Cycle Domination, Independence and Irredundance in graphs

TL;DR

This paper introduces new graph concepts related to cycle independence, domination, and irredundance, establishing their properties and analogs to existing domination inequalities.

Contribution

The paper defines cycle and odd-cycle variants of independence, domination, and irredundance, and develops their theoretical relationships and inequalities.

Findings

Introduces cycle independent and odd-cycle independent sets.

Defines cycle dominating and odd-cycle dominating sets.

Establishes analogs to domination inequality chains for these concepts.

Abstract

A set $S$ of vertices in a graph $G = (V, E)$ is called {\em cycle independent} if the induced subgraph $\langle S\rangle$ is acyclic, and called {\em odd-cycle indepdendet} if $\langle S\rangle$ is bipartite. A set $S$ is {\em cycle dominating} (resp. {\em odd-cycle dominating}) if for every vertex $u \in V \setminus S$ there exists a vertex $v \in S$ such that $u$ and $v$ are contained in a (resp. odd cycle) cycle in $\langle S \setminus \{u\}\rangle$. A set $S$ is {\em cycle irredundant} (resp. odd-cycle irredundant) if for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $u$ and $v$ are in a (resp. odd cycle) cycle of $\langle S \setminus \{u\}\rangle$, but $u$ is not in a cycle of $\langle S \cup \{u\} \setminus \{v\}\rangle$. In this paper we present these new concepts, which relate in a natural way to independence, domination and irredundance in graphs. In particular, we construct analogs to the domination inequality chain for these new concepts.