Opuscula Mathematica
Opuscula Math. 38, no. 6 (2018), 859-870

Minimal unavoidable sets of cycles in plane graphs

Tomáš Madaras
Martina Tamášová
Abstract. A set \(S\) of cycles is minimal unavoidable in a graph family \(\cal{G}\) if each graph \(G \in \cal{G}\) contains a cycle from \(S\) and, for each proper subset \(S^{\prime}\subset S\), there exists an infinite subfamily \(\cal{G}^{\prime}\subseteq\cal{G}\) such that no graph from \(\cal{G}^{\prime}\) contains a cycle from \(S^{\prime}\). In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles.
Keywords: plane graph, polyhedral graph, set of cycles.
Mathematics Subject Classification: 05C10.
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About the authors ▼
Tomáš Madaras
Pavol Jozef Šafárik University, Faculty of Science, Institute of Mathematics, Jesenná 5, 04001 Košice, Slovakia
Martina Tamášová
Pavol Jozef Šafárik University, Faculty of Science, Institute of Mathematics, Jesenná 5, 04001 Košice, Slovakia
About the article ▼
Communicated by Ingo Schiermeyer.
Received: 2017-11-28.
Revised: 2018-06-08.
Accepted: 2018-06-08.
Published online: 2018-07-05.
Cite this article as ▼
Tomáš Madaras, Martina Tamášová, Minimal unavoidable sets of cycles in plane graphs, Opuscula Math. 38, no. 6 (2018), 859-870, https://doi.org/10.7494/OpMath.2018.38.6.859.
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