Abstract

Cover-Incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of incomparability graph and the cover graph of some poset. We give a new characterization of Ptolemaic C-I graphs and prove that each C-I graph contains a Ptolemaic C-I graph as a spanning subgraph. This result is used to present several necessary conditions for a graph G being planar C-I graph. We present a hierarchy of subfamilies of chordal graphs in the class of C-I graphs and prove that many different graph families coincide in the class of C-I graphs.

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