The number of dominating $k$-sets of paths, cycles and wheels

TL;DR

This paper provides a simplified proof for recurrence relations of domination polynomials and counts of dominating k-sets in paths, cycles, and wheels, solving a problem posed in 2015 and extending to multiple graph types.

Contribution

It introduces a shorter proof for recurrence relations and explicitly determines the number of dominating k-sets for paths, cycles, and wheels.

Findings

Recurrence relation for domination polynomial of paths

Explicit formulas for dominating k-sets in paths, cycles, wheels

Solved a 2015 problem on dominating k-sets

Abstract

We give a shorter proof of the recurrence relation for the domination polynomial $\gamma (P_{n},t)$ and for the number $\gamma _{k}(P_{n})$ of dominating $k$-sets of the path with $n$ vertices. For every positive integers $n$ and $k,$ numbers $\gamma _{k}(P_{n})$ are determined solving a problem posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating $k$-sets $\gamma _{k}(C_{n})$ of cycles and $\gamma _{k}(W_{n})$ of wheels with $n$ vertices are computed.