of Cm and Cn has

γb(CmCn) =

  

rad(CmCn) − 1 if m, n both even,

rad(CmCn) otherwise.

Beaudou and Brewster [5] later extended the result on grids. Theorem 2.41. [5] For any pair of integers n ≥ 4 and m ≥ 4,

mp(PnPm) = γb(PnPm),

with the exception of P4P6 where mp(P4P6) = 4 and γb(P4P6) = 5.

The product CnCm is sometimes called a tori. Soh and Koh [58] presented the

tori result of Theorem 2.41 separately in their work.

Later Soh and Koh [57] extended the result for the product of paths. They studied the strong product, the direct product and the lexicographic product (•) of paths. The lexicographic product of G1 and G2 has vertex set {(x1, x2)|x1 ∈ V (G1) and

x2 ∈ V (G2)}. Two vertices (v1, v2) and (u1, u2) are adjacent if either v1u1 ∈ E(G1)

or v1 = u1 and v2u2 ∈ E(G2).

Theorem 2.42. [57] The broadcast domination number of the strong product of Pm

and Pn with m ≥ n ≥ 1 has

γb(Pm Pn) = 1 2 m − m max{p, 3} , where p = 2dn−1₂ e + 1.

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Theorem 2.43. [57] The broadcast domination number of the direct product of Pm

and Pn with m ≥ n ≥ 1 has

γb(Pm× Pn) =            m if n = 1,

n · m+1_n+1 if n ≥ 2, m, n both odd withm+1_n+1 an integer, 2 · m−b_n+1m c 2 otherwise.

Theorem 2.44. [57] The broadcast domination number of the lexicographic product of Pm and Pn has γb(Pm• Pn) =    max{_m 3 , n 3} if m = 1 or n ∈ {1, 2, 3}, max{2m₅ , 2} otherwise.

Koh and Soh [43] continued their work on the Cartesian product of graphs in a subsequent work. The major result in the paper is the theorem below:

Theorem 2.45. [43] For the graph CmPn with m ≥ 3 and n ≥ 2,

γb(CmPn) =    m 2 if n = 2 and m is even, dm 2e + d n 2e otherwise.

2.5 Conjectures and Open Problems

In this section we will gather together the conjectures and suggestions for future work that have been mentioned in this chapter.

Problem 2.46. Classify graphs G that have rad(G) = γb(G).

Conjecture 2.47. [37] The broadcast domination number of a graph G can be found in O(n5) time.

Problem 2.48. Determine the complexity to find the multipacking number of a graph G.

Conjecture 2.49. [4] For any connected graph G, γb(G) ≤ 2mp(G).

Conjecture 2.50. [39] For any connected graph G on n vertices, γb2(G) ≤

2n+4 3 .

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Problem 2.51. Classify graphs G that have γb(G) = mp(G).

Conjecture 2.52. [37] The broadcast domination number for chordal graphs can be found in O(n2_{) time.}

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Chapter 3 Complexity of Limited Broadcasts

in General Graphs

We saw in the previous chapter that the broadcast domination number of a given graph can be found in polynomial time. By contrast, C´aceres et al. [15] showed that for each integer k ≥ 1 it is NP-complete to decide whether a given graph has k-limited broadcast domination number at most B. In this chapter we give a simpler proof of the same result, and also prove that for each integer k ≥ 1, it is NP-complete to decide whether a given graph has k-limited multipacking number at least a given integer M . These are the first NP-complete results for a multipacking problem other than 1-limited multipacking, which is the same as 2-packing.

Theorem 3.1. For each fixed integer k ≥ 1, the problem of deciding whether a given graph G has k-limited broadcast domination number at most a given integer B is NP-complete.

Proof : It is easy to see that the problem is in NP. The transformation is made from the NP-complete problem vertex cover [42]. Suppose we are given a connected graph G with at least two vertices and want to determine if G has a vertex cover of size at most l. Construct an instance, G0, of k-limited broadcast domination as follows.

