Recognising the overlap graphs of subtrees of restricted trees is hard

TL;DR

This paper investigates the computational complexity of recognizing various subclasses of overlap graphs of subtrees in trees with restricted structures, proving NP-completeness for several cases and polynomial recognition for others.

Contribution

It establishes NP-completeness results for recognizing subclasses of SOGs in restricted trees, expanding understanding of their computational complexity.

Findings

Recognition problems are NP-complete for certain subclasses.

Recognition remains polynomial for other subclasses, such as circle graphs.

The paper clarifies the complexity landscape of SOG recognition in restricted trees.

Abstract

The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs in open. The complexities of recognising many subclasses of SOGs are known. We consider several subclasses of SOGs by restricting the underlying tree. For a fixed integer $k \geq 3$, we consider: \begin{my_itemize} \item The overlap graphs of subtrees in a tree where that tree has $k$ leaves \item The overlap graphs of subtrees in trees that can be derived from a given input tree by subdivision and have at least 3 leaves \item The overlap and intersection graphs of paths in a tree where that tree has maximum degree $k$ \end{my_itemize} We show that the recognition problems of these classes are NP-complete. For all other parameters we get circle graphs, well known to be polynomially recognizable.