Theoretical results for Perfect Location signed Roman domination problem

TL;DR

This paper introduces the Perfect Location Signed Roman Domination problem, combining multiple Roman domination concepts, and provides exact values and bounds for various graph classes, advancing theoretical understanding in graph theory.

Contribution

It defines the PLSRD problem, derives exact values for several graph classes, and establishes bounds for general and specific graph families, expanding the theoretical framework of Roman domination.

Findings

Exact PLSRD numbers for complete, bipartite, wheel, path, cycle, ladder, prism, and grid graphs.

Lower bounds for 3-regular graphs.

Upper bounds for flower snark graphs.

Abstract

The study of Roman domination has evolved to encompass a variety of challenging extensions, each contributing to the broader understanding of domination problems in graph theory. This paper explores the Perfect Location Signed Roman Domination (PLSRD) problem, a novel combination of the Perfect Roman, Locating Roman, and Signed Roman Domination paradigms. In PLSRD, each weak vertex, assigned the label -1, must be protected by exactly one strong vertex, with additional limitation that two weak vertices cannot share the same strong vertex, while the total sum of labels in the closed neighborhood of each vertex must remain positive. This paper provides exact values for the PLSRD number in several well-known graph classes, including complete graphs, complete bipartite graphs, wheels, paths, cycles, ladders, prism graphs, and 3 x n grids. Additionally, we establish a lower bound for a general 3 regular graph, as well as the upper bounds for flower snarks graphs, highlighting the intricate interplay between the PLSRD constraints and the structural properties of these graph families.