First, replace each edge of G with kn internally vertex disjoint copies of Pk+1,

where n = |V (G)|. If k is even, then each added path has a unique central vertex. For each path X, add a new vertex vX and a path of length k/2 joining vX to the

central vertex of X. If k is odd, then each added path has two central vertices. For each added path X, add new vertices vX and wX, and add edges between wX and

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The above construction of G0 can be accomplished in polynomial time. We claim that G has a vertex cover of size at most ` if and only if G0 has a dominating k-limited broadcast of cost at most k`.

u

wX

vX

v

Figure 3.1: A replacement of edge uv when k = 3.

u

vX

v

Figure 3.2: A replacement of edge uv when k = 4.

The dashed line in each figure above represents kn − 1 copies of the same repre-sentation shown. Suppose that G has a vertex cover C of size `. For each vertex y in G such that y ∈ C, broadcast from y in G0 with strength k. This gives a dominating k-limited broadcast in G0.

Now suppose that G0 has a dominating k-limited broadcast of cost at most k`. Because the strength of each broadcast vertex is limited to k, the degree 1 vertices (vX) in G0 can only hear a broadcast from a vertex on a shortest path from that

vertex to u or v (each such path has length k). Thus, if neither u nor v broadcasts with strength k, the weight of the broadcast from the vertices in the structure is at least nk > k`. It follows that, in a dominating k-limited broadcast of cost at most k`, the only vertices broadcasting with positive strength are vertices of G. The set of all such vertices is a vertex cover of G of size at most `.

The result now follows.

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deciding whether G has k-limited broadcast domination number at most B is NP-complete.

We have shown that the k-limited broadcast domination problem is NP-complete, now we move on to the k-limited multipacking problem. Recall that a k-limited multipacking is a subset s ⊆ V such that for every u ∈ V and every integer l such that 1 ≤ l ≤ k, |Nl[v] ∩ S| ≤ l, and from Chapter 2 that this problem is the dual of

the LP relaxation of the k-limited broadcast domination integer linear program. Theorem 3.3. For each fixed integer k ≥ 1, the problem of deciding whether a given graph G has k-limited multipacking number at least a given integer M is NP-complete. Proof : Similar to the k-limited broadcast domination problem, note that the k-limited multipacking problem is in NP. First, notice that a 1-limited multipacking is the same as a 2-packing, so the problem is NP-complete for k = 1 [33].

Suppose k ≥ 2. The transformation is from 2-packing.

Case 1: k = 2. We construct an instance G0 of k-limited multipacking as follows. For each vertex v in G, we build a vertex gadget for v in G0. We add three vertices av, bv, cv to v and form a P4 in the sequence of av, bv, v, cv. The graph G0 can be

constructed in polynomial time. We claim that G has a 2-packing of size at least p if and only if G0 has a k-limited multipacking of size at least p + n, where n = |V (G)|. Suppose that G has a 2-packing of size at least p, say P . For each gadget of v in G0, we can always pack vertex av into the 2-limited multipacking P0 of G0. If v ∈ P ,

we can pack cv into P0. Thus P0 is a 2-limited multipacking in G0 with size p + n.

Suppose that G0 has a 2-limited multipacking P0 of size p + n. First notice that for each gadget associated with the vertex v we cannot pack more than two vertices, and if we pack two vertices into P0, the two vertices are av and cv. In P0, if for some

gadget of v we only pack one vertex, we exchange that vertex for av. If some gadget

of v is empty, we can always move a vertex in P0 to av. As a result, we have a packing

where all vertices of the form av have been packed as well as some additional vertices

of the form cv. Denote this new multipacking as P00.

Now for u, v ∈ V (G), if d(u, v) = 1, note at most one cu and cv are in P00 as it

violates the distance 2 constraint of both u and v since au and av are both in P00. If

d(u, v) = 2, and let w be the vertex adjacent to both u and v in G. We cannot pack cu and cv into P00 together as it violates the distance 2 constraint of w since aw is in

P00. So if a vertex v has cv packed into P00, we can pack v into P in G and P is a

