Partitioning an interval graph into subgraphs with small claws

TL;DR

This paper investigates how to partition interval graphs into subgraphs with bounded claw number, providing exact formulas, bounds, and approximation algorithms for the minimum number of such subgraphs needed.

Contribution

It derives exact formulas for the minimum number of induced subgraphs with bounded claw number needed to partition interval graphs, and presents approximation algorithms with proven ratios.

Findings

Exact formula for (n,v) as (n,v) = log_{v+1}(n v + 1).

Bounds on (w,v): log_{v+1} w + 1 (w,v)  log_{v+1} w + 3.

Approximation algorithms with ratio 3 for v  2, and ratio 2 for v  3.

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 $v \ge 3$.