Decycling cubic graphs

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 1 ( n + 2 )/ 4 1 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 1 ( n + 2 )/ 4 1 . In addition, they characterised the structure of minimum decycling sets and their complements. If n - 2 ( mod 4 ) , then G has a decycling set which is independent and its complement induces a tree. If n - 0 ( mod 4 ) , 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. (c) 2024 Elsevier B.V. All rights reserved.