On H-Topological Intersection Graphs

Abstract

Biró, Hujter, and Tuza (1992) introduced the concept of H-graphs, intersection graphs of connected subregions of a graph H thought of as a one-dimensional topological space. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying H-graphs from the point of view of computational problems that are fundamental in theoretical computer science: recognition, graph isomorphism, dominating set, clique, and colorability.Surprisingly, we negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is NP-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T -graphs, for each fixed tree T . For the special case when T is a star S d of degree d, we have an O(n 3.5 )-time algorithm.We give FPTand XP-time algorithms solving the minimum dominating set problem on S dgraphs and H-graphs parametrized by d and the size of H, respectively. As a byproduct, the algorithm for H-graphs adapts to an XP-time algorithm for the independent set and the independent dominating set problems on H-graphs.If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph G has a Helly H-representation, the clique problem can be solved in polynomial time.Further, we show that both the k-clique and the list k-coloring problems are solvable in FPTtime on H-graphs (parameterized by k and the treewidth of H). In fact, these results apply to classes graphs of graphs with treewidth bounded by a function of the clique number.Finally, we observe that H-graphs have at most n O( H ) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to O( H n 2 ).

On the Tractability of Optimization Problems on H-Graphs

“…On the positive side, we show that when H is a tree, then Minimum Dominating Set is FPT parameterized by the size of H. This significantly extends the result from [CTVZ17] for stars to arbitrary trees. We actually prove a slightly more general result, namely that Minimum Dominating Set is FPT on chordal graphs G parameterized by the leafage of the graph, i.e.…”

“…The first steps in this direction were done by Chaplick et al in [CTVZ17] who showed that Minimum Dominating Set is fixed-parameter tractable (FPT) on K 1,d -graphs parameterized by d. In this paper we show that Minimum Dominating Set is W[1]-hard parameterized by H and the size of the solution. Thus the existence of an FPT algorithm for a general graph H is very unlikely.…”

“…It is a natural question whether at least some of these algorithmic results can be extended to more general classes of intersection graphs. Chaplick et al [CTVZ17] and Chaplick and Zeman [CZ17] 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. Most of the algorithms developed on H-graphs in [CTVZ17,CZ17] run in time n f (H) , where n is the number of vertices in the input graph and f is some function.…”

Combinatorial Problems on H-graphs

“…We obtain new algorithmic results on clique, coloring, and isomorphism problem. In a companion paper [7], we studied recognition and dominating set problems on H-graphs. We begin with related graph classes.…”

Intersection Graphs of Non-crossing Paths

“…Finally, an alternative view of host domains has been considered quite recently through the notion of H-graphs [3,5,6,12], i.e., for a fixed graph H, a graph G is an H-graph when it is an intersection graph of connected subgraphs of a subdivision of H. Here, interval graphs are the K 2 -graphs and circular arc graphs are the K 3 -graphs. While there is a natural notion of proper H-graphs [3] and these do indeed restrict H-graphs (for every H), we believe that the more restrictive non-crossing H-graphs might have a nicer structure and lead to easier (and more efficient) algorithms.…”