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Case 2: k ≥ 3. For each vertex v in G we first build the vertex gadget associated with v in G0. Add k − 1 vertices and call these vertices special vertices of v, and then connect all special vertices to v by Pk+1. Secondly, add a new vertex cv and

then connect cv to all vertices introduced in the previous step by Pk. For each of the

special vertices of v, we label the shortest path between v and the special vertex to be v, v1,0, v2,0, . . . , vk−1,0, vk,0, where the special vertex is vk,0. For each of the shortest

path from vm,0 to cv, we label the vertices to be vm,0, vm,1, vm,2, . . . , vm,k−2, cv. Then

we add an edge between vertex vi,j to vi+1,j+1 when we have 1 ≤ i ≤ k − 1 and

j ≤ 0 ≤ k − 3. For all the vertices added in these steps, we call the resulting subgraph induced by the vertices Hv. Note that every vertex in Hv is at distance at

most k − 1 from cv and at distance at most k from v.

v

cv

Figure 3.3: One of the three special vertex structures in Hv when k = 4.

Replace each edge uv ∈ E(G) with a path of length 2k−1₂ + 1. Call u0 the vertex on this path at distance k−1₂ away from v, and call v0 the vertex on this path at distance k−1₂ away from u. Let Puv0 be the path from u to v0 and P_vu0 the path

from v to u0. We will say that Puv0 is in the vertex gadget associated with u, and

Pvu0 is in the vertex gadget associated with v. Then for all vertices of distance x

(1 ≤ x ≤ k−1₂ ) from v on some Pvu0, connect all the vertices together to form a

clique. This completes the vertex gadget associated with v in G0.

From this description, note that the gadget of v and the gadget of u are joined by the edge u0v0. When k is even, we contract this edge so the vertices u0 and v0 are identified, and this vertex exists both in the gadget of v and in the gadget of u. In the case of k even and the case of k odd, we have d(u, v) = k in G0.

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v

u0 v0

u Hv

Figure 3.4: The vertex gadget associated with v in G0 when k = 4.

In the figure shown above, v is a degree 3 vertex in G and u is one of its neighbours in G. The dashed path is the path Pvu0 in the gadget of v and two similar paths are

connected to the gadgets of the other neighbours of v.

Note that the graph G0 as described above can be constructed in polynomial time. We claim that G has a 2-packing of size at least p if and only if G0 has a k-limited multipacking of size at least p + n(k − 1).

Suppose that G has a 2-packing of size at least p, say P . For each vertex gadget associated with v, we can pack all special vertices of v into P0. Then for all vertices v ∈ P , pack v into P0 as well. The set P0 is a k-limited multipacking of size p+n(k−1) in G0.

Suppose that G0 has a k-limited multipacking of size at least p + n(k − 1), say P0. First notice that inside Hv we cannot pack more than k − 1 vertices since cv is

at distance at most k − 1 from every vertex in Hv. Secondly notice that inside the

gadget of v, we cannot pack more than k vertices since v is at distance at most k from every vertex in the gadget.

If the gadget of v has at least k − 1 vertices in P0, we can exchange these k − 1 vertices for the special vertices of v. Then in P0 either we have a vertex remaining in the gadget but not inside Hv, or no vertices are left outside of Hv. If a gadget of

v has a < k − 1 vertices in P0, we can exchange k − 1 − a arbitrary vertices from another gadget that are not inside Hu for some other vertex u and also the a vertices

inside the gadget of v for the special vertices of v.

This gives a multipacking P00 of size p + n(k − 1) vertices with all special vertices occupied. Suppose a vertex x is chosen in P00 and x ∈ Hv for some v ∈ V (G).

If x is in a gadget of v only, then we put v into P in G. If x is in the intersection of the gadgets of v and u (note this only happens when k is even and we contract the edge u0v0), we can put either v or u into P in G. We claim that P is a 2-packing in

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G.

Suppose x is in the gadget of v only. Let u be a vertex adjacent to v in G. We cannot have k vertices in the gadget of u in P00 since d(x, u) ≤ k and every vertex in the gadget of u is at most distance k away from u. Let w be a vertex adjacent to u in G. We cannot have k vertices in the gadget of w in P00 since d(x, u) ≤ k and for every vertex y in the gadget of w but not in Hw, we also have d(y, u) ≤ k. So

we cannot pack y in P00. Thus we do not pack any vertices in G that are at most distance 2 away from v.

