H-join decomposable graphs and algorithms with runtime single exponential in rankwidth

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 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 2))) , and the q -Coloring problem in time O(n(m + 2 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 Rk and show that the graphs of rankwidth at most k are exactly the graphs having an Rk -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+ 9 2 ×k)) , the Minimum Dominating Set problem in time O(n(m+2 3 4 k+ 23 4 × k)) and the q -Coloring problem in time O(n(m+2 q 2 k+ 5q+4 2 ×k×q)) . These are the first algorithms for NP-hard problems whose runtimes are single exponential in the rankwidth1.