Recurrence relations and splitting formulas for the domination polynomial

TL;DR

This paper develops recurrence relations and splitting formulas for the domination polynomial of graphs, enabling more efficient computation and analysis of dominating sets in various graph structures.

Contribution

It introduces new linear recurrence relations and splitting formulas for the domination polynomial, expanding the tools for analyzing dominating sets in graphs.

Findings

Derived linear recurrence relations for D(G,x) for arbitrary graphs.

Established splitting formulas based on articulation vertices.

Provided formulas applicable to various special graph cases.

Abstract

The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G,x) based on articulation vertices, and more generally, on splitting sets of vertices.