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.

翻译：我们研究周围多圆形图及其计算相邻多圆形图的复杂度。 相邻多圆形图是其有共同邻居的脊椎子子子子子的生成功能。 我们引入了一个称为锚宽度的圆形图子子子子子的参数, 并引入了一个算法来计算以多圆形时间运行的周边多圆形图, 如果锚宽度是多圆形的, 则以多圆形宽度计算。 锚宽度是作为常见邻居的圆形不同子组的最大值。 此外, 我们研究圆形图的锚宽度和一些子类的锚宽度, 如带边框叶的圆形比较图和圆形图子。 圆形图的叶值是子树主树代表面的最小叶数。 我们显示, 圆形图的锚宽度在$n ⁇ ell} $\ell} 美元中表示叶值。 这显示, 对于一些计算相邻多圆形图的子类系, 在聚圆形时是可能的。