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Cover-incomparability graphs and chordal graphs. (English) Zbl 1208.05105
Summary: 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. Maxová, P. Pavlíkova and A. Turzík, On the complexity of cover-incomparability graphs of posets, Order 26, No. 3, 229-236 (2009; Zbl 1172.05049)], 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.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
poset; underlying graph; chordal graph; Split graph; block graph
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##### References:
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