Minimal unavoidable sets of cycles in plane graphs

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.