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Edge Dominating Sets and Vertex Covers

100%
Dutton R., Klostermeyer W. F.
Discussiones Mathematicae Graph Theory
|
2013
|
tom 33
|
nr 2
437-456
EN
Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of trees and grid graphs.
2
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Domination, Eternal Domination, and Clique Covering

100%
Klostermeyer W. F., Mynhardt C. M.
Discussiones Mathematicae Graph Theory
|
2015
|
tom 35
|
nr 2
283-300
EN
Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique covering numbers of graph are explored in this paper. Among other results, we characterize bipartite and triangle-free graphs with domination and eternal domination numbers equal to two, trees with equal m-eternal domination and clique covering numbers, and two classes of graphs with equal domination, eternal domination and clique covering numbers.
3
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Eternal Domination: Criticality and Reachability

100%
Klostermeyer W. F., MacGillivray G.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
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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