The domination polynomial of powers of paths and cycles

TL;DR

This paper extends the proof that the domination sequence is unimodal to all powers of paths and cycles, providing a broader understanding of domination properties in graph theory.

Contribution

It generalizes the unimodality of domination sequences from paths and cycles to their arbitrary powers, expanding previous results.

Findings

Confirmed unimodality for powers of paths and cycles

Extended previous conjecture verification to broader graph classes

Provided new insights into domination sequence behavior

Abstract

A dominating set in a graph is a set of vertices with the property that every vertex in the graph is either in the set or adjacent to something in the set. The domination sequence of the graph is the sequence whose $k$th term is the number of dominating sets of size $k$. Alikhani and Peng have conjectured that the domination sequence of every graph is unimodal. Beaton and Brown verified this conjecture for paths and cycles. Here we extend this to arbitrary powers of paths and cycles.