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

TL;DR

This paper characterizes certain chordal graphs and cographs that are Cover-Incomparability graphs and provides linear-time algorithms for recognizing them.

Contribution

It offers a structural characterization of chordal graphs and cographs that are C-I graphs and develops efficient recognition algorithms.

Findings

Chordal graphs with at most two independent simplicial vertices are C-I graphs.

Similar characterization applies to cographs.

Linear-time recognition algorithms are derived for these classes.

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 (Maxov\'{a} 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.