Theoretical Studies of the k-Strong Roman Domination Problem

TL;DR

This paper introduces and analyzes the k-strong Roman domination problem, a new generalization of Roman domination, providing theoretical results, bounds, and connections to other domination problems across various graph classes.

Contribution

It is the first theoretical study of the k-strong Roman domination problem, establishing bounds, exact values, and relations to other domination problems for multiple graph classes.

Findings

Provides bounds and exact values for specific graph classes.

Establishes connections between k-strong Roman domination and other domination problems.

Offers an attainable lower bound for cubic graphs.

Abstract

The concept of Roman domination has been a subject of intrigue for more than two decades with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the k-strong Roman domination. In this variation, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The objective is to find a feasible mapping that assigns an (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while always retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.