A similar argument holds when x is in the gadget of v and u together. As a result, P is a 2-packing of size p in G.

The result now follows.

Corollary 3.4. Given a graph G and positive integers k and M , the problem of deciding whether G has k-limited multipacking number at least M is NP-complete.

In this chapter we proved that the complexity of deciding whether a given graph has k-limited broadcast domination number at most B and deciding whether a given graph has k-limited multipacking number at least M are both NP-complete. While the complexity for the k-limited broadcast domination number was previously de-termined by C´aceres et al. [15], the proof provided here makes use of a simpler construction. The result on the complexity of the k-limited multipacking number is the first to be provided. As shown in Chapter 2, Heggernes and Lokshtanov [36] proved that the time complexity for finding an optimal solution of the broadcast dom-ination problem for a fixed graph is O(n6_{). As of the time of writing, the complexity}

of the multipacking problem is still unknown. Here we remark that the proof of NP-completeness of k-limited multipacking does not work for the general multipacking problem.

Since the complexity of determining the k-limited broadcast domination number and the k-limited multipacking number for general graphs has been settled here, the attention now turns to determining these parameters in specific graph classes. We will tackle problems of this nature in Chapter 5 and Chapter 6 where we focus on strongly chordal graphs, interval graphs, circular arc graphs, proper interval bigraphs and trees.

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Chapter 4 2-Limited Broadcast Domination

on Subcubic Graphs

A subcubic graph is a graph whose maximum degree is at most 3, while a cubic graph (also called a 3-regular graph in the literature) is a graph in which every vertex has degree 3. The motivation for this problem first comes from the paper [55] by Reed. In the paper, Reed proved that for any cubic graph G on n vertices, the domination number γ(G) is at most 3n₈ . He conjectured that the upper bound for domination number on cubic graphs is n₃. Later in 2009, Kostochka and Stodolsky [45] improved the bound to 4n₁₁ for domination on cubic graphs by extending Reed’s idea and taking care of more cases. However Kostochka and Stodolsky [44, 59] also showed that Reed’s conjecture was incorrect by giving a cubic graph with 60 vertices which needs 21 vertices to dominate the whole graph.

The 2-limited broadcast domination variation is the immediate generalization and relaxation of domination. Our interest here is to answer the question: if we allow using cost 1 and 2 (instead of only 1 as in standard domination), can we reach Reed’s bound?

The method we will follow was first seen in the paper by Henning, L¨owenstein and Rautenbach [38]. In the paper, they showed a bound for cubic graph, where instead finding the bound directly on cubic graphs, they first find a stronger result on subcubic graphs. Since the class of cubic graphs is a subclass of the class of subcubic graphs, the bound for subcubic graphs also works for cubic graphs. In this chapter, we first establish an upper bound on the 2-limited broadcast domination number for a subcubic graph G that is (C4, C6)-free and then obtain our result for cubic graphs

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as a corollary. For this purpose, we shall prove the following stronger result, a proof of which is presented below. Here nk(G) represents the number of vertices in G of

degree k.

Theorem 4.1. If G is a (C4, C6)-free subcubic graph, then

9γb,2(G) ≤ 9n0(G) + 6n1(G) + 4n2(G) + 3n3(G).

If G is a cubic graph of order n, then n0(G) = n1(G) = n2(G) = 0 and n = n3(G).

Hence as an immediate consequence of Theorem 4.1, we have the following upper bound on the 2-limited broadcast domination number of a cubic graph.

Corollary 4.2. If G is a cubic graph of order n that is (C4, C6)-free, then γb,2(G) ≤ 1₃n.

In what follows, we present a proof of Theorem 4.1.

For the sake of simplicity, we call a vertex of degree k in G whose neighbours have degrees d1, d2, . . . , dk, respectively, where d1 ≤ d2 ≤ · · · ≤ dk, a (d1, d2, . . . , dk)-vertex.

In particular, every vertex of a cubic graph is a (3, 3, 3)-vertex.

Suppose, to the contrary, that Theorem 4.1 is false. Among all counterexamples to the theorem, let G be chosen to have minimum order. Clearly, G is connected and has order at least 3. For each vertex v in the subcubic graph G, we define its weight as wG(v) =                9 if dG(v) = 0 6 if dG(v) = 1 4 if dG(v) = 2 3 if dG(v) = 3.

