by
Laura Elizabeth Teshima
B.Sc., Thompson Rivers University, 2010
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of
MASTER OF SCIENCE
in the Department of Mathematics and Statistics
c
Laura Elizabeth Teshima, 2012 University of Victoria
All rights reserved. This thesis may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author.
Broadcasts and multipackings in graphs
by
Laura Elizabeth Teshima
B.Sc., Thompson Rivers University, 2010
Supervisory Committee
Dr. Christina M. Mynhardt, Co-supervisor (Department of Mathematics and Statistics)
Dr. Richard C. Brewster, Co-supervisor (Department of Mathematics and Statistics)
Dr. Gary MacGillivray, Departmental Member (Department of Mathematics and Statistics)
Supervisory Committee
Dr. Christina M. Mynhardt, Co-supervisor (Department of Mathematics and Statistics)
Dr. Richard C. Brewster, Co-supervisor (Department of Mathematics and Statistics)
Dr. Gary MacGillivray, Departmental Member (Department of Mathematics and Statistics)
ABSTRACT
A dominating broadcast on a graph G = (V, E) is a function f : V → {0, 1, . . . , diam(G)} with f (v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f (v) from a vertex v with f (v) > 0. The cost of a broadcast f is σ(f ) = P
v∈V f (v), and the broadcast number γb(G) is the minimum cost of a
dominating broadcast. A set X ⊆ V (G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir(G) is the cardinality of a smallest maximal irredundant set of G. In the first half of this thesis, we prove the bound γb(G) ≤ 32ir(G) for any
graph G and provide an infinite class of graphs with γb(G) = 32ir(G).
In the second portion of this thesis, we consider a new dual property to broadcast domination. A set M ⊆ V is called a multipacking of a graph G = (V, E) if, for each v ∈ V and each s such that 1 ≤ s ≤ diam(G), v is within distance s of at most s vertices in M . The multipacking number, denoted mp(G), is the maximum cardinality of a multipacking of G. We prove that for any tree T , γb(T ) = mp(T ).
This generalizes the well-known result of A. Meir and J. W. Moon [Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975), no. 1, 225–233; MR0401519 (53 #5346)] that the 2-packing number of any tree equals its domination number. We then present an algorithm for finding a maximum multipacking of a tree, and show how multipackings can be used to certify the minimality of a dominating broadcast. We conclude with thesis by suggesting some open problem in broadcast domination and multipackings.
Contents
Supervisory Committee ii Abstract iii Table of Contents iv List of Figures vi 1 Introduction 12 Notation and Background 3
2.1 Broadcast Domination . . . 3 2.1.1 Basic Results . . . 4 2.1.2 Shadow Trees . . . 5 2.1.3 Broadcasts in Trees . . . 8 2.1.4 Complexity . . . 9 2.2 Irredundance . . . 9 2.2.1 Complexity . . . 12 2.3 Packings . . . 12
3 A New Upper Bound for the Broadcast Number of a Graph 14 3.1 Proof of Theorem 3.1 . . . 15
3.1.1 Corollaries . . . 19
4 Broadcast Domination and Multipackings in Graphs 24 4.1 Broadcast Domination as an LP Problem . . . 24
4.2 Broadcasts and Multipackings in Trees . . . 28
4.2.1 Lemmas . . . 29
4.2.2 Proof of Theorem 4.4 . . . 38
4.3 An Alternative Proof of Theorem 3.1 . . . 48
5 Conclusions 52
5.1 Future Work . . . 52
List of Figures
Figure 2.1 A graph G with min {rad(G), γ(G)} = 4 and γb(G) = 3. . . 5
Figure 2.2 A tree T with diametrical paths P1 = {a, b, c, d, e, f, g, h, i, j} and P2 = {k, b, c, d, e, f, g, h, r, v}. . . 6
Figure 2.3 The shadow tree ST ,P1 of T in Figure 2.2. . . 6
Figure 2.4 The shadow tree ST ,P2 of T in Figure 2.2. . . 6
Figure 2.5 A labelled shadow tree ST ,P. . . 8
Figure 2.6 A graph H illustrating the subsets of V (H). . . 11
Figure 3.1 A graph H with ir(H) = 2 and γb(H) = 3. . . 17
Figure 3.2 A graph Gk with ir(Gk) = 2k and γb(Gk) = 3k. . . 17
Figure 3.3 A counterexample to the converse of Corollary 3.5. . . 23
Figure 4.1 A graph G with γb(G) = 4, mp(G) = 2. . . 28
Figure 4.2 Illustration of Lemma 4.7. . . 30
Figure 4.3 Illustration of Case 1 of the proof of Lemma 4.8 . . . 33
Figure 4.4 Illustration of Case 2 of the proof of Lemma 4.8. . . 34
Figure 4.5 Trees with k = mp(T ) = γb(T ) for k = 3, 4. . . 38
Figure 4.6 The construction of Case 2 of Theorem 4.4. . . 40
Figure 4.7 Example of Algorithm 4.1. . . 47
Introduction
Domination in graphs is a popular and extensively researched topic [15, 16]. A popu-lar problem in domination is determining how a telecommunications company might position its radio towers. The company is charged with providing radio coverage to a collection of geographic regions. A single tower transmitting with a strength (or cost) of one unit can provide coverage to the region it is located in and all regions immediately adjacent to it. Obviously, the company would like to minimize its ex-penses by erecting as few towers as possible. How should the company distribute its towers to minimize its costs? If we consider each region as a vertex of a graph, where two vertices are joined with an edge if their corresponding geographic regions are ad-jacent, then any minimum dominating set S represents a minimum cost arrangement of radio towers.
Formally, a dominating set S of a graph G = (V, E) is a set such that every vertex of G is either in S or adjacent to some vertex in S. That is, S is a dominating set if and only if for each v ∈ G, either v ∈ S or there exists some u ∈ S such that uv ∈ E. The smallest cardinality of a dominating set is the domination number γ(G). For vertices u and v of G, we say that v dominates u if u = v or u is adjacent to v.
More recently, broadcast domination has emerged as an interesting generalization of domination. Returning to the radio tower problem in domination, broadcast dom-ination allows the company to build its towers with varying signal strength so that a tower may transmit its signal a greater distance, but at a greater cost. Similarly to the ordinary domination case, the solution of the telecommunications company’s quandary is solvable with a minimum cost dominating broadcast f .
A broadcast on a connected graph G is a function f : V → {0, 1, . . . , diam(G)} such that f (v) ≤ e(v) for all v ∈ V , where e(v) = max {d(v, w) : w ∈ V } is the eccentricity of v. We say that f is a dominating broadcast of G if every vertex of G is within distance f (v) from a vertex v such that f (v) > 0. The cost of a broadcast f is σ(f ) = P
v∈V f (v), and the broadcast number of G is γb(G) =
min {σ(f ) : f is a dominating broadcast of G}. A dominating broadcast f of G such that σ(f ) = γb(G) is called a γb-broadcast. If f is a dominating broadcast such that
f (v) ∈ {0, 1} for each v ∈ V , then {v ∈ V : f (v) = 1} is an ordinary dominating set of G.
The focus of this thesis is to present two new results in broadcast domination which have well-known analogues in ordinary domination. In Chapter 2, we review some required notation and relevant background results. The first major result, presented in Chapter 3, is a new upper bound for the broadcast number of a graph in terms of its irredundance number. This tightens the previous implicit bound from Bollob´as and Cockayne [2], which was derived for the ordinary domination number. We then examine a generalization of packings, called multipackings, and their relationship to broadcasting, in Chapter 4. In particular, we extend an often-cited result of Meir and Moon [23], by proving that the multipacking and broadcast numbers of a tree are equal. Finally, in Chapter 5, we summarize our results and suggest some future research directions.
Chapter 2
Notation and Background
In this chapter we examine some previous results in the topics of broadcasting, ir-redundance and packings. We follow the notation of [4, 16], but provide additional definitions in the following chapter as required.
2.1
Broadcast Domination
For any positive integer k and any vertex v, we define the (closed ) k-neighbourhood of v by Nk[v] = {u ∈ V : d(u, v) ≤ k}. The k-neighbourhood of a set X ⊆ V is the
set Nk[X] = Sv∈XNk[v].
Now consider a broadcast f on the graph G. The broadcast vertices of G form the set Vf+ = {v ∈ V : f (v) > 0}. The broadcast neighbourhood of v ∈ Vf+ is the set Nf[v] = Nf (v)[v] = {u ∈ V : d(u, v) ≤ f (v)}. Thus, f is a dominating broadcast of G,
or G is f -dominated, if for each u ∈ V there exists v ∈ Vf+ such that u ∈ Nf[v]; we say
that u hears the broadcast from v. If for each u ∈ V , |v ∈ Vf+ : u ∈ Nf[v] | = 1,
2.1.1
Basic Results
Broadcast domination was introduced by Erwin in his 2001 doctoral dissertation as “cost domination” [12]. Many of these results are summarized in [13]. In his dis-sertation, Erwin established several sharp upper and lower bounds for the broadcast number, including the following often-referenced result.
Proposition 2.1. [12] For every nontrivial connected graph G, diam(G) + 1
3
≤ γb(G) ≤ min {rad(G), γ(G)} . (2.1)
A simple result for paths nicely demonstrates the previous proposition. Proposition 2.2. [12] For any integer n ≥ 2,
γb(Pn) = γ(Pn) =
ln 3 m
.
Erwin also characterized graphs with γb(G) ≤ 3.
Proposition 2.3. [12] Let G be a connected graph. If min {rad(G), γ(G)} = k, where 1 ≤ k ≤ 3, then γb(G) = k.
These propositions highlight three classes of graphs defined in terms of their broad-cast number.
