Neighborhood inclusion posets and tree representations for chordal and dually chordal graphs

Abstract

This paper is motivated by the following problem: given the family ofclique trees of a chordal graph G, determine whether it is also the family of compatible trees of some dually chordal graph H. A relationship isestablished between this problem and neighborhood inclusion posets, i.e.,the posets where the vertex-neighborhoods of graphs are ordered by inclusion. The problem of determining whether a given poset is the neighborhoodinclusion poset of some graph is proved to be NP-complete. Finally, a polynomial time reduction is found from Neighborhood Poset Recognition to onevariant of the initially stated problem, thus proving that the latter is alsoNP-complete.