On $k$-rainbow domination in middle graphs

TL;DR

This paper introduces the concept of $k$-rainbow domination in middle graphs, establishing bounds, exact values for specific graph classes, and exploring related parameters like the domatic number.

Contribution

It initiates the study of $k$-rainbow domination in middle graphs, providing bounds, exact values for certain classes, and analyzing related parameters such as the domatic number.

Findings

Determined the middle 3-rainbow domination number for some graph classes.

Provided bounds for the middle 3-rainbow domination number of trees based on matching number.

Calculated the 3-rainbow domatic number for middle graphs of paths and cycles.

Abstract

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.