Neighborhood and Domination Polynomials of Graphs

Abstract

The neighborhood complex of a graph is the family of subsets of open neighborhoods of its vertices. The neighborhood polynomial is the ordinary generating function for the number of sets of the neighborhood complex with respect to their cardinality. This paper provides a new representation of the neighborhood polynomial as a sum over complete bipartite subgraphs of a graph. Using the close relation between the domination polynomial and the neighborhood polynomial of a graph, we can also give a new presentation of the domination polynomial. Finally we show that finding the number of certain double cliques of a graph is sufficient to determine the number of dominating sets of a graph.