Generalized 3-Rainbow Domination in Graphs and Honeycomb System (HC(n))

Abstract

In this paper, we introduce the concept of the generalized 3 -rainbow dominating function of a graph G . This function assigns an arbitrary subset of three colors to each vertex of the graph with the condition that every vertex (including its neighbors) must have access to all three colors within its closed neighborhood. The minimum sum of assigned colors over all vertices of G is defined as the g 3 -rainbow domination number, denoted by γ g 3 r . We present a linear-time algorithm to determine a minimum generalized 3-rainbow dominating set for several graph classes: trees, paths ( P n ) , cycles ( C n ) , stars ( K 1 , n ) , generalized Petersen graphs ( G P ( n , 2 ) , GP ( n , 3 ) ) , and honeycomb networks ( H C ( n ) ) .