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

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. Submitted: July 2011 Reviewed: October 2011 Revised: November 2011 Reviewed: January 2012 Revised: January 2012 Accepted: January 2012 Final: January 2012 Published: January 2012 Article type: Regular paper Communicated by: C. D. Tóth Supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). An extended abstract of this paper appeared in the proceedings of IWOCA 2011. E-mail addresses: [email protected] (Bojan Mohar) [email protected] (Petr Škoda) 226 Mohar and Škoda Minimizing the Number of Label Transitions