On partitioning interval graphs into proper interval subgraphs and related problems

Abstract

In this paper, we establish that any interval graph (resp. circular-arc graph) with n vertices admits a partition into at most ⌈log 3 n⌉ (resp. ⌈log 3 n⌉ + 1) proper interval subgraphs, for n > 1. The proof is constructive and provides an efficient algorithm to compute such a partition. On the other hand, this bound is shown to be asymptotically sharp for an infinite family of interval graphs. In addition, some results are derived for related problems.

Partitioning an interval graph into subgraphs with small claws

“…Extending the two previous results [5,9], we determine exact values of µ(k, v) and κ(n, v) for all v ≥ 1:…”

“…But this cannot hold, since we proved in Theorem 3 that κ(4, 2) = κ(5, 2) = κ(6, 2) = 3; see our Lemmas 5, 6, and 7. We refer to Figure 7 for a counterexample that identifies a flaw in the proof of [9,Lemma 2.3]. This flaw also affects several other results, including an upper bound on κ(n, 2) [9, Proposition 2.7].…”

“…Thus κ(n, 1) = ⌊log 2 (n+1)⌋. Gardi [9,Proposition 2.2] showed that for all n ≥ 1, there exists an interval graph with n vertices that admits no vertex partition into less than ⌊log 3 (2n + 1)⌋ proper interval subgraphs. Thus κ(n, 2) ≥ ⌊log 3 (2n + 1)⌋.…”