Reconfiguration of Vertex Covers in a Graph

Takehiro ITO  Hiroyuki NOOKA  Xiao ZHOU  

IEICE TRANSACTIONS on Information and Systems   Vol.E99-D   No.3   pp.598-606
Publication Date: 2016/03/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.2015FCP0010
Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science---Developments of the Theory of Algorithms and Computation---)
combinatorial reconfiguration,  even-hole-free graph,  graph algorithm,  vertex cover,  

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Suppose that we are given two vertex covers C0 and Ct of a graph G, together with an integer threshold k ≥ max{|C0|, |Ct|}. Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms C0 into Ct 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.