A set $S\subseteq V$ of a graph $G=(V,E)$ is a dominating set if each vertex has a neighbor in $S$ or belongs to $S$. Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\geq 1$, if $G$ has a dominating set of size at most $k$. It is well known that this problem is $\mathsf{NP}$-complete even for claw-free graphs. We give a complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter $d$. We show that the problem is $\mathsf{NP}$-complete for every fixed $d\ge 3$ and polynomial time solvable for $d\le 2$. To prove the case $d=2$, we show that Minimum Maximal Matching can be solved in polynomial time for $2K_2$-free graphs.

翻译：设图 $G=(V,E)$ 的顶点子集 $S\subseteq V$ 构成支配集，当且仅当每个顶点在 $S$ 中有邻居或属于 $S$。支配集问题要求判定：给定图 $G$ 和整数 $k\geq 1$，$G$ 是否存在规模不超过 $k$ 的支配集。众所周知，即使对于无爪图，该问题也是 $\mathsf{NP}$-完备的。本文针对直径为 $d$ 的无爪图类，给出了支配集问题的复杂度二分性结果。我们证明：对于每个固定的 $d\ge 3$，该问题为 $\mathsf{NP}$-完备；对于 $d\le 2$，则该问题可在多项式时间内求解。为证明 $d=2$ 的情形，我们证明了最小极大匹配问题可在多项式时间内求解于 $2K_2$-自由图。