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Eternal Domination: Criticality and Reachability

100%
Klostermeyer W. F., MacGillivray G.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
63-77
EN
We show that for every minimum eternal dominating set, D, of a graph G and every vertex v ∈ D, there is a sequence of attacks at the vertices of G which can be defended in such a way that an eternal dominating set not containing v is reached. The study of the stronger assertion that such a set can be reached after a single attack is defended leads to the study of graphs which are critical in the sense that deleting any vertex reduces the eternal domination number. Examples of these graphs and tight bounds on connectivity, edge-connectivity and diameter are given. It is also shown that there exist graphs in which deletion of any edge increases the eternal domination number, and graphs in which addition of any edge decreases the eternal domination number.
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Structure of the set of all minimal total dominating functions of some classes of graphs

100%
Kumar K., MacGillivray G.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 3
407-423
EN
In this paper we study some of the structural properties of the set of all minimal total dominating functions ($𝔉_T$) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $𝔉_T(G)$ for some classes of graphs.
3
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Analogues of cliques for oriented coloring

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Klostermeyer W., MacGillivray G.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
|
nr 3
373-387
EN
We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.
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