Graph Operations and Neighborhood Polynomials

TL;DR

This paper studies the neighborhood polynomial of graphs, providing formulas, algorithms, and complexity results for its computation under various graph operations.

Contribution

It introduces an explicit formula for neighborhood polynomials after vertex attachment and shows polynomial-time computability in k-degenerate graphs.

Findings

Explicit formula for neighborhood polynomial after vertex attachment

Recursive algorithm for calculating neighborhood polynomial

Polynomial-time computability in k-degenerate graphs

Abstract

The neighborhood polynomial of graph $G$ is the generating function for the number of vertex subsets of $G$ of which the vertices have a common neighbor in $G$. In this paper, we investigate the behavior of this polynomial under several graph operations. Specifically, we provide an explicit formula for the neighborhood polynomial of the graph obtained from a given graph $G$ by vertex attachment. We use this result to propose a recursive algorithm for the calculation of the neighborhood polynomial. Finally, we prove that the neighborhood polynomial can be found in polynomial-time in the class of $k$-degenerate graphs.