Recognizing Proper Tree-Graphs

TL;DR

This paper studies the complexity of recognizing proper H-graphs, showing fixed-parameter tractability when H is a tree and NP-completeness for certain multigraphs, thus clarifying the computational boundaries.

Contribution

It establishes fixed-parameter tractability for recognizing proper H-graphs when H is a tree and NP-completeness for some multigraphs, advancing understanding of their computational complexity.

Findings

Recognition is fixed-parameter tractable for proper T-graphs when T is a tree.

Recognition becomes NP-complete for certain multigraphs with 4 vertices and 5 edges.

Provides algorithms and complexity results for proper H-graph recognition.

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 Bir\'o, 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^{\mathcal{O}(t^2 \log t)} \cdot n^3 $ 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.