Type I: (radial ) graphs with γb(G) = rad(G),
Type II: (1-cap) graphs with γb(G) = γ(G),
Type III: graphs with γb(G) < min {rad(G), γ(G)}.
Thus, Proposition 2.3 shows that all graphs with min {rad(G), γ(G)} ≤ 3 are radial or 1-cap. Figure 2.1 depicts Erwin’s example which shows that a similar state-ment does not hold for a graph with min {rad(G), γ(G)} = 4. Much of the research
done on broadcast domination has been in classifying various graph families as radial or 1-cap. In the following sections, we examine the classification of trees.
1 2
Figure 2.1: A graph G with min {rad(G), γ(G)} = 4 and γb(G) = 3.
Another useful result for investigating broadcasts on graphs is from Dunbar et al. concerning efficient broadcasts.
Proposition 2.4. [11] Every graph G has a γb-broadcast which is efficient.
Proposition 2.4 allows for the examination of broadcasts on graphs without having to worry about the possibility of overlapping broadcast neighbourhoods of broadcast vertices.
2.1.2
Shadow Trees
We now examine a useful construct for investigating the broadcast number of trees, called shadow trees. Let P : v0, ..., vd be a diametrical path of a tree T with
diam(T ) = d. For each vi ∈ V (P ), let Ui be the set of all vertices of T that are
connected to vi by a (possibly trivial) path internally disjoint from P . Let ui be
a vertex in Ui at maximum distance from vi, and let Bi be the vi − ui path. The
shadow tree ST ,P of T with respect to P is the subtree of T induced by
Sd
i=0V (Bi).
If T ∼= ST,P for some diametrical path P of T , then T is also called a shadow tree.
Note that a shadow tree has maximum degree at most three. In Figures 2.2, 2.3 and 2.4 we illustrate a tree T with diametrical paths P1 and P2, and the non-isomorphic
a b c d e f g h i j
k l m n o p
q r
s t u v
Figure 2.2: A tree T with diametrical paths P1 = {a, b, c, d, e, f, g, h, i, j} and P2 =
{k, b, c, d, e, f, g, h, r, v}.
a b c d e f g h i j
l o
q s
t
Figure 2.3: The shadow tree ST,P1 of T in Figure 2.2.
k b c d e f g h r v
a o
i s
Figure 2.4: The shadow tree ST ,P2 of T in Figure 2.2.
Shadow trees are particularly useful in studying broadcasts on trees. Herke and Mynhardt demonstrated the following.
Theorem 2.5. [19] For any shadow tree ST of T , γb(ST) = γb(T ).
That is, the broadcast numbers of a tree and any of its shadow trees are equal. This result simplifies the study of broadcasts on trees by allowing the removal of certain branches and leaves.
Before considering more results on shadow trees, we introduce some additional notation concerning their structure. Consider a shadow tree ST ,P. If Ui− {vi} 6= ∅,
we call vi a branch vertex and the vi− ui path Bi a branch. Furthermore, for αi =
d(vi, ui) ≥ 1, the tree ∆i induced by {vi−αi, . . . , vi−1} ∪ V (Bi) ∪ {vi+1, . . . , vi+αi} is
called the triangle at i. If the vertex subset {vi−αi, . . . , vi, . . . , vi+αi} of the triangle
∆i is contained in the vertex subset {vj−αj, . . . , vj, . . . , vj+αj} of the triangle ∆j, then
∆i is called a nested triangle. In Figure 2.3, the vertex set ∆g = {f, g, h, o} forms
the triangle at g. Likewise, ∆h = {f, g, h, i, j, q, t} induces the triangle at h. Since
{f, g, h} ⊆ {f, g, h, i, j}, ∆g is a nested triangle. A free edge is an edge of ST ,P that
is not in any triangle; note that all free edges of ST ,P lie on P . The edges cd, de and
ef in Figure 2.3 are free edges.
The triangles of ST ,P are labeled in order of their occurrence on P and are denoted
∆i1, ∆i2, . . . , ∆ic. For simplicity, we abuse notation and denote ∆i1 as ∆1, and ∆ic
as ∆c. A free edge on P that comes before ∆1 is called a leading free edge; likewise,
a free edge that comes after ∆c is called a trailing free edge. If e is a free edge of
ST ,P, we also call e a free edge of T with respect to P . A set M of edges of the
diametrical path P of the tree T is a split-P set if each component T0 of T − M has a positive even diameter and P0 = T0∩ P is a diametrical path of T0. A split-set of
T is a split-P set for some diametrical path P of T . An edge in any split-set of T is a split-edge. In Figures 2.3-2.4, the sets M1 = {cd} and M2 = {ef } are split-P1 sets.
Thus, cd and ef are split-edges. The requirement that P0 = T0∩ P be a diametrical path of T0 implies that each split-edge is a free edge. However, not all free edges are split-edges. For example, the free edge de in Figures 2.2-2.4 is not a split-edge because the components of the trees after the deletion of de have odd diameters.
Herke and Mynhardt provide the following result which demonstrates how split-sets determine the broadcast number of a tree.
Theorem 2.6. [19] If M is a split-set of maximum cardinality m of a tree T , then γb(T ) = diam(T ) − m 2 .
Note that ldiam(T )−m2 m= diam(T )−m2 unless m = 0 and diam(T ) is odd.
v0 v2 vc vd uc,1 uc,α Qc Bc P ef el
Figure 2.5: A labelled shadow tree ST ,P.
We illustrate the following notation in Figure 2.5. Let c be the highest index such that vc is a branch vertex of T . The subpath Qc: vc, ..., vd of P is called the trailing
endpath of T . The branch of T that starts at vc is the path Bc: vc= uc,0, uc,1, ..., uc,α
of length α, and is called the last branch of T . The triangle ∆c associated with
Bc is called the last triangle of T . For brevity we also write Bc and Qc for V (Bc)
and V (Qc), respectively. We denote the lengths of Bc and Qc by `(Bc) and `(Qc),
respectively; note that α = `(Bc) ≤ `(Qc) = d − c. The first and last edges of ∆c
on P are ef = vc−αvc−α+1 and e` = vc+α−1vc+α, respectively. We say that the leaf
uc,α of Bc binds ef (e`, respectively) if T − uc,α has a split-set that contains ef (e`,
respectively), but T has no such split-set.
2.1.3
Broadcasts in Trees
Dunbar, Hedetniemi and Hedetniemi [10] first considered the characterization of trees that are Type I and Type II in an unpublished manuscript in 2003. In 2008, Seager [27] published her characterization of broadcast types of caterpillars. A complete
characterization of radial trees was given by Herke in her 2009 thesis.
Theorem 2.7. [18] A tree T is radial if and only if it has no nonempty split-set. Mynhardt and Wodlinger [25] expanded upon these results by providing eight conditions for a tree to be uniquely radial ; that is, a tree with γb(T ) = rad(T ) and
whose only minimum cost broadcast is a broadcast from a central vertex with cost equal to the radius of T . In his 2011 thesis, Lunney [21] continued the characterization of broadcasts in trees by determining a large class of 1-cap trees.
2.1.4
Complexity
Determining the domination number of a general graph is NP-hard. Indeed, deter-mining γ(G) for many classes of well-known graphs, including bipartite graphs and chordal graphs, remains NP-hard [16]. Thus, since broadcast domination is a gener-alization of domination, it was originally speculated that finding γb(G) would also be
NP-hard. However, in 2006 Heggernes and Lokshtanov [17] published an algorithm for finding γb(G) in O(n6) time, for a graph on n vertices.
Trees are one of the few classes of graphs with linear algorithms for finding γ(T ). This algorithm, for computing γ(T ) for an arbitrary unweighted tree, was discovered by Cockayne, Goodman and Hedetniemi in 1975 [5]. In 2007, Dabney [8] found a linear algorithm for γb(T ) for an arbitrary unweighted tree. This result was published
in [9].
2.2
Irredundance
The graph property irredundance is closely related to domination. Formally, a set X ⊆ V (G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; it is possible that y = x. An irredundant set
X is maximal irredundant if, for any v ∈ V − X, X ∪ {v} is not irredundant. The irredundance number ir(G) is the cardinality of a smallest maximal irredundant set of G. A maximal irredundant set of cardinality ir(G) is an ir-set.
Irredundant sets are often viewed in terms of their private neighbourhoods. For v ∈ X, a vertex u is called a private neighbour of v with respect to X, if N [u] ∩ X = {v}. Notice that it is possible that u = v; in this case, we call v a self-private neighbour. The private neighbourhood of v with respect to X is pn(v, X) = {u : N [u] ∩ X = {v}}. The external private neighbourhood of v with respect to X is the set epn(v, X) = pn(v, X) − {v}. Note the alternative definition that X is irredundant if and only if pn(v, X) 6= ∅ for all v ∈ X.
Irredundance was introduced by Cockayne, Hedetniemi and Miller in 1978 [7]. They observed the following proposition, which allows the use of irredundance to test the minimality of a dominating set.
Proposition 2.8. [7] A dominating set S is minimal dominating if and only if it is irredundant.
They also demonstrated the following inequality, a portion of the well-known domination chain.
Proposition 2.9. [7] For any graph G, ir(G) ≤ γ(G).
The following year, Bollob´as and Cockayne [2] showed that for any graph G,
γ(G) ≤ 2 ir(G) − 1. (2.2)
When investigating irredundant sets, it is helpful to view the vertices of a graph G as partitioned into four subsets (see Figure 2.6):
R
Y
Z X
C
Figure 2.6: A graph H illustrating the subsets of V (H). X, an irredundant set
Y =S
x∈Xepn(x, X), the set of external private neighbours of vertices in X
R = V − N [X], the set of vertices not dominated by X
C = V − (X ∪ Y ∪ R), the set of vertices in V − X that are dominated by two or more vertices in X.
