Rainbow domination in the lexicographic product of graphs

TL;DR

This paper investigates the 2-rainbow domination number in the lexicographic product of graphs, establishing bounds and exact values based on domination invariants, with special consideration for specific cases.

Contribution

It provides sharp bounds and exact formulas for the 2-rainbow domination number of lexicographic graph products, extending understanding of domination parameters in graph theory.

Findings

Established sharp lower and upper bounds for the 2-rainbow domination number.

Derived exact values for the 2-rainbow domination number in most cases.

Identified exceptional cases where the bounds do not apply.

Abstract

Let k be a positive integer and let f be a map from V(G) to the set of all subsets of {1,2,3,...,k}. The function f is called a k-rainbow dominating function of G provided that whenever u is a vertex of G such that f(u) is the empty set, then for each integer r in {1,2,3,...,k} there is a neighbor x of u such that f(x) contains r. The k-rainbow domination number of G is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function of G. The k-rainbow domination number of G is the ordinary domination number of the Cartesian product of G and a complete graph of order k. We focus on the 2-rainbow domination number of the lexicographic product of graphs and prove sharp lower and upper bounds for this number. In fact, we prove the exact value of the 2-rainbow domination number of the lexicographic product of G with H in terms of domination invariants of G, except for the case when H has 2-rainbow domination number 3 and there is a minimum 2-rainbow dominating function of H such that some vertex in H is assigned the label {1,2}.