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- W3199062009 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 $checkkappa(w,v)$ 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 $kappa(n,v) = lfloorlog_{v+1} (n v + 1)rfloor$, and that $lfloorlog_{v+1} wrfloor + 1 le checkkappa(w,v) le lfloorlog_{v+1} wrfloor + 3$. 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 $v ge 3$." @default.
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- W3199062009 date "2021-09-23" @default.
- W3199062009 modified "2023-09-27" @default.
- W3199062009 title "Partitioning an interval graph into subgraphs with small claws" @default.
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