Isomorphism of (mis)labeled graphs

Abstract

For similarity measures of labeled and unlabeled graphs, we study the complexity of the graph isomorphism problem for pairs of input graphs which are close with respect to the measure. More precisely, we show that for every fixed integer k we can decide in quadratic time whether a labeled graph G can be obtained from another labeled graph H by relabeling at most k vertices. We extend the algorithm solving this problem to an algorithm determining the number ℓ of vertices that must be deleted and the number k of vertices that must be relabeled in order to make the graphs equivalent. The algorithm is fixed-parameter tractable in k+ℓ Contrasting these tractability results, we also show that for those similarity measures that change only by finite amount d whenever one edge is relocated, the problem of deciding isomorphism of input pairs of bounded distance d is equivalent to solving graph isomorphism in general. © 2011 Springer-Verlag Berlin Heidelberg.