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 κ(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 κ(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 κ(n, v) = ⌊log v+1 (nv + 1)⌋, and that ⌊log v+1 w⌋ + 1 ≤ κ(w, v) ≤ ⌊log v+1 w⌋ + 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 ≤ v ≤ 2, and 2 when v ≥ 3.

Vertebrate interval graphs

“…Thus for k ≥ 2, k-PARTITION(1) in interval graphs admits an algorithm running in O(k • n 2k+1 ) time. It is unknown [9] whether there exist k ≥ 2 and v ≥ 2 such that k-PARTITION(v) in interval graphs admits an polynomial-time algorithm.…”