THE TREE-ACHIEVING SET AND NON-SEPARATING INDEPENDENT SET PROBLEM OF SUBCUBIC GRAPHS

The decycling number del(G) (respectively, tree-achieving number del(T)(G)) of a graph G is the smallest number of vertices whose deletion yields a forest (respectively, tree). Obviously, del(T)(G)>=del(G) for all graphs. A graph is cubic (respectively, subcubic ) if every vertex has degree three (respectively, at most three). A non-separating independent set is an independent vertices set whose deletion yields a connected subgraph. The nsis number Z(G) is the maximum cardinality of a non-separating independent set. In this article, we present a sufficient and necessary condition for del(T)(G)=del(G) in cubic graphs. That is del(T)(G)=del(G) if and only if there exists a Xuong-tree [J.L. Gross and R.G. Rieper, Local extrema in genus-stratified graphs, J. Graph Theory 15 (1991) 153-171] T-X of G such that every odd component of G-E(T-X) contains at least three edges. Further, we give a formula for Z(G) in subcubic graphs: there is a Xuong-tree T-X of G such that alpha(1)(T-X)=Z(G), where alpha(1)(T-X) is the independence number of the subgraph of G induced by leaves of T-X.