Decremental optimization of vertex-colouring under the reconfiguration framework

Suppose that we are given a positive integer k, and a k-(vertex-)colouring of a given graph G. Then we are asked to find a colouring of G using the minimum number of colours among colourings that are reachable from by iteratively changing a colour assignment of exactly one vertex while maintaining t...

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Bibliographic Details
Published in:International journal of computer mathematics. Computer systems theory Vol. 8; no. 1; pp. 80 - 92
Main Authors: Yanagisawa, Yusuke, Suzuki, Akira, Tamura, Yuma, Zhou, Xiao
Format: Journal Article
Language:English
Published: Taylor & Francis 02.01.2023
Subjects:
Combinatorial reconfiguration
graph algorithms
graph colouring
linear-time algorithms
ISSN:2379-9927, 2379-9935
Online Access:Get full text
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Summary:Suppose that we are given a positive integer k, and a k-(vertex-)colouring of a given graph G. Then we are asked to find a colouring of G using the minimum number of colours among colourings that are reachable from by iteratively changing a colour assignment of exactly one vertex while maintaining the property of k-colourings. 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 colours.
ISSN:2379-9927
2379-9935
DOI:10.1080/23799927.2023.2185543