The Neighborhood Polynomial of Chordal Graphs

TL;DR

This paper introduces the concept of anchor width in chordal graphs and presents a polynomial-time algorithm for computing the neighborhood polynomial when this parameter is bounded, highlighting complexity differences among subclasses.

Contribution

It defines anchor width for chordal graphs and provides an algorithm to compute the neighborhood polynomial efficiently under certain conditions.

Findings

Anchor width is at most n^ell for chordal graphs with leafage l.

Computing the neighborhood polynomial is polynomial-time for subclasses with bounded anchor width.

It is NP-hard to compute the neighborhood polynomial for general chordal graphs.

Abstract

We study the neighborhood polynomial and the complexity of its computation for chordal graphs. The neighborhood polynomial of a graph is the generating function of subsets of its vertices that have a common neighbor. We 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. The anchor width is the maximal number of different sub-cliques of a clique which appear as a common neighborhood. Furthermore we study the anchor width for chordal graphs and some subclasses such as 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. We show that the anchor width of a chordal graph is at most $n^{\ell}$ where $\ell$ denotes the leafage. This shows that for some subclasses computing the neighborhood polynomial is possible in polynomial time while it is NP-hard for general chordal graphs.