For a subset X ⊆ V (G), we define the weight of X in G as wG(X) =

X

v∈X

wG(v),

and so wG(X) is the sum of the weights of the vertices in X. We define the weight of

G as

w(G) = wG(V (G)) =

X

v∈V (G)

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Since G is a counterexample to the theorem, we have 9γb,2(G) > w(G). As

observed earlier, G is connected and has order at least 3. If G has order 3, then γb,2(G) = γ(G) = 1 and w(G) ≥ 9, a contradiction. If G has order 4, then either G

has a dominating vertex, in which case γb,2(G) = γ(G) = 1 and w(G) ≥ 12, or G is a

path P4, in which case γb,2(G) = 2 and w(G) ≥ 20. In both cases, 9γb,2(G) ≤ w(G),

a contradiction. Hence, the graph G contains at least five vertices.

For a subset X ⊆ V (G), we define X = V (G) \ X and we define the boundary of X, denoted ∂G(X), to be the set of all vertices in X that have a neighbour in X.

Thus, G[X] = G − X and ∂G(X) = {v ∈ X : dX(v) ≥ 1}. If X is a proper subset of

V (G) and v ∈ X, then the degree of v in G − X is at most its degree in G, implying that the weight of v in G − X is at least as large as its weight in G. Further, if v has at least one neighbour in X (that is, if v belongs to the boundary ∂G(X)), then

the degree of v in G − X is less than its degree in G, and therefore the vertex v has a higher weight in G − X than in G. We define the resulting weight increase of the vertex v ∈ X by

ΦG(v; X) = wG−X(v) − wG(v).

If the set X is clear from context, we simply write ΦG(v) rather than ΦG(v; X) for

notational convenience. To illustrate the weight increase of a vertex, let X ⊂ V (G) and consider an arbitrary vertex v ∈ X and let G0 = G − X. If v has no neighbour in X, then ΦG(v) = 0. If dG(v) = 3 and dG0(v) = 2, then Φ_G(v) = 4 − 3 = 1. If

dG(v) = 3 and dG0(v) = 0, then Φ_G(v) = 9 − 3 = 6. If d_G(v) = 2 and d_G0(v) = 1, then

ΦG(v) = 6 − 4 = 2. If dG(v) = 1 and dG0(v) = 0, then Φ_G(v) = 9 − 6 = 3. We refer

to the sum of the weight increases of vertices in X as the cost of removing X from G and denote it by

ΦG(X) =

X

v∈X

ΦG(v).

Letting G0 = G − X and noting that V (G0) = X, we have ΦG(X) = X v∈X (wG0(v) − w_G(v)) = wG0(X) − w_G(X) = w(G0) − (w(G) − wG(X)), = w(G0) − w(G) + wG(X).

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We state this formally as follows.

Fact 4.3. If X ⊂ V (G) and G0 = G − X, then ΦG(X) = w(G0) − w(G) + wG(X).

We call an edge an exit edge of X in G if it joins a vertex in X to a vertex in X and we denote the number of exit edges of X in G by ξG(X). We share the

weight increase, ΦG(v), of the vertex v among the dX(v) exit edges of X in G that

are incident with v. To each exit edge e of X in G incident with the vertex v ∈ X we assign the cost of

ΦG(e; X) =

ΦG(v)

d_X(v).

If the set X is clear from context, we simply write ΦG(e) rather than ΦG(e; X)

for notational convenience. We note that ΦG(X) =

X

e∈[X,X]

ΦG(e).

We shall frequently use the following three facts about the cost of removing a set of vertices from the graph G. Recall that if X ⊆ V (G), then we denote G[X] simply by GX. We first present an upper bound on ΦG(X) in terms of the number of exit

edges of X in G.

Fact 4.4. Let X ⊂ V (G) and let G0 = G − X. If e is an exit edge of X in G and v is the vertex of X incident with the edge e, then

ΦG(e) =    3 if dG(v) = 1 5 2 if dG(v) = 2 and dG0(v) = 0,

and ΦG(e) ≤ 2 otherwise.

