Ptolemaic and Chordal Cover-Incomparability Graphs

Cover-incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of the incomparability graph and the cover graph of some poset. C-I graphs captured attention as an interesting class of graphs from posets. It is known that the recognition of C-I graphs is NP-complete (Maxová et al., Order 26 (3), 229–236, 2009 ). Hence, the problem of finding a particular graph family of C-I graphs whose recognition complexity is polynomial is interesting. We present a new forbidden subgraph characterization of Ptolemaic C-I graphs and a linear time algorithm for its recognition. The characterization of chordal C-I graph is an unsolved problem in this area for quite some time. In this paper, we characterize the family of chordal C-I graphs.