Abstract
This paper is motivated by the following problem: given the family of clique trees of a chordal graph G, determine whether it is also the family of compatible trees of some dually chordal graph H. A relationship is established 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 neighborhood inclusion poset of some graph is proved to be NP-complete. Finally, a polynomial time reduction is found from Neighborhood Poset Recognition to one variant of the initially stated problem, thus proving that the latter is also NP-complete.
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