Spartan Bipartite Graphs Are Essentially Elementary

Authors Neeldhara Misra , Saraswati Girish Nanoti

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Author Details

Neeldhara Misra
  • Department of Computer Science and Engineering, Indian Institute of Technology, Gandhinagar, India
Saraswati Girish Nanoti
  • Department of Mathematics, Indian Institute of Technology, Gandhinagar, India


The authors are grateful to several anonymous reviewers for helpful comments on earlier versions of this manuscript.

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Neeldhara Misra and Saraswati Girish Nanoti. Spartan Bipartite Graphs Are Essentially Elementary. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph G and is denoted by evc(G). It is clear that evc(G) is at least mvc(G), the size of a minimum vertex cover of G. We say that G is Spartan if evc(G) = mvc(G). The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on 2n vertices where every edge belongs to a perfect matching, an easy strategy is to have n guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Algorithm design techniques
  • Bipartite Graphs
  • Eternal Vertex Cover
  • Perfect Matchings
  • Elementary
  • Spartan


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