Variants of the Domination Number for Flower Snarks

TL;DR

This paper investigates various domination parameters in flower snarks, a class of 3-regular graphs, providing new exact values for multiple domination variants beyond the classical domination number.

Contribution

It extends existing research by determining several new domination parameters for flower snarks, including independent, 2-, total, connected, upper, secure, and weak Roman domination numbers.

Findings

Independent domination number determined for flower snarks.

2-domination and total domination numbers calculated.

Connected and secure domination numbers established.

Abstract

We consider the flower snarks, a widely studied infinite family of 3--regular graphs. For the Flower snark $J_n$ on $4n$ vertices, it is trivial to show that the domination number of $J_n$ is equal to $n$. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.