Abstract
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 Bíró, 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 logarithmically-bounded boolean-width (via mim-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 parameterized by the size of H. For Maximum Clique we show that it admits a polynomial kernel parameterized by H and the solution size.
Highlights
The notion of H-graph was introduced in the work of Bíró et al [4] on precoloring extensions of graphs
Chaplick et al [9] and Chaplick and Zeman [10] initiated the systematic study of algorithmic properties of H-graphs. They showed that a number of fundamental optimization problems like Maximum Independent Set and Minimum Dominating Set are solvable in polynomial time on H-graphs for any fixed H
The existence of an fixed-parameter tractable (FPT) algorithm for a general graph H is very unlikely. (We refer to books [11, 12] for definitions from parameterized complexity and algorithms.) We prove a similar lower bound for Maximum Independent Set parameterized by the size of H plus the solution size
Summary
The notion of H-graph was introduced in the work of Bíró et al [4] on precoloring extensions of graphs. It is wellknown that on interval, chordal, circular-arc, and other graphs with “simple” intersection models many NP-hard optimization problems are solvable in polynomial time, see e.g. the book of Golumbic [19] for an overview It is a natural question whether at least some of these algorithmic results can be extended to more general classes of intersection graphs. Pipelining the bound on the number of minimal separators in H-graphs with meta-algorithmic results of Fomin et al [16], we obtained another wide class of problems solvable in polynomial time on H-graphs Examples of such problems are Treewidth, Minimum Feedback Vertex Set, Maximum Induced Subgraph excluding a planar minor, and various packing problems.
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