Decremental Optimization of Vertex-Coloring Under the Reconfiguration Framework

Abstract

Suppose that we are given a positive integer k, and a k-(vertex-)coloring of a given graph G. Then we are asked to find a coloring of G using the minimum number of colors among colorings that are reachable from by iteratively changing a color assignment of exactly one vertex while maintaining the property of k-colorings. In this paper, we give linear-time algorithms to solve the problem for graphs of degeneracy at most two and for the case where . These results imply linear-time algorithms for series-parallel graphs and grid graphs. In addition, we give linear-time algorithms for chordal graphs and cographs. On the other hand, we show that, for any , this problem remains NP-hard for planar graphs with degeneracy three and maximum degree four. Thus, we obtain a complexity dichotomy for this problem with respect to the degeneracy of a graph and the number k of colors.