Minimizing the Number of Label Transitions Around a Nonseparating Vertex of a Planar Graph

Abstract

We study the minimum number of label transitions around a given vertex v0 in a planar multigraph G, in which the edges incident with v0 are labelled with integers 1, . . . , l, and the minimum is taken over all embeddings of G in the plane. For a fixed number of labels, a lineartime fixed-parameter tractable algorithm that computes the minimum number of label transitions around v0 is presented. If the number of labels is unconstrained, then the problem of deciding whether the minimum number of label transitions is at most k is NP-complete.