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

• Authors: Binh-Minh Bui-Xuan;Jan Arne Telle;Martin Vatshelle
• Venue: Discrete Applied Mathematics
• Year: 2010

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^(^@r^(^H^)^^^2^))), and the q-Coloring problem in time O(n(m+2^O^(^q^@r^(^H^)^^^2^))), where @r(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^kxk^2)), the Minimum Dominating Set problem in time O(n(m+2^3^4^k^^^2^+^2^3^4^kxk^3)) and the q-Coloring problem in time O(n(m+2^q^2^k^^^2^+^5^q^+^4^2^kxk^2^qxq)). These are the first algorithms for NP-hard problems whose runtimes are single exponential in the rankwidth.