Recognizing Proper Tree-Graphs

RWTH Publications (RWTH Aachen)

Abstract

We investigate the parameterized complexity of the recognition problem for the proper H-graphs. The H-graphs are the intersection graphs of connected subgraphs of a subdivision of a multigraph H, and the properness means that the containment relationship between the representations of the vertices is forbidden. The class of H-graphs was introduced as a natural (parameterized) generalization of interval and circular-arc graphs by Biró, Hujter, and Tuza in 1992, and the proper H-graphs were introduced by Chaplick et al. in WADS 2019 as a generalization of proper interval and circular-arc graphs. For these graph classes, H may be seen as a structural parameter reflecting the distance of a graph to a (proper) interval graph, and as such gained attention as a structural parameter in the design of efficient algorithms. We show the following results. - For a tree T with t nodes, it can be decided in 2^{𝒪(t² log t)} ⋅ n³ time, whether an n-vertex graph G is a proper T-graph. For yes-instances, our algorithm outputs a proper T-representation. This proves that the recognition problem for proper H-graphs, where H required to be a tree, is fixed-parameter tractable when parameterized by the size of T. Previously only NP-completeness was known. - Contrasting to the first result, we prove that if H is not constrained to be a tree, then the recognition problem becomes much harder. Namely, we show that there is a multigraph H with 4 vertices and 5 edges such that it is NP-complete to decide whether G is a proper H-graph.

Authors 4

1. Steven Chaplick

   Maastricht University

   Affiliation as printed

   Maastricht University, The Netherlands

   Maastricht University

2. Petr A. Golovach

   University of Bergen

   Affiliation as printed

   Department of Informatics, University of Bergen, Norway

   University Of Bergen;

3. Tim A. Hartmann Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen, Germany

   RWTH Aachen University

4. Dušan Knop

   Czech Technical University in Prague

   Affiliation as printed

   Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic

   Czech Technical Univ. in Prague