IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Online ISSN : 1745-1337
Print ISSN : 0916-8508
Special Section on Discrete Mathematics and Its Applications
On the Eternal Vertex Cover Numbers of Generalized Trees
Hisashi ARAKIToshihiro FUJITOShota INOUE
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2015 Volume E98.A Issue 6 Pages 1153-1160

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Abstract

Suppose one of the edges is attacked in a graph G, where some number of guards are placed on some of its vertices. If a guard is placed on one of the end-vertices of the attacked edge, she can defend such an attack to protect G by passing over the edge. For each of such attacks, every guard is allowed either to move to a neighboring vertex, or to stay at where she is. The eternal vertex cover number τ(G) is the minimum number of guards sufficient to protect G from any length of any sequence of edge attacks. This paper derives the eternal vertex cover number τ(G) of such graphs constructed by replacing each edge of a tree by an arbitrary elementary bipartite graph (or by an arbitrary clique), in terms of easily computable graph invariants only, thereby showing that τ(G) can be computed in polynomial time for such graphs G.

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© 2015 The Institute of Electronics, Information and Communication Engineers
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