Formula omitted]-join decomposable graphs and algorithms with runtime single exponential in rankwidth

Abstract

We introduce H -join decompositions of graphs, indexed by a fixed bipartite graph H . These decompositions are based on a graph operation that we call a H -join, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertices of H , and connecting classes by the pattern H . H -join decompositions are related to modular, split and rank decompositions. Given an H -join decomposition of an n -vertex m -edge graph G , we solve the Maximum Independent Set and Minimum Dominating Set problems on G in time O ( n ( m + 2 O ( ρ ( H ) 2 ) ) ) , and the q -Coloring problem in time O ( n ( m + 2 O ( q ρ ( H ) 2 ) ) ) , where ρ ( H ) is the rank of the adjacency matrix of H over GF(2). Rankwidth is a graph parameter introduced by Oum and Seymour, based on ranks of adjacency matrices over GF(2). For any positive integer k we define a bipartite graph R k and show that the graphs of rankwidth at most k are exactly the graphs having an R k -join decomposition, thereby giving an alternative graph-theoretic definition of rankwidth that does not use linear algebra. Combining our results we get algorithms that, for a graph G of rankwidth k given with its width k rank-decomposition, solves the Maximum Independent Set problem in time O ( n ( m + 2 1 2 k 2 + 9 2 k × k 2 ) ) , the Minimum Dominating Set problem in time O ( n ( m + 2 3 4 k 2 + 23 4 k × k 3 ) ) and the q -Coloring problem in time O ( n ( m + 2 q 2 k 2 + 5 q + 4 2 k × k 2 q × q ) ) . These are the first algorithms for NP-hard problems whose runtimes are single exponential in the rankwidth. 1 1 For a polynomial function p o l y we call 2 p o l y ( k ) single exponential in k .