Theoretical results for Perfect Location signed Roman domination problem

TL;DR

This work studies the Perfect Location Signed Roman Domination (PLSRD) problem, defining a function $f:V\to\{-1,1,2\}$ that must satisfy the (C1)-(C3) constraints and yields a total weight $f(V)$; the weak vertices (-1) must be uniquely protected by a neighboring 2 and two distinct weak vertices cannot share a 2-neighbor. The authors derive exact values and tight bounds for a broad spectrum of graph families, including $K_n$, $K_{p,n}$, paths, cycles, ladders, prisms, and grids like $Grid_{3\times n}$, and establish a lower bound for 3-regular graphs via $\lceil 3n/5\rceil$ and a 2-packing-based upper bound. They provide constructive labelings that achieve these bounds, and for prism and flower-snark graphs present case-based and parity-dependent results. The results illuminate how the interplay of perfect, locating, and signed domination shapes optimal weight and guarding structure, and they set the stage for algorithmic developments and broader explorations in related graph classes.

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.

Figures (4)

• Figure 1: An illustration of labeling of a tree, based on a 2-packing set $S$. Enlarged vertices belong to $S$. Four leaf vertices from $S$ are labeled with -1. At the right side of the graph two support vertices from $S$ are labeled with 2, while one leaf vertex per each is labeled with -1. Finally, the remaining vertex from $S$ in the central part of the graph is labeled with -1, one its neighbor with 2, while others are labeled with 1. The rest of vertices is labeled with 1.
• Figure 2: Segments of several different configurations illustrating the mutual positioning of vertices $v$, $u$, and $v'$, labeled as -1, 1, and 1, respectively.
• Figure 3: The labeling of the flower snarks graph $J_9$ derived from the definition of the function $f$. The "central" vertices $a_4,b_4,c_4$ and $d_4$ are highlighted, with their respective labels shown in brackets.
• Figure 4: The labeling of the flower snarks graph $J_{11}$ derived from the definition of the function $f$. The "central" vertices $a_6,b_6,c_6$ and $d_6$ are highlighted, with their respective labels shown in brackets.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• Proposition 4
• proof