Furthermore, let Z be the set of isolated vertices of G[X]. Cockayne, Grobler, Hedetniemi and McRae [6] provide a useful necessary and sufficient condition for an irredundant set to be maximal irredundant.
Theorem 2.10. [6] An irredundant set X is maximal irredundant if and only if
A: for each v ∈ N [R] there exists x ∈ X such that pn(x, X) ⊆ N [v].
If A holds, we say that v annihilates x, and we call A the annihilation property. Henceforth we let X denote a maximal irredundant set of G. Then for each r ∈ R there exists x ∈ X such that d(r, x) = 2, that is, R ⊆ N2[X].
If x, x0 ∈ X, then R(x) and R(x0) may or may not be disjoint. Moreover, by the
annihilation property,
for each r ∈ R there exists x ∈ X such that epn(x, X) ⊆ N (r).
Note that r ∈ R(x) in this case. Hence
R = [
x∈X
R(x). (2.3)
2.2.1
Complexity
In his 1984 dissertation, Pfaff demonstrated that finding ir(G) for an arbitrary graph is NP-hard [26]. Determining ir(G) for other common graph classes, including split-graphs (and hence chordal split-graphs) [20], line split-graphs [22], and k-regular split-graphs [14], has also been shown to be NP-hard. In 1985, Bern, Lawler and Wong [1] constructed an O(n) algorithm for determining ir(T ) for a tree on n vertices.
2.3
Packings
A set of vertices S is a 2-packing if for each pair of vertices u, v ∈ S, N [u] ∩ N [v] = ∅. The 2-packing number of a graph G, ρ(G), is the cardinality of a maximum 2-packing. The definition provides the following immediate bound.
Theorem 2.11. For any graph G, ρ(G) ≤ γ(G).
For trees, Meir and Moon [23] proved the equality of the 2-packing and domination numbers in 1975.
The development of multipackings will be discussed later in Chapter 4. For now, we provide only the basic definitions. For a graph G and an integer k such that 1 ≤ k ≤ diam(G), a set M of vertices of G is called a k-multipacking if, for each v ∈ V and each integer s such that 1 ≤ s ≤ k, the set Ns[v] contains at most s
vertices in M . Formally, M is a k−multipacking if for all v ∈ V , |Ns[v] ∩ M | ≤ s for
all 1 ≤ s ≤ k. The k-multipacking number mpk(G) is the maximum cardinality of a k-multipacking of G. A 1-multipacking is simply an ordinary 2-packing, and mp1(G) is the 2-packing number ρ(G). If M is a k-multipacking, where k = diam(G), we call M a multipacking, and the k-multipacking number the multipacking number, denoted mp(G).
Chapter 3
A New Upper Bound for the
Broadcast Number of a Graph
In this chapter we provide a new upper bound for the broadcast number of a graph in terms of its irredundance number. By combining the irredundance and domination number bound of Bollob´as and Cockayne (2.2) with Erwin’s bound for the broadcast number (2.1), we obtain the following implicit bound:
γb(G) ≤ min {γ(G), rad(G)} ≤ 2 ir(G) − 1
for any graph G.
However, this bound is really based upon γ(G), not γb(G). By ignoring the
mid-dleman, we obtain a new tight bound for γb(G). The work in this chapter has been
accepted for publication in [3].
Theorem 3.1. For any graph G, γb(G) ≤ 32ir(G). Moreover, for each k ∈ Z+ there
exists a connected graph Gk such that ir(Gk) = 2k and γb(Gk) = 3k.
provides a tight lower bound for ir(G) in terms of the more easily computed broadcast number γb(G).
3.1
Proof of Theorem 3.1
To prove our new upper bound, we first establish two lemmas. We follow the notation defined in Section 2.2.
Lemma 3.2. For any z ∈ Z and any r ∈ R(z), there exists x ∈ X − Z such that r ∈ R(x).
Proof. By the definition of R, r /∈ N [X]. By the annihilation property of the maximal irredundant set X, r annihilates some x ∈ X. Thus r ∈ R(x). Since r is not adjacent to x, x /∈ N [r]. Since r annihilates x, pn(x, X) ⊆ N [r]. It follows that x /∈ pn(x, X), that is, x is not an isolated vertex of G[X]. Therefore x ∈ X − Z as required. By Lemma 3.2 and (2.3), R = [ x∈X−Z R(x), that is, R ⊆ N2[X − Z]. (3.1)
Lemma 3.3. If a graph G has an ir-set X such that G[X] has only isolated vertices, then γb(G) ≤ ir(G).
Proof. By Lemma 3.2, R = ∅. Hence X dominates G so that γb(G) ≤ γ(G) = ir(G).
Theorem 3.1 For any graph G, γb(G) ≤ 32 ir(G). Moreover, for each k ∈ Z+ there
exists a connected graph Gk such that ir(Gk) = 2k and γb(Gk) = 3k.
Proof. Let X be an ir-set of G. We proceed by examining three types of components of G[X] and their 2-neighbourhoods. To simplify notation we denote the vertex set of each component Xi of G[X] also by Xi. Below we define a broadcast f in three steps
such that Vf+ ⊆ X, and for each Xi 6= K2, Pv∈Xif (v) ≤ |Xi|, while Pv∈Xif (v) = 3
if Xi = K2. Hence σ(f ) ≤ 32|X|.
Type 1: Xi is a component with order n ≥ 3.
Let v be a central vertex of Xi. Then, for each x ∈ Xi,
d(v, x) ≤ rad(Xi) ≤ rad(Pn) =
n − 1 2
.
Define f (v) = n and f (u) = 0 otherwise. Since the distance from v to any other vertex in N2[Xi] is at most
n−1
2 + 2 ≤ n (since n ≥ 3), N2[Xi] is f -dominated.
Type 2: Xi = K2.
Choose arbitrary v ∈ Xi, define f (v) = 3 and f (u) = 0 otherwise. Then N2[Xi] ⊆
Nf[v].
Type 3: Xi = {z}. Then z ∈ Z.
Define f (z) = 1. Then N [z] = Nf[v].
If X = Z, then by Lemma 3.3 we are done. If X 6= Z, then by (3.1), R ⊆ N2[X −
Z]. Hence each vertex in R hears a broadcast from a vertex in a Type 1 or 2 component of G[X]. Also, C ⊆ N [X], hence each vertex in C hears a broadcast from a vertex in X. It follows that f is a dominating broadcast of G and so γb(G) ≤ σ(f ) ≤ 32|X|.
Let H be the graph depicted in Figure 3.1. Since H has no universal vertex (a vertex adjacent to every other vertex of H), ir(H) ≥ 2. Also, X = {v2, v3} is
v0 v0 0 v 0 5 v5 v1 v4 v2 v3 v02 3 X
Figure 3.1: A graph H with ir(H) = 2 and γb(H) = 3.
pn(v3, X) = {v4}, X ∪ {y} is redundant for each y ∈ N [R] = {v0, v00, v1, v4, v5, v50}.
This shows that ir(H) = 2.
v0 v0 v5 v6 v11 v6k−6 v6k−1 0 v50 v60 v110 v6k−60 v6k−10 v1 v4 v7 v10 v6k−5 v6k−2 v2 v3 v8 v9 v6k−4 v6k−3 v02 v08 v6k−40 3 X 3 3
Figure 3.2: A graph Gk with ir(Gk) = 2k and γb(Gk) = 3k.
The function f : V (H) → {0, 3} defined by f (v20) = 3 and f (u) = 0 otherwise is a dominating broadcast of H, hence γb(H) ≤ 3. Suppose H has a dominating
broadcast g with cost 2. Let w be the vertex that broadcasts to v0. Since g(w) ≤ 2,
w ∈ {v0, v00, v1, v2}, hence w does not broadcast to v5. Therefore g(w) = 1, w ∈
{v0, v1}, and there exists a vertex w0 6= w with g(w0) = 1 that broadcasts to v5.
As for w, w0 ∈ {v4, v5}. Now v02 hears no broadcast, a contradiction. Therefore
γb(H) = 3.
correspond-ing to the vertex v6−j (v06−j, where applicable) of H by v6i−j (v06i−j). Form the graph
Gk by joining v6i−1 to v6i, i = 1, ..., k − 1. See Figure 3.2.
As for H, X =Sk
i=1{v6i−4, v6i−3} is a maximal irredundant set of Gkof cardinality
2k, hence ir(Gk) ≤ 2k. We show that γb(Gk) = 3k; it will follow that ir(Gk) ≥ 2
3γb(Gk) = 2k.
The function f : V (Gk) → {0, 3} defined by f (u) = 3 if u ∈ {v6i−40 : i = 1, ..., k}
and f (u) = 0 otherwise is a dominating broadcast of Gk, hence γb(Gk) ≤ 3k. For
any dominating broadcast f0 of Gk, if u ∈ V (Hi) hears a broadcast from a vertex
w ∈ V (Hi), we say that u hears a local broadcast, otherwise we say that u hears only
foreign broadcasts. Denote the set of vertices that hear only foreign broadcasts of f0 by For(f0).
Among all γb-broadcasts of Gk, let g be one such that |For(g)| is minimum. If
For(g) = ∅, then for each i = 1, ..., k and each vertex u ∈ V (Hi), u hears a local
broadcast, and as with H we show thatP
w∈V (Hi)g(w) = 3 for each i = 1, ..., k. Thus
σ(g) = 3k and we are done.
