The Forest Number of (n,m)-Graphs

Abstract

Let G = (V,E) be a graph and F ⊆ V. Then F is called an induced forest of G if G[F] is acyclic. The forest number, denoted by f(G), of G is defined by

$$f(G):=\max\{ |F| : F \;{\rm is \;an \;induced \;forest\; of} \; G\}.$$

We proved that if G runs over the set of all graphs of order n and size m, then the values f(G) completely cover a line segment $\left[x, y \right]$ of positive integers. Let $\mathcal{G}(n,m)$ be the set of all graphs of order n and size m and $\mathcal{C\!G}(n,m)$ be the subset of $\mathcal{G}(n,m)$ consisting of all connected graphs. We are able to obtain the extremal results for the forest number in the class $\mathcal{G}(n, m)$ and $\mathcal{C\!G}(n, m)$.