E.J. Cockayne’s research while affiliated with University of Victoria and other places

A broadcast on a graph GG is a function f:V→Z+∪{0}f:V→Z+∪{0}. The broadcast number of GG is the minimum value of ∑v∈Vf(v)∑v∈Vf(v) among all broadcasts ff for which each vertex of GG is within distance f(v)f(v) from some vertex vv with f(v)≥1f(v)≥1. This number is bounded above by the radius and the domination number of GG. We show that to characterize trees with equal broadcast and domination numbers it is sufficient to characterize trees for which all three of these parameters coincide.

A vertex set X of a simple graph is called OO-irredundant if for each v∈X, N(v)-N(X-{v})≠∅. Basic results for maximal OO-irredundant set of a graph are obtained.

This work is concerned with a facilities location problem in networks modelled by n-vertex undirected graphs G = (V, E), where V = {v(1), ... , v(n)}. Let r = (r(1), ... , r(n)) and s = (s(1), ... , s(n)) be n-tuples of nonnegative integers. Suppose that at most r(i) units of some commodity may be located at the vertex v(i), while there must be at least s(i) units in the vicinity (i.e. in the closed neighbourhood) of v(i). Consider the function f : V -> N (the set of nonnegative integers) where f (v(i)) is the number of units placed at v(i). If f satisfies the above requirements, then f is called an s-dominating r-function of G. In this paper we initiate the theory (called < r, s >-domination) of such functions. Special cases include (basic) domination, k-tuple domination and {k}-domination. Extensions of the graph-theoretic concepts of independence, irredundance, packing and domatic numbers are also considered. The well-studied inequality chain for independence, domination and irredundance parameters is generalised.

For any permutation ππ of the vertex set of a graph GG, the generalized prism πGπG is obtained by joining two copies of GG by the matching {uπ(u):u∈V(G)}{uπ(u):u∈V(G)}. Denote the domination number of GG by γ(G)γ(G). If γ(πG)=γ(G)γ(πG)=γ(G) for all ππ, then GG is called a universal fixer. The edgeless graphs are the only known universal fixers, and are conjectured to be the only universal fixers. We prove that claw-free graphs are not universal fixers.

A necessary and sufficient condition for an open irredundant set of vertices of a graph to be maximal is obtained. This result is used to show that the smallest cardinality amongst the maximal open irredundant sets in an nn-vertex isolate-free graph with maximum degree ΔΔ is at least n(3Δ−1)/(2Δ3−5Δ2+8Δ−1)n(3Δ−1)/(2Δ3−5Δ2+8Δ−1) for Δ≥5Δ≥5, n/8n/8 for Δ=4Δ=4 and 2n/112n/11 for Δ=3Δ=3. The bounds are the best possible.

We show that the double domination number of an n-vertex, isolate-free graph with minimum degree δ is bounded above by n(ln(δ+1)+lnδ+1)/δ. This result improves a previous bound obtained by J. Harant and M. A. Henning [Discuss. Math., Graph Theory 25, No. 1–2, 29–34 (2005; Zbl 1073.05049)]. Further, we show that for fixed k and large δ the k-tuple domination number is at most n(lnδ+(k-1+o(1))lnlnδ)/δ, a bound that is essentially best possible.

A vertex subset X of a simple graph is called OC-irredundant (respectively CO-irredundant) if for each y∈X, N(v)-N[X-{v}]≠∅ (respectively N[v]-N[X-{v}]≠∅). Sharp bounds involving order and maximum degree for the minimum cardinality of a maximal OC-irredundant set and a maximal CO-irredundant set of a tree are obtained and extremal trees are exhibited.

A new strategy, called secure total domination, for placing guards in order to protect a graph, is introduced. Some properties of the strategy for arbitrary graphs are determined, we evaluate the minimum number of guards (termed the secure total domination number and denoted by γ st ) for the path P n and obtain a sharp lower bound for γ st for n-vertex forests with maximum degree at most Δ.

Suppose that at most r units of some commodity may be positioned at any vertex of a graph G = (V,E) while at least s ( r) units must be present in the vicinity (i.e. closed neighbourhood) of each vertex. Suppose that the function f : V 7!{0,...,r}, whose values are the numbers of units stationed at vertices, satisfies the above require- ment. Then f is called an s-dominating r-function. We present an algorithm which finds the minimum number of units required in such a function and a function which attains this minimum, for any tree.