Hence suppose For(g) 6= ∅. Let u0 ∈ For(g)∩V (Hi0) for some i0and let w ∈ V (Hj)
broadcast to u0. Assume without loss of generality that j > i0. Then there exists an index i ≤ i0 and a vertex u ∈ For(g) ∩ V (Hi) at maximum distance from w. If v6i−10
is such a vertex u, then so is v6i−3, if v06i−4 is such a u, then so is v6i−4, and if v6i−60
is such a vertex, then so is v6i−6. Therefore we may assume u = v` for some `. We
may similarly assume that w = vm for some m. Furthermore, we may assume that
g(vs) = 0 for all ` < s < m, otherwise there exists a dominating broadcast g0 of Gk
such that σ(g0) < σ(g), which is a contradiction.
We consider the different values of `. In each case we define a dominating broadcast h such that σ(h) = σ(g) and |For(h)| < |For(g)|, thus producing a contradiction on the choice of g. In most cases we use vm+t for some t ≥ 1 as a broadcast vertex of
h instead of w = vm; note that m + t ≤ 6k, otherwise we can obtain a lower cost
dominating broadcast.
Case 1 ` = 6i − 6. Then g(w) ≥ 6. Let h(v6i−3) = 3, h(w) = 0, h(vm+3) = g(w) − 3,
and h(v) = g(v) otherwise.
Case 2 ` ∈ {6i − 5, 6i − 4}. Then g(w) ≥ 4. Let h(v6i−3) = 2, h(w) = 0, h(vm+2) =
g(w) − 2, and h(v) = g(v) otherwise.
Case 3 ` = 6i − 3. Then g(w) ≥ 3. Let h(v6i−2) = 1, h(w) = 0, h(vm+1) = g(w) − 1,
and h(v) = g(v) otherwise.
Case 4 ` ∈ {6i − 2, 6i − 1}. Since v` ∈ For(g), v6i−10 does not hear a broadcast from
vq, q < 6i − 1. Since v6i−10 is g-dominated, g(v 0
6i−1) = 1, and thus ` 6= 6i − 2; we need
only consider ` = 6i − 1. Let h(v6i−2) = 1, h(v6i−10 ) = 0 and h(v) = g(v) otherwise.
Let p be the largest index such that vp or v0p hears the broadcast g from w. Then
this vertex also hears the broadcast h from vm+t in each of Cases 1 – 4. All other
vertices on v`−vppaths, and the end-vertices attached to these paths, hear a broadcast
from either vm+t or a vertex of Hi, and certainly v` hears a broadcast from a vertex
of Hi in each case. Therefore h is a dominating broadcast such that σ(h) = σ(g) and
|For(h)| < |For(g)|. This completes the proof of Theorem 3.1.
3.1.1
Corollaries
Having proven our new bound in Theorem 3.1, we now present a pair of corollaries concerning graphs with γb(G) ≤ ir(G) and graphs with γb(G) = 32ir(G). In doing so,
we often define a broadcast f on a component of an ir-set as for the Type 1, 2 or 3 components in the proof of Theorem 3.1. We then simply say that we use a Type i assignment for f , where i ∈ {1, 2, 3}.
Corollary 3.4. If a graph G has an ir-set X such that every nontrivial component of G[X] has order at least three, then γb(G) ≤ ir(G).
Proof. By using a Type 1 or Type 3 assignment for each component of G[X], we define a dominating broadcast f with σ(f ) = |X|.
Corollary 3.5. Suppose γb(G) = 32ir(G). Then for any ir-set X of G,
(i) G[X] = mK2 for some integer m,
(ii) each vertex c ∈ C is adjacent to exactly two vertices u, v ∈ X, where uv ∈ E(G), and c is adjacent to no vertices in N2[x] where x ∈ X − {u, v},
(iii) for each u ∈ X there exists r ∈ R(u) such that r /∈ R(w) for any w ∈ X − {u}, (iv) for each edge uv of G[X] there exists c ∈ C adjacent to u and v,
(v) for any two vertices u, v ∈ X, no vertex in pn(u, X) is adjacent to all vertices in pn(v, X).
Proof. Assume γb(G) = 32ir(G).
(i) Suppose G has an ir-set X such that G[X] mK2. As in the proof of Theorem
1, by using a Type 2 assignment on each K2 component of G[X] and a Type 1 or 3
assignment otherwise, we obtain a dominating broadcast f of G with σ(f ) < 32ir(G), a contradiction.
(ii) Suppose c ∈ C is adjacent to u, u0 ∈ X, where uu0 ∈ E(G). Then there exist/ distinct vertices v, v0 ∈ X adjacent to u and u0, respectively. Define a broadcast
f so that f (c) = 4, all other components have Type 2 assignments, and f (w) = 0 otherwise. Then f is a dominating broadcast with σf(G) < 32ir(G), a contradiction.
Case 1 c is adjacent to some c0 ∈ C. By the above, c0is adjacent to some component
u0v0 of G[X]. Define a broadcast f so that f (c) = 4; for each component in G[X] other than uv and u0v0, use Type 2 assignments, and f (w) = 0 otherwise. Then all vertices in N2[{u, v}] ∪ N2[{u0, v0}] hear the broadcast from c, and all other vertices
in G are broadcast to by a Type 2 assignment.
Case 2 c is adjacent to some y0 ∈ pn[u0], where u0v0is a component of G[X]. Define a
broadcast f so that f (y0) = 4; for each component in G[X] other than uv and u0v0, use Type 2 assignments, and f (w) = 0 otherwise. Since N2[{u, v}] ∪ N2[{u0, v0}] ⊆ N4[y0],
G is dominated by this broadcast.
Case 3 c is adjacent to some r ∈ R(u0), where u0v0 is a component of G[X]. Define a broadcast f so that f (c) = 3 and f (v0) = 2; for each edge in G[X] other than uv and u0v0, use Type 2 assignments, and f (w) = 0 otherwise. The vertices in N2[{u, v}],
N2[v0] and N [u0] hear the broadcast from c or v0. We need only examine R(u0). If
R(u0) = {r}, we are done, so let r0 ∈ R(u0) − {r}. By the annihilation property,
r0 must annihilate some w ∈ X. If w = u0 then d(r0, r) ≤ 2, and thus r0 hears the broadcast from c. If w ∈ {u, v}, then d(r0, c) ≤ 3, so it again hears the broadcast from c. If w = v0, then d(r0, v0) = 2 and it hears the broadcast from v0. If w /∈ {u, v, u0, v0},
then r0 hears a Type 2 broadcast from some other component of G[X].
In each case f is a dominating broadcast with σf(G) < 32 ir(G), a contradiction.
(iii) Suppose no such r exists and assume u is matched to v in G[X]. Let f (v) = 2, use Type 2 assignments for all other components of G[X], and let f (w) = 0 for all other vertices of G. Then u, pn(u, X), pn(v, X) and R(v) all hear the broadcast from v. By our assumption, for all r0 ∈ R(u), r0 ∈ R(x) for some other x ∈ X. Thus,
the Type 2 assignment on the component containing x dominates r0. Therefore, G is f -dominated and σ(G) < 32ir(G). This again contradicts the assumption of γb(G) = 32ir(G).
(iv) Suppose uv is an edge of G[X] such that no vertex in C is adjacent to u and v. Note that pn(u, X) = epn(u, X) and pn(v, X) = epn(v, X). If x ∈ pn(u, X) is not adjacent to a vertex in R, then no vertex in R(u) annihilates u. By the annihilation property each r ∈ R(u) annihilates a vertex in X − {u}, that is, r ∈ R(w) for some w ∈ X − {u}, contradicting (iii). Thus each vertex in pn(u, X) and, similarly, each vertex in pn(v, X) is adjacent to a vertex in R.
Choose any a ∈ pn(u, X) and any b ∈ pn(v, X) and let f (a) = f (b) = 1. Use Type 2 assignments for all other components of G[X], and let f (w) = 0 otherwise. We see immediately that N2[X − {u, v}] is f -dominated and only need to verify that
N2[{u, v}] − N2[X − {u, v}] is f -dominated.
Consider z ∈ N2[{u, v}] − N2[X − {u, v}]. Since no vertex in C is adjacent to u
and v, z ∈ pn(u, X) ∪ pn(v, X) ∪ R, and thus, z ∈ N [R]. By the annihilation property and the choice of z, pn(w, X) ⊆ N [z] for w ∈ {u, v} and thus z hears the broadcast from either a or b. Therefore f is a dominating broadcast with σ(f ) < 32ir(G), which once again is a contradiction.
(v) Suppose u, v ∈ X and there is some a ∈ pn(u, X) that is adjacent to all vertices in pn(v, X). If uv is an edge in G[X], define a broadcast f by f (a) = 2; for each component of G[X] other than uv, use Type 2 assignments, and f (w) = 0 otherwise. Then σf(G) < 32ir(G). Immediately, N [u], N [v] and R(v) are all subsets of N2[a],
and hear the broadcast from a. We need only examine R(u). Let r ∈ R(u). By the annihilation property, r annihilates some w ∈ X. If w = u, then r ∈ N [a] ⊆ N2[a]. If
w = v, then r is adjacent to all vertices in pn(v, X) and hence r ∈ N2[a]. Otherwise,
w ∈ X − {u, v}, in which case r hears the broadcast from w or its unique neighbour in X. In all cases f is a dominating broadcast.
Now consider the case when u and v are not adjacent. By (i), let u0, v0 ∈ X be the vertices adjacent to u and v respectively. Define f by f (u) = 3 and f (v0) = 2;
for any other edge in G[X], use Type 2 assignments, and f (w) = 0 otherwise. Then N2[X − {v}] is dominated and we only need to check the vertices in R(v). Consider
r ∈ R(v). By definition, r is adjacent to some b ∈ pn(v, X), which in turn is adjacent to a. Thus d(r, u) ≤ 3 and r hears the broadcast from u. Moreover, σ(f ) < 32ir(G). This contradiction completes the proof of the corollary.
