Decycling cubic graphs

TL;DR

This paper advances understanding of decycling sets in cubic graphs by proving new structural guarantees and relaxing connectivity conditions, revealing a surprising link between decycling number and maximum genus.

Contribution

It proves that near-independent decycling sets with a tree complement always exist in certain cubic graphs and extends previous results to weaker connectivity assumptions.

Findings

Guarantees the existence of near-independent decycling sets with a tree complement.

Relates decycling number to the maximum genus of cubic graphs.

Weakens connectivity conditions needed for previous decycling results.

Abstract

A set of vertices of a graph $G$ is said to be decycling if its removal leaves an acyclic subgraph. The size of a smallest decycling set is the decycling number of $G$. Generally, at least $\lceil(n+2)/4\rceil$ vertices have to be removed in order to decycle a cubic graph on $n$ vertices. In 1979, Payan and Sakarovitch proved that the decycling number of a cyclically $4$-edge-connected cubic graph of order $n$ equals $\lceil (n+2)/4\rceil$. In addition, they characterised the structure of minimum decycling sets and their complements. If $n\equiv 2\pmod4$, then $G$ has a decycling set which is independent and its complement induces a tree. If $n\equiv 0\pmod4$, then one of two possibilities occurs: either $G$ has an independent decycling set whose complement induces a forest of two trees, or the decycling set is near-independent (which means that it induces a single edge) and its complement induces a tree. In this paper we strengthen the result of Payan and Sakarovitch by proving that the latter possibility (a near-independent set and a tree) can always be guaranteed. Moreover, we relax the assumption of cyclic $4$-edge-connectivity to a significantly weaker condition expressed through the canonical decomposition of 3-connected cubic graphs into cyclically $4$-edge-connected ones. Our methods substantially use a surprising and seemingly distant relationship between the decycling number and the maximum genus of a cubic graph.