Cover-incomparability graphs and chordal graphs

Authors:
Boštjan Brešar;Manoj Changat;Tanja Gologranc;Joseph Mathews;Antony Mathews
Affiliations:
Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia;C-GRaF, Department of Futures Studies, University of Kerala, Trivandrum-695034, India;Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia;C-GRaF, Department of Futures Studies, University of Kerala, Trivandrum-695034, India;Department of Mathematics, St.Berchmans College, Changanassery - 686 101, Kerala, India
Venue:
Discrete Applied Mathematics
Year:
2010

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Abstract

The problem of recognizing cover-incomparability graphs (i.e. the graphs obtained from posets as the edge-union of their covering and incomparability graph) was shown to be NP-complete in general [J. Maxova, P. Pavlikova, A. Turzik, On the complexity of cover-incomparability graphs of posets, Order 26 (2009) 229-236], while it is for instance clearly polynomial within trees. In this paper we concentrate on (classes of) chordal graphs, and show that any cover-incomparability graph that is a chordal graph is an interval graph. We characterize the posets whose cover-incomparability graph is a block graph, and a split graph, respectively, and also characterize the cover-incomparability graphs among block and split graphs, respectively. The latter characterizations yield linear time algorithms for the recognition of block and split graphs, respectively, that are cover-incomparability graphs.