The converse of Corollary 3.5 is not true. The graph G in Figure 3.3 satisfies conditions (i) – (v) but has γb(G) = 5 and ir(G) = 4.
5
X
Figure 3.3: A counterexample to the converse of Corollary 3.5.
Theorem 3.1 shows that the ratio γb/ ir is bounded above by 32. However, neither
the ratio ir /γb nor the difference ir −γb is bounded above. To see this, consider a
tree T obtained from the star K1,m by subdividing each edge exactly once. Then
Chapter 4
Broadcast Domination and
Multipackings in Graphs
In this chapter we introduce the notion of a multipacking of a graph by formulating broadcast domination as a linear programming problem and then considering its dual. We generalize the famous result of Meir and Moon [23] and show the equality of the multipacking and broadcast numbers of a tree. Finally, we provide an alternative proof to the second half of Theorem 3.1 using multipackings.
4.1
Broadcast Domination as an LP Problem
Like many other graph-theoretic parameters, broadcast domination can be considered as an integer programming (IP) problem. Its fractional relaxation linear program (LP), or primal linear program (PLP), has a dual linear program (DLP) whose IP formulation provides a lower bound for the broadcast number (see (4.1) below). While this bound may be weak in general, we use it to establish γb(Gk) for the graph Gk in
Theorem 3.1. This work has been accepted for publication in [3].
of LP-duality in domination-related problems.
A dominating broadcast on a graph G can also been viewed as a covering of G with k-neighbourhoods centred at each of the broadcast vertices. Thus a broadcast can be seen as a collection B = {Nk[v]} such that for each u ∈ V there exists some
Nk[v] ∈ B with u ∈ Nk[v]. For this covering form of broadcasting, we denote the cost
of the broadcast B by
σB =
X
Nk[v]∈B
k.
Finding the minimum σB is a natural IP. With each Nk[v] we associate an indicator
variable xk,v ∈ {0, 1}, where xk,v= 1 if f (v) = k 0 otherwise.
The IP objective function is given by
minX
v∈V
X
1≤k≤e(v)
k · xk,v.
There is one constraint for each vertex u. We define Bu = {(k, v) : u ∈ Nk[v]}, the
set of k-neighbourhoods that contain u. Our IP constraints require that each u be in at least one selected k-neighbourhood. That is, for each u ∈ V ,
X
(k,v)∈Bu
The fractional relaxation LP is given by minX v∈V X 1≤k≤e(v) k · xk,v s.t. X (k,v)∈Bu xk,v ≥ 1 for each u ∈ V xk,v ≥ 0.
The dual LP has one variable yu for each vertex u. It is
maxX
u∈V
yu
s.t. X
u∈Nk[v]
yu ≤ k for each Nk[v], k = 1, . . . , e(v)
yu ≥ 0.
That is to say, we assign a weight yu to each u ∈ V so that, for each k ∈ {1, ..., e(u)},
the total weight in the k-neighbourhood of u does not exceed k. Thus, when we consider the 1-neighbourhood of u, it follows that 0 ≤ yu ≤ 1. In the case that yu ∈
{0, 1} this simplifies to choosing a set of vertices Y so that each k-neighbourhood of u has at most k vertices in Y . The following proposition is an immediate consequence of these concepts.
Proposition 4.1. Let Y ⊆ V such that |Y ∩ Nk[v]| ≤ k for each v ∈ V and each
k = 1, 2, . . . , e(v). Then γb ≥ |Y |.
Proof. Let f be a γb-broadcast and let v ∈ V+. Then Nf[v] contains at most f (v)
vertices from Y . Since V =S
γb = X v∈Vf+ f (v) ≥ X v∈Vf+ |Nf[v] ∩ Y | ≥ |Y |.
The set Y described in Proposition 4.1 is a multipacking of G, as defined in Section 2.3. Hence, by Proposition 4.1,
γb(G) ≥ mp(G). (4.1)
For some graphs it turns out that γb(G) = mp(G), that is, the PLP and the
DLP have optimum integer solutions. However, for other graphs γb(G) > mp(G).
This is the case when the PLP and DLP have fractional optimal solutions. Con-sider the example of C5, where x1,v = yv = 13 and x2,v = 0 for each vertex v. Then
P
v∈V
P2
k=1k · xk,v =
P
u∈V yu = 53. Thus these labellings, which satisfy the
con-straints, are primal and dual optimal, and minP
v∈V
P2
k=1k · xk,v = maxP yu = 53.
However, γb(C5) = 2 and mp(C5) = 1.
The following result is immediate from the definition of multipackings.
Proposition 4.2. If G is 1-cap (γ(G) = γb(G)) and does not have an efficient γ-set,
then mp(G) < γb(G).
This proposition is again nicely demonstrated with C5. Although γb(C5) = γ(C5),
C5 does not have an efficient γ-set. Hence, mp(C5) < γb(C5). We can generalize this
result for all cycles.
Proposition 4.3. For any cycle Cn with n ≥ 3, mp(Cn) = γb(Cn) if and only if
n ≡ 0 (mod 3).
n ≡ 0 (mod 3) have efficient γ-sets; let M be such a set. Consider any v ∈ V . Since M is efficient and dominating, |N [v] ∩ M | = 1. Pick any 2 ≤ s ≤ e(v). Suppose v ∈ M . If s ≡ 0 (mod 3), then |Ns[v] ∩ M | = s; otherwise, if s 6≡ 0 (mod 3),
then |Ns[v] ∩ M | < s. Now suppose v 6∈ M . If s = 2, |Ns[v] ∩ M | = s; otherwise,
|Ns[v] ∩ M | < s. It follows that M is a multipacking of Cn.
(⇐) The converse follows immediately from Proposition 4.2, since cycles with n 6≡ 0 (mod 3) do not have efficient γ-sets.
Our initial investigations suggested that γb(G) − mp(G) ≤ 1. However, the graph
G in Figure 4.1 is an example with γb(G) = 4, mp(G) = 2. This is currently the only
graph we know of with γb(G) − mp(G) ≥ 2.
Figure 4.1: A graph G with γb(G) = 4, mp(G) = 2.
4.2
Broadcasts and Multipackings in Trees
In Chapter 2, we introduced a famous result of Meir and Moon which we restate here. Theorem 2.12 [23] For any tree T , ρ(T ) = γ(T ).
Since multipackings generalize 2-packings in the same manner that broadcast dom-ination generalizes ordinary domdom-ination, it seems natural that the analogue of Meir and Moon’s result should hold for multipackings and broadcasts. We prove next that this is indeed the case. The work in Sections 4.2.1 - 4.2.2 has been submitted for publication in [24].
Theorem 4.4. For any tree T , γb(T ) = mp(T ).
Before we prove Theorem 4.4, we require the formulation of some technical lem-mas.
4.2.1
Lemmas
Let T be a tree with γb(T ) = k. Then diam(T ) ≥ 2k − 1 since γb(T ) ≤ rad(T ) (as
observed in [12]) and 2 rad(T )−1 ≤ diam(T ). If diam(T ) ≥ 3(k−1), then mp(T ) ≥ k, as shown below in Lemma 4.5.
Lemma 4.5. For any graph G, if diam(G) ≥ 3(k − 1), then mp(G) ≥ k.
Proof. Let G have a diametrical path P : v0, ..., vd, where d ≥ 3(k − 1). Define
Vi = {v : d(v, v0) = i} for each 1 ≤ i ≤ d. We claim that the set M = {vi : i ≡ 0(mod 3),
0 ≤ i ≤ 3(k − 1)} is a multipacking of G. By our choice of M , any vertex vi of P
satisfies |Ns[vi] ∩ M | ≤ s for all s ≥ 1. Consider any 1 ≤ r ≤ d and any v ∈ Vr. Since
vr ∈ Vr is on P and M ⊆ V (P ), Ns[v] ∩ M ⊆ Ns[vr] ∩ M . Thus |Ns[v] ∩ M | ≤ s for
all s ≥ 1. It follows that mp(G) ≥ |M | = k.
Hence to prove Theorem 4.4 we only need to consider trees T with γb(T ) = k and
2k − 1 ≤ diam(T ) ≤ 3k − 4.
• Let Tk,d be the set of all trees T such that diam(T ) = d, γb(T ) = k and
γb(T − `) = k − 1 for each leaf ` of T .
Since γb(ST ,P) = γb(T ) (Theorem 2.5), each T ∈ Tk,d is a shadow tree. Also, if
ui,αi is the leaf on the branch Bi of a nested triangle ∆i, then an edge e is a free edge
of T − ui,αi if and only if e is a free edge of T . By Theorem 2.6 and the fact that
every split-edge is a free edge, γb(T − ui,αi) = γb(T ). Thus T ∈ Tk,d has no nested
Lemma 4.6. If the shadow tree T has no trailing free edges, then γb(T −uc,α) = γb(T ).
Proof. If T has no trailing free edges, then ef = vd−2αvd−2α+1 and e` = vd−1vd.
Suppose to the contrary that γb(T − uc,α) < γb(T ). Since diam(T − uc,α) = diam(T ),
Theorem 2.6 implies that T − uc,α has a maximum split-set of larger cardinality than
a maximum split-set of T . Since ef and e` are the only free edges of T − uc,α that
are not free edges of T , uc,α binds ef or e`. But e` is not a split-edge of T − uc,α
because vd is an isolated vertex of T − uc,α − e` and thus the subgraph induced by
{vd} does not have positive diameter, and ef is not a split-edge of T − uc,α because
the component of T − uc,α− ef that contains vdhas odd diameter 2α − 1 and no other
split-edges. Thus any split-set of T − uc,α is a split-set of T , a contradiction.
