Ptolemaic and Chordal Cover-Incomparability Graphs

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, ≤) 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.

Composition and product of cover-incomparability graphs

“…Several authors considered the problem of characterizing C-I graphs from well-known graph families: split graphs, block graphs [6], cographs [7], Ptolemaic graphs [12], distance-hereditary graphs [12], and k-trees [10]. C-I graphs are identified among the planar graphs and chordal graphs along with new characterizations of Ptolemaic graphs, respectively in [4] and [3]. It is also interesting to note that every C-I graph has a Ptolemaic C-I graph as a spanning subgraph [4].…”