The decycling number of a line graph

Abstract

The decycling number of a graph G, denoted by $\nabla \left(G\right)$, is the number of vertices in a minimum decycling set of G. The line graph of G is denoted by $L\left(G\right)$. In this paper we show that $\nabla \left(L\right(G\left)\right)=\beta \left(G\right)+\mu \left(G\right)-1$, where $\beta \left(G\right)$ is the cycle rank of G and $\mu \left(G\right)$ is the path partition number of G. In particular, $\nabla \left(L\right(G\left)\right)=\beta \left(G\right)$ if and only if G has a Hamilton path, and $\nabla \left(L\right(G\left)\right)\le \frac{2n}{3}-\frac{2}{3}$ if G is a cubic graph with n vertices, where $n\ge 10$. If $L\left(G\right)$ is a planar graph, then we prove that $\nabla \left(L\right(G\left)\right)\le \frac{\left|V\right(L\left(G\right)\left)\right|}{2}$, which means that the conjecture proposed by Albertson and Berman in 1979 that the decycling number of any planar graph H is at most $\frac{\left|V\right(H\left)\right|}{2}$ holds for a planar line graph. If G is a connected graph of order n which is 2-cell embedded on the orientable surface ${\sum }_g$ (or the non-orientable surface ${\sum }_k^{\prime }$), then we show that $\nabla \left(L\right(G\left)\right)\le 2n+l-7+6g$ (or $2n+l-7+3k$) if G has a spanning tree with l leaves. Our bounds are tight for $l=2$.