Averaging $2$-Rainbow Domination and Roman Domination

TL;DR

This paper proves a conjecture relating the sum of 2-rainbow and Roman domination numbers to the order of a graph, characterizing extremal cases and confirming the inequality for graphs with minimum degree at least 2.

Contribution

The paper proves Fujita and Furuya's conjecture on the sum of domination numbers and characterizes all extremal graphs achieving equality.

Findings

Confirmed the conjecture for graphs with minimum degree at least 2.

Characterized all extremal graphs where the inequality holds with equality.

Extended understanding of domination parameters in graph theory.

Abstract

For a graph $G$, let $\gamma_{r2}(G)$ and $\gamma_R(G)$ denote the $2$-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) proved $\gamma_{r2}(G)+\gamma_R(G)\leq \frac{6}{4}n(G)$ for a connected graph $G$ of order $n(G)$ at least $3$. Furthermore, they conjectured $\gamma_{r2}(G)+\gamma_R(G)\leq \frac{4}{3}n(G)$ for a connected graph $G$ of minimum degree at least $2$ that is distinct from $C_5$. We characterize all extremal graphs for their inequality and prove their conjecture.