Lemma 4.6 shows that all trees in Tk,d have at least one trailing free edge and,
similarly, at least one leading free edge. In the next lemma we demonstrate how any multipacking M of a shadow tree T can be transformed into a new multipacking M0 of the same size.
v0 vc vd v0 vc vd Qc Bc Qc Bc A multipacking M on T A multipacking M0 on T Figure 4.2: Illustration of Lemma 4.7.
Lemma 4.7. Let M be a multipacking of a shadow tree T and suppose M contains the x vertices vi1, ..., vix of Qc, where c ≤ i1 ≤ · · · ≤ ix ≤ d, and the y vertices uc,j1, ..., uc,jy
of Bc− {vc}, where 1 ≤ j1 ≤ · · · ≤ jy ≤ α. Then there exists a multipacking M0 of
T such that |M0| = |M | and M0 contains the x vertices v
d, vd−3, ..., vd−3x+3 of Qc and
Proof. Define M0 by
M0 = (M − Bc− Qc) ∪ {vd, vd−3, ..., vd−3x+3} ∪ {uc,α, uc,α−3, ..., uc,α−3y+3}.
That is, M0 is obtained from M by “pushing out” the vertices of M ∩ Bc and M ∩ Qc
as far as possible towards the leaf, using the leaf and every third vertex from the leaf until the correct number of vertices have been used. This technique is illustrated in Figure 4.2. It is clear that |M | = |M0|. If M = M0, then we are done. Thus assume
M 6= M0. We show that M0 is a multipacking of T .
Let T0 be the component of T − {vcvc+1, vcuc,1} that contains vc. For each vertex
u ∈ V (T0), each r = 0, ..., x−1, and each ` = 0, ..., y−1, dT(u, vd−3r) ≥ dT(u, vix−r) and
dT(u, uc,α−3`) ≥ dT(u, uc,jy−`). Hence if T has a vertex u such that |Ns[u] ∩ M
0| ≥ s + 1
for some integer s, then u ∈ {vc+1, ..., vd} ∪ {uc,1, ..., uc,α}.
First assume that there exists such a vertex u ∈ {vc+1, ..., vd}. Let m be the
smallest index, c + 1 ≤ m ≤ d, for which there exists an integer s such that |Ns[vm] ∩ M0| ≥ s + 1. By the choice of M0, s ≥ 2. Define
X = {vm+s−1, vm+s} if m + s ≤ d {vd} if m + s − 1 = d ∅ otherwise.
Now Ns−1[vm−1] ∪ X = Ns[vm], and again by the choice of M0, X contains at most
one vertex from M0. Hence |Ns−1[vm−1] ∩ M0| ≥ s, contradicting the choice of m. A
similar contradiction follows if u ∈ {uc,1, ..., uc,α}. Therefore M0 is a multipacking of
T .
In the next lemma we consider two trees T and T0, each with diameter d. We assume that both these trees have a diametrical path P : v0, ..., vd labelled with the
same labels. Examples of T and T0 are shown in Figures 4.3 and 4.4.
Lemma 4.8. Let T and T0 be shadow trees of diameter d such that the last branch Bc = vc, uc,1, ..., uc,α of T has length α = d−c−1, the last branch Bc0 = vc0, uc0,1, ..., uc0,α0
of T0 has length α0 = d − c0 − 2, where c0 = c − 1 (and thus α = α0), and
T − {uc,1, ..., uc,α} ∼= T0 − {uc−1,1, ..., uc−1,α}. Let M0 be a multipacking of T0 that
satisfies the conditions of Lemma 4.7. Define M ⊆ V (T ) by
M = (M0− (Bc−1− {vc−1})) ∪ {uc,i : uc−1,i∈ M0, i ≥ 1} if |{vc, uc−1,2} ∩ M0| ≤ 1
(M0− Bc−1− {vc}) ∪ {uc,i : uc−1,i ∈ M0} ∪ {vc−1} otherwise.
Then M is a multipacking of T . Moreover, γb(T ) = γb(T0).
Proof. Except for the first edge e0f = vd−2α−2vd−2α−1 of ∆c−1 in T0, which is a free
edge of T but not of T0, and the last edge e` = vd−2vd−1 of ∆c in T , which is a free
edge of T0 but not of T , an edge e of P is a free edge of T if and only if e is a free edge of T0. But e0f is not a split-edge of T , nor is e` a split-edge of T0, because the
component of T − e0f (T0 − e`, respectively) that contains vd has odd diameter. By
Theorem 2.6, γb(T ) = γb(T0).
Now consider the sets M and M0. Since `(Bc−1) = `(Qc−1) − 2, `(Qc−1) ≡
k (mod 3) for some k ∈ {0, 1, 2} and α = `(Bc−1) = `(Bc) ≡ k + 1 (mod 3).
Let T1 be the component of T − vc−1vc that contains vc−1. We consider two cases,
depending on the elements of M0.
Case 1 |{vc, uc−1,2} ∩ M0| ≤ 1. Since M0 satisfies the conditions of Lemma 4.7,
M0∩ V (Bc−1) ⊆ {uc−1,k+1, uc−1,k+4, ..., uc−1,α},
M ∩ V (Bc) ⊆ {uc,k+1, uc,k+4, ..., uc,α} and (4.2)
v0 vc−1vc vd v0 vc−1vc vd
A multipacking M0 on T0 A multipacking M on T Figure 4.3: Illustration of Case 1 of the proof of Lemma 4.8
This situation is illustrated in Figure 4.3. For each u ∈ V (T1) and each uc,i ∈ M ,
dT(u, uc,i) > dT0(u, uc−1,i). Hence if T has a vertex u such that |Ns[u] ∩ M | ≥ s + 1
for some integer s, then u ∈ Bc∪ Qc.
First assume there exists an integer s such that |Ns[vc] ∩ M | ≥ s + 1 in T . By the
choice of M and M0, s ≥ 2. The only vertex of M that is in the s-neighbourhood of vc in T for which there is no corresponding vertex of M0 in the s-neighbourhood of
vc−1 in T0 is vc+s, if vc+s ∈ M . In T0, |Ns[vc−1] ∩ M0| ≤ s. Since |Ns[vc] ∩ M | ≥ s + 1
in T , we deduce the following three facts:
(i) |Ns[vc−1] ∩ M0| = s,
(ii) there is no vertex of M0 in T1 at distance s from vc−1, and
(iii) vc+s ∈ M and thus vc+s ∈ M0.
Since M0 is a multipacking, vc+s−1 ∈ M/ 0. Moreover, by (4.2), s ≡ k − 1 (mod 3).
Now (4.2) also implies that uc−1,s ∈ M/ 0. Thus
|Ns−1[vc−1] ∩ M0| = |Ns[vc−1] ∩ M0| = s,
which is impossible since M0 is a multipacking. Hence |Ns[vc] ∩ M | ≤ s for all s ≥ 1.
If there exists a vertex u ∈ {vc+1, ..., vd} ∪ {uc,1, ..., uc,α} such that |Ns[u] ∩ M | ≥
v0 vc−1vc vd v0 vc−1vc vd
A multipacking M0 on T0 A multipacking M on T Figure 4.4: Illustration of Case 2 of the proof of Lemma 4.8.
Case 2 {vc, uc−1,2} ⊆ M0. By Lemma 4.7, `(Qc−1) ≡ 1 (mod 3), α = `(Bc−1) =
`(Bc) ≡ 2 (mod 3),
M0∩ V (Bc−1) = {uc−1,2, uc−1,5, ..., uc−1,α},
M ∩ V (Bc) = {uc,2, uc,5, ..., uc,α},
M0∩ V (Qc−1) = {vc, vc+3, ..., vd} and (4.3)
M ∩ V (Qc−1) = {vc−1, vc+3, ..., vd}.
This situation is illustrated in Figure 4.4. Thus vc−1 ∈ M − M0. Since M0 is a
multipacking and {vc, uc−1,2} ⊆ N2[vc−1] ∩ M0, {vc−2, vc−3} ∩ M0 = ∅. Hence,
N [vc−1] ∩ M = N2[vc−1] ∩ M = {vc−1},
and for each s ≥ 3,
|Ns[vc−1] ∩ M | ≤ |Ns[vc−1] ∩ M0| ≤ s.
Suppose there exists u ∈ V (T1) − {vc−1} such that in T , |Ns[u] ∩ M | ≥ s + 1 for
some integer s. Since M0 is a multipacking of T0, |Ns[u] ∩ M0| ≤ s in T0. Let w be
The only vertex x ∈ M such that u is closer to x in T than to its counterpart x0 ∈ M0 in T0 is x = v
c−1, and then x0 = vc. It follows that |Ns[u] ∩ M | = s + 1 and
|Ns[u] ∩ M0| = s in T0. Therefore
Ns[u] ∩ {vc−1, vc} = {vc−1}. (4.4)
Since vc ∈ N/ s[u] and dT0(u, uc−1,2) > d(u, vc), it also follows that uc−1,2 ∈ N/ s[u].
Now consider Ns+1[w]. By (4.4), vc−1 ∈ Ns[u], hence the definition of w implies that
vc, uc−1,2 ∈ Ns+1[w]. Since vc, uc−1,2 ∈ M0 by assumption and vc ∈ N/ s[u] by (4.4), it
follows that
|Ns+1[w] ∩ M0| ≥ |Ns[u] ∩ M0| + 2 = s + 2,
contrary to M0 being a multipacking. Hence |Ns[u] ∩ M | ≤ s for all u ∈ V (T1) and
all s ≥ 1.
