# Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

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 (defender) moves 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 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 NP-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. This result also holds for split graphs. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.

## Authors

• 15 publications
• 1 publication
01/17/2022

### Eternal vertex cover number of maximal outerplanar graphs

Eternal vertex cover problem is a variant of the classical vertex cover ...
10/30/2020

### Monitoring the edges of a graph using distances

We introduce a new graph-theoretic concept in the area of network monito...
12/12/2018

### On Graphs with Minimal Eternal Vertex Cover Number

The eternal vertex cover problem is a variant of the classical vertex co...
03/19/2020

### Minimum Scan Cover with Angular Transition Costs

We provide a comprehensive study of a natural geometric optimization pro...
11/05/2019

### Angle Covers: Algorithms and Complexity

Consider a graph with a rotation system, namely, for every vertex, a cir...
09/10/2019

### Approximating Vertex Cover using Structural Rounding

In this work, we provide the first practical evaluation of the structura...
11/14/2017

### Sequences of radius k for complete bipartite graphs

A k-radius sequence for a graph G is a sequence of vertices of G (typica...
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