Proof : We show that every exit edge of X in G contributes at most 3 to the cost, ΦG(X), of removing X from G. Let G0 = G − X and let e be an arbitrary exit edge of

X in G. Let v be the vertex of X incident with the edge e. We show that ΦG(e) ≤ 3,

with equality if and only if dG(v) = 1. If dG(v) = 1, then dG0(v) = 0, implying that

ΦG(v) = 9 − 6 = 3, dX(v) = 1 and ΦG(e) = 3. If dG(v) = 2 and dG0(v) = 1, then

ΦG(v) = 6 − 4 = 2, dX(v) = 1 and ΦG(e) = 2. If dG(v) = 2 and dG0(v) = 0, then

ΦG(v) = 9 − 4 = 5, dX(v) = 2 and ΦG(e) = 5/2. If dG(v) = 3 and dG0(v) = 2, then

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ΦG(v) = 6 − 3 = 3, dX(v) = 2 and ΦG(e) = 3/2. If dG(v) = 3 and dG0(v) = 0, then

ΦG(v) = 9 − 3 = 6, dX(v) = 3 and ΦG(e) = 2. Thus if dG(v) = 3, then ΦG(e) ≤ 2. In

all cases, ΦG(e) ≤ 3, with equality if and only if dG(v) = 1.

By Fact 4.4, every exit edge of X in G contributes at most 3 to the cost, ΦG(X),

of removing X from G. Hence as an immediate consequence of Fact 4.4, we have the following result.

Fact 4.5. If X ⊂ V (G), then ΦG(X) ≤ 3 ξG(X), with equality if and only if every

exit edge of G is incident with a vertex of degree 1 that belongs to X.

To aid us in our counting arguments, for each vertex v ∈ X, we define EX(v) to

be the set of exit edges of X in G incident with v and we define ΨG(v; X) =

X

e∈EX(v)

ΦG(e; X).

As before, if the set X is clear from context, we simply write ΨG(v) rather than

ΨG(v; X) for notational convenience. Thus,

ΦG(X) = X v∈X ΨG(v) = X e∈[X,X] ΦG(e) = X v∈X ΦG(v).

For convenience, in this chapter we will use 2LD-broadcast to denote a 2-limited dominating broadcast.

Fact 4.6. Let X ⊂ V (G) and let G0 = G − X. If f is a 2LD-broadcast in GX and f0

is a γb,2-broadcast of G0, then the following holds.

• (a) f ∪ f0 _{is a 2LD-broadcast in G.}

• (b) ΦG(X) > wG(X) − 9f (X).

• (c) 3 ξG(X) > wG(X) − 9f (X).

Proof : Part (a) is immediate since the union of a 2LD-broadcast in GX and a

2LD-broadcast in G0 is a 2LD-broadcast of G. By Part (a), the union of a γb,2-broadcast

of GX and a γb,2-broadcast of G0 is a 2LD-broadcast of G, implying that γb,2(G) ≤

γb,2(GX) + γb,2(G0). Since G is a counterexample to the theorem, we have w(G) <

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not a counterexample to the theorem, and so 9γb,2(G0) ≤ w(G0). Since f is a

2LD-broadcast in GX, we note that γb,2(GX) ≤ f (X). These observations, together with

Fact 4.3, imply that w(G) < w(G0) + 9f (X) = ΦG(X) + w(G) − wG(X) + 9f (X), or,

equivalently, ΦG(X) > wG(X) − 9f (X). This completes the proof of Part (b). Part

(c) follows immediately from Part (b) and Fact 4.5.

We now return to the proof of Theorem 4.1. In what follows we present a series of claims describing some structural properties of G which culminate in the implication of its nonexistence.

Claim 1. No vertex in G has two neighbours of degree 1.

Proof : Suppose, to the contrary, that G contains such a vertex v. Since G contains at least five vertices, dG(v) = 3. Let NG(v) = {v1, v2, v3} where dG(v1) = dG(v2) = 1.

