Abstract
The Eternal Vertex Cover problem is a dynamic variant of the vertex cover problem. We have a two player game in which guards are placed on some vertices of a graph. In every move, one player (the attacker) attacks an edge. In response to the attack, the second player (the defender) moves some of the guards along the edges of the graph in such a manner that at least one guard moves along the attacked edge. If such a movement is not possible, then the attacker wins. If the defender can defend the graph against an infinite sequence of attacks, then the defender wins.
The minimum number of guards with which the defender has a winning strategy is called the eternal vertex cover number of the graph G. On general graphs, the computational problem of determining the minimum eternal vertex cover number is \(\mathsf {NP}\)-hard and admits a 2-approximation algorithm and an exponential kernel. The complexity of the problem on bipartite graphs is open, as is the question of whether the problem admits a polynomial kernel.
We settle both these questions by showing that Eternal Vertex Cover is \(\mathsf {NP}\)-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.
Dedicated to the memory of Professor Rolf Niedermeier.
The second author acknowledges support from the SERB-MATRICS grant MTR/2017/001033 and IIT Gandhinagar. The third author acknowledges support from CSIR.
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Notes
- 1.
We refer the reader to Sect. 2 for the definition of the notion of a polynomial kernel.
- 2.
Here, as elsewhere, we drop the index referring to the underlying graph if the reference is clear.
- 3.
The notion of generalized trees in the context of eternal vertex cover was considered by Araki et al. (2015). Such graphs are characterized by the following property: every block is an elementary bipartite graph or a clique having at most two cut-vertices in it. Note that a cobipartite graph with four vertices in both parts with two disjoint edges across the parts is not a generalized tree.
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Acknowledgments
The authors thank Pratik Tale for pointing out a correction in a previous version of this paper, and to the anonymous reviewers of CSR for their very detailed comments that have helped improve the presentation of the paper. In particular, we note that the suggestion in Remark 1 was pointed out by one of the reviewers.
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Babu, J., Misra, N., Nanoti, S.G. (2022). Eternal Vertex Cover on Bipartite Graphs. In: Kulikov, A.S., Raskhodnikova, S. (eds) Computer Science – Theory and Applications. CSR 2022. Lecture Notes in Computer Science, vol 13296. Springer, Cham. https://doi.org/10.1007/978-3-031-09574-0_5
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