On the tractability of optimization problems on H-graphs

For a graph H, a graph G is an H-graph if it is an intersection graph of connected subgraphs of some subdivision of H. H-graphs naturally generalize several important graph classes like interval graphs or circular-arc graph. This class was introduced in the early 1990s by B́ıró, Hujter, and Tuza. Recently, Chaplick et al. initiated the algorithmic study of H-graphs by showing that a number of fundamental optimization problems like Maximum Clique, Maximum Independent Set, or Minimum Dominating Set are solvable in polynomial time on H-graphs. We extend and complement these algorithmic findings in several directions. First we show that for every fixed H, the class of H-graphs is of logarithmicallybounded boolean-width. Pipelined with the plethora of known algorithms on graphs of bounded boolean-width, this describes a large class of problems solvable in polynomial time on H-graphs. We also observe that H-graphs are graphs with polynomially many minimal separators. Combined with the work of Fomin, Todinca and Villanger on algorithmic properties of such classes of graphs, this identify another wide class of problems solvable in polynomial time on H-graphs. The most fundamental optimization problems among the problems solvable in polynomial time on H-graphs are Maximum Clique, Maximum Independent Set, and Minimum Dominating Set. We provide a more refined complexity analysis of these problems from the perspective of parameterized complexity. We show that Maximum Independent Set and Minimum Dominating Set are W[1]-hard being parameterized by the size of H plus the size of the solution. On the other hand, we prove that when H is a tree, then Minimum Dominating Set is fixed-parameter tractable (FPT) parameterized by the size of H. For Maximum Clique we show that it admits a polynomial kernel parameterized by H and the solution size. ∗Emails of authors: {fedor.fomin, petr.golovach}@ii.uib.no, [email protected]. The third author has been supported by the Polish National Science Centre grant PRELUDIUM DEC2013/11/N/ST6/02706 and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, ERC consolidator grant DISTRUCT, agreement No 648527. †Department of Informatics, University of Bergen, Norway. ‡Technische Universität Berlin, Germany. 1 ar X iv :1 70 9. 09 73 7v 2 [ cs .D S] 1 6 Fe b 20 18