We note that dG(v3) ∈ {2, 3}. Let X = {v, v1, v2} and let f : X → {0, 1, 2} be the

2LD-broadcast in GX that assigns the values 1, 0, 0 to v, v1, v2, respectively. We note

that f (X) = 1, wG(X) = 15 and ξG(X) = 1, and so 3 ξG(X) = 3 < 6 = 15 − 9 =

wG(X) − 9f (X), contradicting Fact 4.6(c).

Claim 2. There is no (1, 2)-vertex in G.

Proof : Suppose, to the contrary, that G contains such a vertex v. Let NG(v) =

{v1, v2} where dG(v1) = 1 and dG(v2) = 2. Let v3 be the neighbour of v2 different

from v. Let X = {v, v1, v2} and let f : X → {0, 1, 2} be the 2LD-broadcast in GX that

assigns the values 1, 0, 0 to v, v1, v2, respectively. We note that f (X) = 1, wG(X) = 14

and ξG(X) = 1, and so 3 ξG(X) = 3 < 5 = 14 − 9 = wG(X) − 9f (X), contradicting

Fact 4.6(c).

Claim 3. There is no (1, 3)-vertex in G.

{v1, v2} where dG(v1) = 1 and dG(v2) = 3. Let X = {v, v1} and let f : X → {0, 1, 2}

be the 2LD-broadcast in GX that assigns the value 0 and 1 to v and v1, respectively.

We note that f (X) = 1, wG(X) = 10 and ΦG(v2) = wG−X(v2) − wG(v2) = 4 − 3 = 1.

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Claim 4. There is no (1, 2, 2)-vertex in G.

{v1, v2, v3} where dG(v1) = 1 and dG(v2) = dG(v3) = 2. If v2v3 ∈ E(G), then the

graph G is determined and has order 4, a contradiction. Hence, v2v3 ∈ E(G). Let/

u2 and u3 be the neighbours of v2 and v3, respectively, different from v. Since G is

C4-free, we note that u2 6= u3. Let X = {v, v1, v2, v3} and let f : X → {0, 1, 2} be

the 2LD-broadcast in GX that assigns the value 1 to v and the value 0 to v1, v2 and

v3. We note that f (X) = 1, wG(X) = 17 and ξG(X) = 2, and so 3 ξG(X) = 6 < 8 =

17 − 9 = wG(X) − 9f (X), contradicting Fact 4.6(c).

Claim 5. There is no (1, 2, 3)-vertex in G.

{v1, v2, v3} where dG(v1) = 1, dG(v2) = 2 and dG(v3) = 3. Let X = {v, v1, v2, v3} and

let f : X → {0, 1, 2} be the 2LD-broadcast in GX that assigns the value 1 to v and the

value 0 to v1, v2 and v3. We note that f (X) = 1 and wG(X) = 16. If v2v3 ∈ E(G),

then ξG(X) = 1, and so 3 ξG(X) = 3 < 7 = 16 − 9 = wG(X) − 9f (X), contradicting

Fact 4.6(c). Hence, v2v3 ∈ E(G). Let u/ 2 be the neighbour of v2 different from v,

and let x3 and y3 be the neighbours of v3 different from v. Since G is C4-free, we

note that the vertices u2, x3 and y3 are distinct. Renaming x3 and y3 if necessary,

we may assume that dG(x3) ≥ dG(y3). By Claim 3, there is no (1, 3)-vertex in G,

implying that dG(u2) ≥ 2 and ΦG(u2) ≤ 2. By Claim 1, the vertex v3 has at most one

neighbour of degree 1, implying that dG(x3) ≥ 2 and ΦG(x3) ≤ 2. Since dG(y3) ≥ 1,

we note that ΦG(y3) ≤ 3. Thus, ΦG(X) = ΦG(u2) + ΦG(x3) + ΦG(y3) ≤ 2 + 2 + 3 =

7 = 16 − 9 = wG(X) − 9f (X), contradicting Fact 4.6(b).

Claim 6. δ(G) ≥ 2.

Proof : Suppose, to the contrary, that G contains a vertex, v1 say, of degree 1. Let v

be the neighbour of v1. Since G contains at least five vertices, we note that dG(v) ≥ 2.

