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Which distance-hereditary graphs are cover-incomparability graphs? (English) Zbl 1286.05039
Summary: In this paper we deal with cover-incomparability graphs of posets, or briefly C-I graphs. These are graphs derived from posets as the edge-union of their cover graph and their incomparability graph. We answer two recently posed open questions. Which distance-hereditary graphs are C-I graphs? Which Ptolemaic (i.e. chordal distance-hereditary) graphs are C-I graphs? It follows that C-I graphs can be recognized efficiently in the class of all distance-hereditary graph whereas recognizing C-I graphs in general is known to be NP-complete.

MSC:
05C12 Distance in graphs
05C75 Structural characterization of families of graphs
06A06 Partial orders, general
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