Recognizing Graph Search Trees☆,☆☆

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Abstract

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show NP-completeness. We complement our results by providing linear time algorithms for these searches on split graphs.

Keywords

Search tree recognition
LBFS
LDFS
MNS
MCS

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A full version of this paper can be found in [Beisegel, J., C. Denkert, E. Köhler, M. Krnc, N. Pivač, R. Scheffler and M. Strehler, Recognizing Graph Search Trees (2018), preprint on arXiv: https://arxiv.org/abs/1811.09249]. The proofs of statements marked with (⋆) can be found there.

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The work of this paper was done in the framework of a bilateral project between Brandenburg University of Technology and University of Primorska, financed by German Academic Exchange Service and the Slovenian Research Agency (BI-DE/17-19-18).

1

The author gratefully acknowledges the European Commission for funding the InnoRenew CoE project (Grant Agreement #739574) under the Horizon2020 Widespread-Teaming program and the Republic of Slovenia (Investment funding of the Republic of Slovenia and the European Union of the European Regional Development Fund).

2

Funded in part by the Slovenian Research Agency (research programs P1-0285 and J1-9110 and Young Researchers Grant).

© 2019 The Author(s). Published by Elsevier B.V.