On Signed Rainbow Domination in Graphs

This article introduces the concept of k-signed rainbow domination (k-SRD), a novel extension of signed domination in graphs, with significant theoretical and practical implications. This study introduces the concept of k-signed rainbow domination, a novel extension of signed domination in graphs, with significant theoretical and practical implications. By investigating the properties and behavior of k-signed rainbow domination, this study aims to provide a deeper understanding of graph domination and its applications. We derive bounds for the k-signed rainbow domination number in terms of the degree and order of a graph, and characterize graphs with extremal values. Our work generalizes several results related to signed domination in graphs, obtaining the exact values of the k-signed rainbow domination number for specific classes of graphs. We also explored the conditions under which the k-signed rainbow domination number is zero or negative, and established a relationship between signed rainbow domination and rainbow domination in graphs. In addition, we provide bounds on the 2-SRD number for the generalized Petersen graph P(n,1) and discuss several real-world applications, demonstrating the practical relevance of our research. We present an algorithm for computing the k-SRD number, accompanied by a detailed complexity analysis. Overall, this study contributes significantly to the field of graph theory and domination, offering new insights and perspectives for future research.