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Which \(k\)-trees are cover-incomparability graphs? (English) Zbl 1284.05063
Summary: In this paper we deal with cover-incomparability graphs of posets. It is known that the class of cover-incomparability graphs is not closed on induced subgraphs which makes the study of structural properties of these graphs difficult. In this paper we introduce the notion of \(s\)-subgraph which enables us to define forbidden \(s\)-subgraphs (i.e. graphs that cannot appear as \(s\)-subgraphs of any cover-incomparability graph). We show that the family of minimal forbidden \(s\)-subgraphs is infinite even for cover-incomparability unit-interval graphs. Using the notion of \(s\)-subgraph we also answer the question which \(k\)-trees are cover-incomparability graphs and which chordal graphs without \(K_4\) are cover-incomparability graphs.

MSC:
05C05 Trees
06A06 Partial orders, general
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