Suppose now there exists an integer s such that |Ns[vc] ∩ M | ≥ s + 1 in T . By
the choice of M and M0, s ≥ 2. We now proceed as in Case 1 and obtain facts (i) – (iii); specifically, by (4.3), d(vc−1, vc+s) = s + 1 ≡ 1 (mod 3), so that s ≡ 0 (mod 3).
Then (4.3) implies that uc−1,s ∈ M/ 0. Thus
|Ns−1[vc−1] ∩ M0| = |Ns[vc−1] ∩ M0| = s,
which is again impossible since M0 is a multipacking. Hence |Ns[vc] ∩ M | ≤ s for all
s ≥ 1.
If there exists a vertex u ∈ {vc+1, ..., vd} ∪ {uc,1, ..., uc,α} such that |Ns[u] ∩ M | ≥
s + 1 for some s, we once again obtain a contradiction as in the proof of Lemma 4.7. Hence, for all v ∈ V (T ) and all s ≥ 1, |Ns[v] ∩ M | ≤ s. Therefore M is a multipacking
of T and mp(T ) ≥ mp(T0).
Lemma 4.9 now follows immediately from Lemma 4.8.
Lemma 4.9. Let T and T0 be shadow trees of diameter d such that the last branch Bc = vc, uc,1, ..., uc,α of T has length d − c − 1, the last branch Bc0 = vc0, uc0,1, ..., uc0,α0
of T0 has length d−c0−2, where c0 = c−1 (and thus α = α0), and T −{u
c,1, ..., uc,α} ∼=
T0− {uc−1,1, ..., uc−1,α}. Then γb(T ) = γb(T0) and mp(T ) ≥ mp(T0).
Lemma 4.10. Let T be a shadow tree with multipacking M . Then, for each vertex v ∈ V (T ) and each s ≥ d(v, vd) + 2, |Ns[v] ∩ M | < s.
Proof. We proceed by examining all vertices v ∈ V (T ), using the usual vertex-labelling scheme outlined above. Since M is a multipacking, |Ns[v] ∩ M | ≤ s for all
s.
Suppose v = vi is on the diametrical path. If vi = v0, then Ns[v0] = Nd[v0] = V
for any s ≥ d(v0, vd) + 2. Hence |Ns[v0] ∩ M | = |Nd[v0] ∩ M | ≤ d < s. Consider
v = vi for i ∈ {1, 2, . . . , d}. Since vi is on the diametrical path and s ≥ d(vi, vd) + 2,
vd ∈ Ns−1[vi−1] and thus Ns−1[vi−1] = Ns[vi]. Hence
|Ns[vi] ∩ M | = |Ns−1[vi−1] ∩ M | ≤ s − 1.
Now suppose v = ui,j is on a branch. Since s > d(ui,j, vd) > d(vi, vd), vd ∈ Ns[vi]
and thus ui,αi ∈ Ns[vi]. Therefore Ns[ui,j] ⊂ Ns[vi] and so
|Ns[ui,j] ∩ M | ≤ |Ns[vi] ∩ M | < s,
as shown above.
Lemma 4.11. Let T ∈ Tk,d have at least three trailing free edges. If M0 is a
multi-packing of the tree T0 = T − {vd−2, vd−1, vd}, then M = M0 ∪ {vd} is a multipacking
of T .
Proof. For each v ∈ T we show that |Ns[v] ∩ M | ≤ s for all s ∈ Z+.
Case 1 v ∈ T − {vd−2, vd−1, vd}. If s ≤ d(v, vd−3), then Ns[v] ∩ M = Ns[v] ∩ M0. But
M0 is a multipacking, hence |Ns[v] ∩ M | ≤ s. If s = d(v, vd−3) + 1, then since vd−2 is
not in M , Ns[v] ∩ M0 = Ns[v] ∩ M and hence |Ns[v] ∩ M | ≤ s. If s ≥ d(v, vd−3) + 2,
then by Lemma 4.10, |Ns[v] ∩ M0| < s and so |Ns[v] ∩ M | ≤ s.
Case 2 v ∈ {vd−2, vd−1, vd}. Since {vd−2, vd−1, vd} ∩ M = {vd}, the case s = 1
is immediate. For s = 2 or 3, Ns[v] ⊆ N [vd−4] ∪ {vd−2, vd−1, vd}. Since M0 is a
multipacking, |N [vd−4] ∩ M | ≤ 1. Thus
|Ns[v] ∩ M | ≤ | (N [vd−4] ∪ {vd−2, vd−1, vd}) ∩ M | = 2.
Otherwise, s ≥ 4 and Ns[v] ⊆ Ns[vd−3]. By Case 1,
|Ns[v] ∩ M | ≤ |Ns[vd−3] ∩ M | ≤ s.
The figures used in the proof of the next lemma also illustrate some of the ideas in the proof of Theorem 4.4.
Lemma 4.12. If T ∈ Tk,d, where k ≤ 4 and 2k − 1 ≤ d ≤ 3k − 4, then mp(T ) = k.
Proof. By Lemma 4.6 we need only consider trees with at least one leading and at least one trailing free edge. The result is vacuously true if k ≤ 2. We exhaustively demonstrate in Figure 4.5 that k = mp(T ) = γb(T ) for k = 3, 4. The large (red, if
is shown. (i) γb= 3, diam = 5 3 (ii) γb= 4, diam = 7 4 4 4 4 (iii) γb= 4, diam = 8 4 4 4
Figure 4.5: Trees with k = mp(T ) = γb(T ) for k = 3, 4.
4.2.2
Proof of Theorem 4.4
We restate the theorem for convenience.Theorem 4.4 For any tree T , γb(T ) = mp(T ).
Proof. The proof is by induction on |V (T )|. The statement is clearly true for trees of order at most 3. Suppose it holds for all trees of order at most n − 1 and let T be a tree such that |V (T )| = n and γb(T ) = k. If T has a leaf ` such that γb(T − `) = k,
then by the induction hypothesis T − ` has a multipacking M of cardinality k, which is also a multipacking of T . Hence assume γb(T − `) = k − 1 for each leaf `. Then
T ∈ Tk,d for some d. As stated above, T is a shadow tree without nested triangles.
By Lemma 4.6, T has at least one trailing free edge.
Case 1 T has at least three trailing free edges. Let T0 = T − {vd−2, vd−1, vd} and
Any γb-broadcast f0 of T0 can be extended to a broadcast f of T such that σ(f ) =
σ(f0) + 1 by broadcasting from vd−1 with a cost of 1. Hence γb(T0) = k − 1 (otherwise
σ(f ) < γb(T )). By the induction hypothesis, |M0| = k − 1, and by Lemma 4.11,
M = M0 ∪ {vd} is a multipacking of T . Hence mp(T ) = k.
Case 2 T has two trailing free edges. Let T00 = T − {vd−1, vd}. As in Case 1,
γb(T00) = k − 1. Note that T00 has no trailing free edges. Thus by Lemma 4.6,
γb(T00 − uc,α) = k − 1. Since T00 − vd−2 = T∼ 00 − uc,α, γb(T00 − vd−2) = k − 1, and
by the induction hypothesis T00− vd−2 has a multipacking M0 of cardinality k − 1.
Let M = M0 ∪ {vd}; we claim that M is a multipacking of T . This construction is
illustrated in Figure 4.6.
Suppose to the contrary that there exists v ∈ V (T ) such that |Ns[v] ∩ M | ≥ s + 1
for some s ≥ 1. If v ∈ T − {vd−2, vd−1, vd} then we obtain a contradiction as in Case
1 in the proof of Lemma 4.11. Thus assume that v ∈ {vd−2, vd−1, vd}. By definition
of M , |Ns[v] ∩ M | ≤ s for any s ≤ 2. For any s ≥ 3, Ns[v] ⊆ Ns[vd−3]. However,
by the above, |Ns[vd−3] ∩ M | ≤ s. Therefore |Ns[v] ∩ M | ≤ s. It follows that M is a
multipacking of T and mp(T ) = k.
Case 3 T has one trailing free edge. Suppose vc−1 is a branch vertex of T and let
x be the leaf of Bc−1. If `(Bc−1) < `(Bc), then ∆c−1 is a nested triangle, which is
not the case. If `(Bc−1) > `(Bc), then ∆c is a nested triangle, which is also not the
case. If `(Bc−1) = `(Bc) = α, then the last edge e0` of ∆c−1 is an edge of ∆c and
thus not a free edge of T − x. The first edge e0f of ∆c−1 is vd−2α−2vd−2α−1. Since
d(vd−2α−1, vd) = 2α + 1, which is odd, e0f is not a split-edge of T − x. Therefore the
leaf x of Bc−1 does not bind any edges of P . By Theorem 2.6, γb(T − x) = γb(T ),
which, again, is not the case. Hence vc−1 is not a branch vertex of T .
Let T0 be the tree obtained by deleting the branch Bc (but not vc) and adding
v0 vc vd−2 vd
uc,α
v0 vc vd−2
uc,α
(i) T with two trailing free-edges (ii) T00 = T − {vd−1, vd}
v0 vc uc,α vd−3 v0 vc uc,α vd−3 (iii) T0 = T00− {vd−2} (iv) M0 on T0 v0 vc vd uc,α (v) M on T
two trailing free edges and, as shown in Case 2, mp(T0) = k. Again by Lemma 4.9, mp(T ) ≥ k and we are done.
4.2.3
Finding a Maximum Multipacking of a Tree
In Subsection 4.2.2, we proved that mp(T ) = γb(T ) for any tree T . We also noted in
Subsection 2.1.4 that γb(T ) (and mp(T )) can be found in linear time [8]. However,
we have not yet addressed the problem of finding a maximum multipacking. In this subsection we apply the insight gained in the proof of Theorem 4.4 to develop an algorithm for finding a maximum multipacking of a tree.
