Partitioning an interval graph into subgraphs with small claws

Abstract

The claw number of a graph $G$ is the largest number $v$ such that $K_{1,v}$ is an induced subgraph of $G$. Interval graphs with claw number at most $v$ are cluster graphs when $v = 1$, and are proper interval graphs when $v = 2$. Let $\kappa(n,v)$ be the smallest number $k$ such that every interval graph with $n$ vertices admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. Let $\check\kappa(w,v)$ be the smallest number $k$ such that every interval graph with claw number $w$ admits a vertex partition into $k$ induced subgraphs with claw number at most $v$. We show that $\kappa(n,v) = \lfloor\log_{v+1} (n v + 1)\rfloor$, and that $\lfloor\log_{v+1} w\rfloor + 1 \le \check\kappa(w,v) \le \lfloor\log_{v+1} w\rfloor + 3$. Besides the combinatorial bounds, we also present a simple approximation algorithm for partitioning an interval graph into the minimum number of induced subgraphs with claw number at most $v$, with approximation ratio $3$ when $1 \le v \le 2$, and $2$ when $v \ge 3$.