# Recognising the overlap graphs of subtrees of restricted trees is hard

and Pergel, M. (2019) Recognising the overlap graphs of subtrees of restricted trees is hard. Acta Mathematica Universitatis Comenianae, 88(3), pp. 651-658.

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## Abstract

The overlap graphs of subtrees in a tree (SOGs) generalise many other graphs classes with set representation characterisations. The complexity of recognising SOGs is open. The complexities of recognising many subclasses of SOGs are known. Weconsider 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.

Item Type: Articles Published Yes Enright, Dr Jessica Enright, J., and Pergel, M. College of Science and Engineering > School of Computing Science Acta Mathematica Universitatis Comenianae Comenius University, Faculty of Mathematics, Physics and Informatics 0862-9544 Copyright © 2019 Acta Mathematica Universitatis Comenianae First published in Acta Mathematica Universitatis Comenianae 88(3): 651-658 Reproduced with the permission of the Publisher

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