On the k -Component Independence Number of a Tree

Abstract

Let $G$ be a graph and $k\ge 1$ be an integer. A subset $S$ of vertices in a graph $G$ is called a $k$ -component independent set of $G$ if each component of $G\left[S\right]$ has order at most $k$ . The $k$ -component independence number, denoted by ${\alpha }_c^k\left(G\right)$ , is the maximum order of a vertex subset that induces a subgraph with maximum component order at most $k$ . We prove that if a tree $T$ is of order $n$ , then ${\alpha }_k\left(T\right)\ge \left(k/\left(k+1\right)\right)n$ . The bound is sharp. In addition, we give a linear-time algorithm for finding a maximum $k$ -component independent set of a tree.