Order DOI 10.1007/s11083-008-9097-1 Cover-Incomparability Graphs of Posets Boštjan Brešar · Manoj Changat · Sandi Klavžar · Matjaž Kovše · Joseph Mathews · Antony Mathews Received: 8 November 2007 / Accepted: 11 September 2008 © Springer Science + Business Media B.V. 2008 Abstract Cover-incomparability graphs (C-I graphs, for short) are introduced, whose edge-set is the union of edge-sets of the incomparability and the cover graph of a poset. Posets whose C-I graphs are chordal (resp. distance-hereditary, Ptolemaic) are char- acterized in terms of forbidden isometric subposets, and a general approach for studying C-I graphs is proposed. Several open problems are also stated. B. Brešar · M. Kovše Department of Mathematics and Computer Science, FNM, University of Maribor, Koroška 160, 2000 Maribor, Slovenia B. Brešar e-mail: bostjan.bresar@uni-mb.si M. Kovše e-mail: matjaz.kovse@gmail.com B. Brešar · S. Klavžar · M. Kovše Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia M. Changat · A. Mathews Department of Futures Studies, University of Kerala, Trivandrum, 695034, India M. Changat e-mail: mchangat@gmail.com A. Mathews e-mail: sonykandans@yahoo.co.in S. Klavžar (B ) Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia e-mail: sandi.klavzar@fmf.uni-lj.si S. Klavžar University of Maribor, Maribor, Slovenia J. Mathews Chingamparampil, Vazhapally, Changanacherry, 686103 Kerala, India e-mail: jose_chingam@yahoo.co.in