By Theorem 2.5, the shadow tree ST ,P of T with diametrical path P has
γb(ST ,P) = γb(T ). It follows that any maximum multipacking M of ST ,P will also
be a maximum multipacking of T . Thus our algorithm need only consider shadow trees. Furthermore, as in the proof of Theorem 4.4, we limit our algorithm to trees in Tk,d, and hence ignore all nested triangles.
We again rely heavily upon the notation developed in Subsection 2.1.2. Notice that vc, vd, Qc, Bc and ∆c depend upon the choice of P for a diametrical path. For
∆i1, the first triangle of T , we define Q1 = {v0, v1, . . . , vi1} as the leading endpath of
T . Similarly to Qc, Q1 is dependent upon the choice of P .
We first explain the steps of the algorithm informally.
• Begin with a shadow tree T with no nested triangles; we will find M , a maximum multipacking of T .
• Create an empty set of vertex pairs S. As the algorithm progresses, we will store vertices in S that will need to be checked at the end.
• Otherwise, repeat the following process of reducing T (and P ) until T is P1, P2
or P3, adding vertices to M at each step.
– Check to ensure that P is still a diametrical path. If at any stage of the process, Bc is the first (and hence only) branch and l(Bc) > l(Q1), then
change P to be P = Bc∪ Qc. Now, the vertices formerly in Q1 become
vertices on a branch.
– If a nested triangle ∆ij at vij results, delete Bij −vij from T .
– Perform one of the following steps depending upon the number of trailing free edges of T .
Case 1 T has at least three trailing free edges. Add vd to M and delete
{vd, vd−1, vd−2} from P and T . Thus the vertex that was previously labelled
vd−3 is now labelled vd.
Case 2 T has two trailing free edges. Add vdto M and delete {vd, vd−1}
from P and T . Thus T now has no trailing free edges. As mentioned in Case 2 of the proof of Theorem 4.4, γb(T ) = γb(T − uc,α), and T − uc,α ∼=
T − vd, so delete vd (formerly vd−2) from T . This makes l(Qc) < l(Bc),
so switch Qc and Bc in P . Thus the vertices formerly on Qc now become
branch vertices, and the vertices formerly on Bc become vertices on the
endpath.
Case 3 T has one trailing free edge. In Case 3 of the proof of The-orem 4.4 we show that there is no branch Bc−1 in T . Let T0 be the
tree obtained from T by deleting Bc (but not vc) and adding the branch
Bc−1 = {uc−1,1, uc−1,2,, . . . , uc−1,α}. That is, T0 = T − (Bc− {vc}) ∪ Bc−1.
Recall that γb(T ) = γb(T0) and that T0 has two trailing free edges. Add
2 of Lemma 4.8) we will need to swap vc for vc−1 ∈ M . Now set T to T0.
In the next iteration of the algorithm, T has at least two trailing free edges and M can be expanded using Case 1 or 2.
Case 4 T has no trailing free edges. By Lemma 4.6, γb(T −uc,α) = γb(T );
delete uc,α from T . Now T has at least one trailing free edge. In the next
iteration, T will have at least two trailing free edges. Once T has been reduced to P1, P2 or P3, add v0 to M .
• Lastly, we must check the vertices in S against M . For each (u, v) ∈ S If u ∈ M , remove u from M and add v to M instead.
Then we are done.
We summarize these steps in Algorithm 4.1 below. An example of this process is given in Figure 4.7.
Algorithm 4.1: FindTreeMP finds a maximum multipacking of a tree Input: Shadow tree T with no nested triangles, diametrical path
P = {v0, v1, . . . vd}
Output: A maximum multipacking M of T
1 M ← ∅ 2 S ← ∅ 3 while T 6= P1, P2, P3 do 4 if c = i1 and l(Bc) > l(Q1) then 5 P ← P − Q1 + Bc 6 comment: Q1 becomes Bc 7 end
8 if ∆ij is a nested triangle then
9 T ← T − (Bij −vij )
10 end
11 if number of trailing free edges ≥ 3 then 12 M ← M ∪ {vd}
13 P ← P − {vd, vd−1, vd−2} 14 T ← T − {vd, vd−1, vd−2}
15 else if number of trailing free edges = 2 then 16 M ← M ∪ {vd}
17 P ← P − {vd, vd−1, vd−2} 18 T ← T − {vd, vd−1, vd−2} 19 P ← P − Qc+ Bc
20 comment: Qc becomes Bc
21 else if number of trailing free edges = 1 then 22 S ← S ∪ {(vc, vc−1)} 23 T ← T ∪ {uc−1,1, uc−1,2, . . . , uc−1,α} 24 T ← T − (Bc− {vc}) 25 else 26 T ← T − {uc,α} 27 end 28 end 29 M ← M ∪ {v0};
30 forall the (u, v) ∈ S do 31 if u ∈ M then
32 M ← (M − {u}) ∪ {v}
33 end
34 end 35 return M
a b c d e f g h i j k l m
n o
p q r s
t u v w x
(1) A tree T0 with diametrical path P = {a, b, c, . . . , m}
a b c d e f g h i j k l m
n
q r
t v
(2) Create shadow tree T = ST0,P of T0 with no nested triangles. Add m to M .
a b c d e f g h i j n q r t v (3) Delete {k, l, m} from T . a b c d e f g h i j n q r t v
a b c d e f g n q r t v (5) Delete {h, i, j} from T . a b c d e f r v n q g t
(6) Swap Qc and Bc. P becomes P − Bc∪ Qc.
a b c d e f r v
n
q t
(7) Delete the nested triangle ∆f.
a b c d e f r v
n
q t
a b c d e n q t (9) Delete {f, r, v} from T . a b c d q t n e (10) Swap Qc and Bc. a b c d q t n e
(11) Shift Bc. Add (d, c) to S. Add t to M .
a b c e
n
(12) Delete {d, q, t} from T .
n a b c e
(13) Shift Bc. Add (b, a) to S. Add e to M .
n a
(14) Delete {b, c, e} from T . Add n to M .
(15) M = {m, j, v, t, e, n}, S = {(g, f ), (e, d), (d, c), (b, a)}. Swap e and d in M .
a b c d e f g h i j k l m
n o
p q r s
t u v w x
(16) A maximum multipacking M of T0. Figure 4.7: Example of Algorithm 4.1.
4.3
An Alternative Proof of Theorem 3.1
Having examined the origins of multipackings and their relationship with broadcasting on trees, we now present a useful application of multipackings for calculating γb(G).
In Theorem 3.1, we presented a new bound for γb(G) in terms of ir(G), and showed
the existence of a graph Gkwith ir(Gk) = 2k and γb(Gk) = 3k, for any positive integer
k. However, in this proof we had to demonstrate that our broadcast f was indeed a minimum dominating broadcast, a somewhat long and cumbersome endeavor.
Now that we have established in Section 4.1 that for any graph G, mp(G) ≤ γb(G),
if we can find a maximum multipacking with cardinality equal to the cost of our dominating broadcast, we can verify that our broadcast is a minimum dominating broadcast. In this subsection we give a maximum multipacking of Gk, and thereby
confirm that γb(Gk) = 3k. v0 v5 v6 v11 v6k−6 v6k−1 v00 v50 v60 v110 v6k−60 v6k−10 v1 v4 v7 v10 v6k−5 v6k−2 v2 v3 v8 v9 v6k−4 v6k−3 v20 v80 v06k−4 X
Figure 4.8: A graph Gk with ir(Gk) = 2k and γb(Gk) = 3k.
Consider the vertex set Mk = v00, v 0 2, v 0 5, v 0 6, v 0 8, v 0 11, . . . , v 0 6k−4, v 0 6k−2, v 0 6k−1 of Gk.
In Figure 4.8, Mk is depicted as the large yellow (if viewed in colour) vertices.
Lemma 4.13. Let v be a vertex of Gk. For any s > min {d(v, v0), d(v, v6k−1)}, if
Proof. Consider arbitrary v ∈ V (Gk) and assume min {d(v, v0), d(v, v6k−1)} =
d(v, v6k−1), i.e., s > d(v, v6k−1).
Let U = {u ∈ V (Gk) : d(u, v) = s}.
If |U | ≤ 1, the result follows, hence suppose |U | ≥ 2. Since s > d(v, v6k−1), U lies “to
the left” of v and is one of the sets {v6i−3, v6i−10 }, {v6i−4, v6i−40 }, {v6i−6, v6i−60 }, for some
1 ≤ i ≤ k. In each case |U ∩Mk| = 1. Hence |Ns[v]∩Mk| = |Ns−1[v]∩Mk|+1 and again
the result follows. By symmetry the result also holds if min {d(v, v0), d(v, v6k−1)} =
d(v, v0).
Claim 4.14. The set Mk = v00, v02, v50, v06, v80, v011, . . . , v06k−4, v6k−20 , v6k−10 is a
multi-packing of the graph Gk in Figure 4.8.
Proof. Suppose to the contrary that there is some v ∈ V (Gk) such that |Ns[v]∩Mk| ≥
s + 1 for some s.
Case 1 v is on the diametrical path P = {v0, v1, v2, v3, . . . , v6k−1}. Let v = vi, where
(a) i is the smallest index and
(b) subject to (a), s the smallest distance
such that |Ns[vi] ∩ Mk| ≥ s + 1. Clearly, |Ns[v0] ∩ Mk| ≤ s for all s, hence v0 6= vi.
Likewise, vi 6= v6k−1. Also, by our choice of Mk, |N [vi] ∩ Mk| ≤ 1 for each i, thus
s ≥ 2. Let ` = i − s if i − s ≥ 0 0 otherwise. Then d(v`, vi) ≤ s.
