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Independent Set Reconfiguration in Cographs and their Generalizations
Author(s) -
Bonsma Paul
Publication year - 2016
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21992
Subject(s) - combinatorics , mathematics , cograph , split graph , chordal graph , independent set , reachability , discrete mathematics , maximal independent set , pathwidth , interval graph , induced path , clique sum , indifference graph , bounded function , graph , longest path problem , 1 planar graph , line graph , mathematical analysis
We study the following independent set reconfiguration problem, called TAR‐Reachability :given two independent sets I and J of a graph G , both of size at least k , is it possible to transform I into J by adding and removing vertices one‐by‐one, while maintaining an independent set of size at least k throughout? This problem is known to be PSPACE‐hard in general. For the case that G is a cograph on n vertices, we show that it can be solved in time O ( n 2 ) , and that the length of a shortest reconfiguration sequence from I to J is bounded by 4 n − 2 k (if it exists). More generally, we show that if G is a graph class for which (i) TAR‐Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in G using disjoint union and complete join operations. Chordal graphs and claw‐free graphs are given as examples of such a class G .