On Mixed Domination in Generalized Petersen Graphs

TL;DR

This paper investigates the mixed domination number in Petersen graphs, providing explicit constructions for optimal sets in certain cases and establishing new upper bounds for the class.

Contribution

It introduces a method for constructing optimal mixed dominating sets in Petersen graphs for specific parameters and offers new bounds for the general case.

Findings

Explicit constructions for $ ext{P}(n, 1)$ and $ ext{P}(n, 2)$

New upper bounds for mixed domination numbers in Petersen graphs

Enhanced understanding of mixed domination in generalized Petersen graphs

Abstract

Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination number $\gamma_{md}(G)$ of the graph is the size of the smallest mixed dominating set of $G$. We present an explicit method for constructing optimal mixed dominating sets in Petersen graphs $P(n, k)$ for $k \in \{1, 2\}$. Our method also provides a new upper bound for other Petersen graphs.