By Claim 1, the vertex v1 is the unique neighbour of v of degree 1. If dG(v) = 2,

then the vertex v is either a (1, 2)-vertex or a (1, 3)-vertex, contradicting Claim 2 or Claim 3. Hence, dG(v) = 3. If v has a neighbour of degree 2, then the vertex

v is either a (1, 2, 2)-vertex or a (1, 2, 3)-vertex, contradicting Claim 4 or Claim 5. Hence, v has no neighbour of degree 2, implying that v is a (1, 3, 3)-vertex. Thus, NG(v) = {v1, v2, v3} where dG(v1) = 1 and dG(v2) = dG(v3) = 3.

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Suppose that v2v3 ∈ E(G). In this case, we let X = {v, v1, v2, v3} and let f : X →

{0, 1, 2} be the 2LD-broadcast in GX that assigns the value 1 to v and the value 0

to v1, v2 and v3. We note that f (X) = 1, wG(X) = 15 and ξG(X) = 2, implying

that 3 ξG(X) = 6 = 15 − 9 = wG(X) − 9f (X), contradicting Fact 4.6(c). Hence,

v2v3 ∈ E(G). Since G is C/ 4-free, we note that v is the only common neighbour of v2

and v3.

Let v2 be a (a1, a2, a3)-vertex and let v3 be a (b1, b2, b3)-vertex. Renaming v2

and v3 if necessary, we may assume that (a1, a2, a3) is the same or smaller in

lexi-cographical order than (b1, b2, b3); that is, either (a1, a2, a3) = (b1, b2, b3) or ai < bi

for the first i ∈ [3] where ai and bi differ. By Claim 4 or Claim 5, if a1 = 1, then

(a1, a2, a3) = (1, 3, 3). If a1 ≥ 2, then since v2 has at least one neighbour of degree 3,

namely the vertex v, we note that (a1, a2, a3) ∈ {(2, 2, 3), (2, 3, 3), (3, 3, 3)}.

Analo-gously, (b1, b2, b3) ∈ {(1, 3, 3), (2, 2, 3), (2, 3, 3), (3, 3, 3)}. We proceed further with the

following series of subclaims. Claim 6.1. b1 ≥ 2.

Proof : Suppose, to the contrary, that b1 = 1, implying that (b1, b2, b3) = (1, 3, 3).

Since (a1, a2, a3) is the same or smaller in lexicographical order than (b1, b2, b3), this

implies that both v2 and v3 are (1, 3, 3)-vertices. Let w2 and w3 be the neighbours of

v2 and v3, respectively, of degree 1. Let X = {v, v1, v2, v3, w2, w3} and let f : X →

{0, 1, 2} be the 2LD-broadcast in GX that assigns the value 2 to v and the value 0

to the remaining vertices of X. We note that f (X) = 2, wG(X) = 27 and ξG(X) =

2, implying that, 3 ξG(X) = 6 < 9 = 27 − 18 = wG(X) − 9f (X), contradicting

Fact 4.6(c).

Claim 6.2. a1 ≥ 2.

Proof : Suppose, to the contrary, that a1 = 1, implying that (a1, a2, a3) = (1, 3, 3).

Let w be the neighbour of v2 of degree 1, and let x be the third neighbour of v2

different from v and w. Thus, dG(x) = 3. Let y and z be the two neighbours of v3

different from v. As observed earlier, the vertex v is the only common neighbour of v2 and v3, and so the vertices x, y and z are distinct. By Claim 6.1, dG(y) ≥ 2 and

dG(z) ≥ 2.

Suppose that dG(y) = dG(z) = 3. In this case, we let X = {v, v1, v2, v3, w} and

f : X → {0, 1, 2} be the 2LD-broadcast in GX that assigns the value 2 to v and the

Limited broadcast domination

Contents

List of Figures

Introduction

Chapter 2

Background

2.1

General Broadcast Domination Theory

2.2

Other Types of Broadcasts

2.3

Broadcast Domination and Multipacking on

Trees

2.4

Broadcast Domination on Different Graph

Classes

2.5

Conjectures and Open Problems

Chapter 3

Complexity of Limited Broadcasts

in General Graphs

Chapter 4

2-Limited Broadcast Domination

on Subcubic Graphs