Recognition of chordal graphs and cographs which are Cover-Incomparability graphs

TL;DR

It is proved that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs, and a similar result is obtained for cographs.

Abstract

Cover-Incomparability graphs (C-I graphs) are an interesting class of graphs from posets. A C-I graph is a graph from a poset $P=(V,\le)$ with vertex set $V$, and the edge-set is the union of edge sets of the cover graph and the incomparability graph of the poset. The recognition of the C-I graphs is known to be NP-complete (Maxová et al., Order 26(3), 229--236(2009)). In this paper, we prove that chordal graphs having at most two independent simplicial vertices are exactly the chordal graphs which are also C-I graphs. A similar result is obtained for cographs as well. Using the structural results of these graphs, we derive linear time recognition algorithms for chordal graphs and cographs which are C-I graphs.

Figures (7)

• Figure 1: (a) A poset $P$, (b) incomparability graph of the poset $P$, and (c) cover-incomparability graph of the poset $P$.
• Figure 2: A chordal graph and two of its clique trees
• Figure 3: General structure of a chordal C-I cograph $G$
• Figure 4: General structure of a C-I cograph $G$
• Figure 5: Claw

Theorems & Definitions (34)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Remark 1
• Lemma 4
• proof
• Corollary 1
• Remark 2
• Lemma 5