The Neighborhood Polynomial of Chordal Graphs
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The neighborhood polynomial of a graph G is the generating function of subsets of vertices in G that have a common neighbor. In this paper we study the neighborhood polynomial and the complexity of its computation for chordal graphs. We will show that it is -hard to compute the neighborhood polynomial on general chordal graphs. Furthermore we will introduce a parameter for chordal graphs called anchor width and an algorithm to compute the neighborhood polynomial which runs in polynomial time if the anchor width is polynomially bounded. Finally we will show that we can bound the anchor width for chordal comparability graphs and chordal graphs with bounded leafage. The leafage of a chordal graphs is the minimum number of leaves in the host tree of a subtree representation. In particular, interval graphs have leafage at most 2. This shows that the anchor width of interval graphs is at most quadratic.
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