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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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##### References:
 [1] Brešar, B.; Changat, M.; Gologranc, T.; Mathews, J.; Mathews, A., Cover-incomparability graphs and chordal graphs, Discrete Appl. Math., 158, 1752-1759, (2010) · Zbl 1208.05105 [2] Brešar, B.; Changat, M.; Klavžar, S.; Kovše, M.; Mathews, J.; Mathews, A., Cover-incomparability graphs of posets, Order, 25, 335-347, (2008) · Zbl 1219.06004 [3] Diestel, R., (Graph Theory, Graduate Texts in Mathematics, (2005), Springer-Verlag) · Zbl 1074.05001 [4] Jamison-Waldner, R. E., Convexity and block graphs, Congr. Numer., 33, 129-142, (1981) · Zbl 0495.05056 [5] Maxová, J.; Pavlíková, P.; Turzík, D., On the complexity of cover-incomparability graphs of posets, Order, 26, 229-236, (2009) · Zbl 1172.05049 [6] Maxová, J.; Turzík, D., Which distance-hereditary graphs are cover-incomparability graphs?, Discrete Appl. Math., 161, 13-14, 2095-2100, (2013) · Zbl 1286.05039
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