Efficient $k$-limited Dominating Broadcasts in Product Graphs

TL;DR

This work studies the smallest $k$ for which a graph $G$ admits an efficient $k$-limited dominating broadcast, denoted $mcr(G)$, unifying efficient domination with broadcast domination. It proves NP-completeness of deciding $mcr(G)$ for general graphs and derives exact $mcr$ and related broadcast parameters for key graph classes such as paths, cycles, and star subdivisions, as well as for lexicographic and strong graph products. A central contribution is the construction of a family of trees $T_k$ with $mcr(T_k)=k$, establishing that every positive integer $k$ is realizable, and a detailed NP-hardness reduction from EXACT 3-SAT to $k$-ELDB using gadget-based graphs. The paper also analyzes how $mcr$ behaves under graph products, providing monotonicity results and case distinctions that connect component radii to product graphs, with implications for resource allocation and network design in structured graphs.

Abstract

In a graph $ G $, a subset of vertices $ S $ is called an efficient dominating set (EDS) if every vertex in the graph is uniquely dominated by exactly one vertex in $ S $. A graph is said to be efficiently dominatable if it contains an EDS. Additionally, a function $ f: V(G) \rightarrow \{0, 1, 2, \dots, k\} $ is termed a $ k $-limited dominating broadcast if, for every vertex $ u \in V(G) $, there exists a vertex $ v $, with $ f(v) \geq 1$ such that $ d(u, v) \leq f(v) $. A vertex $u$ is said to be dominated by a vertex $v$. In this work, we unify these two concepts to explore the notion of efficient $k$-limited broadcast domination in graphs. A $ k $-limited dominating broadcast $f$ is called an efficient $k$-limited dominating broadcast ($k$-$ELDB$) if each vertex in the graph is dominated exactly once. The minimum value of $k$ for which the given graph $G$ has $k$-$ELDB$ is defined as $mcr(G)$. We prove determining $mcr(G)$ is NP-Complete for general graphs and explore the $mcr(G)$ values and other related parameters on standard graphs and their products.

Figures (2)

• Figure 1: Graphs of $T_k$
• Figure 2: Reduction graph $G_1$ for an example of CNF, where $C_1=\{u_1,u_2,u_3\}, C_2=\{u_1,\overline{u_2},\overline{u_5}\}$, etc...

Theorems & Definitions (42)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4