Reconfiguration of Vertex Covers in a Graph

Abstract

Suppose that we are given two vertex covers $C_{0}$ and $C_{t}$ of a graph G, together with an integer threshold $k\ge \max \{\left| C_0 \right| , \left| C_t \right| \}$. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms $C_{0}$ into $C_{t}$ such that each vertex cover in the sequence is of cardinality at most $k$ and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on $k$ for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.