Locating-dominating coalitions in graphs

TL;DR

This paper introduces the concept of locating-dominating coalitions in graphs, explores their existence, bounds, and exact values for certain graph classes, and studies the computational complexity of related decision problems.

Contribution

It initiates the study of locating-dominating coalitions, providing bounds, characterizations, exact values for specific graphs, and analyzing the complexity of associated decision problems.

Findings

Characterized graphs with maximum coalition number

Established bounds on the coalition number

Determined exact coalition numbers for certain graph classes

Abstract

A set $D$ of vertices in a graph $G = (V, E)$ is a locating-dominating set (LD-set) if it is dominating and every two vertices $u$, $v$ of $V\setminus D$ satisfy $N(u) \cap D \neq N(v) \cap D$. Two disjoint sets $A,B\subset V(G)$ form a locating-dominating coalition (for short, an LD-coalition) in $G$ if none of them is an LD-set in $G$ but their union $A\cup B$ is an LD-set. A locating-dominating coalition partition (for short, an LDC-partition) is a vertex partition $\Pi$ such that every set of $\Pi$ is not an LD-set in $G,$ but forms an LD-coalition with another set of $\Pi$. The locating-domination coalition number of $G$, denoted by $C_{L}(G),$ equals the maximum cardinality of an LDC-partition of $G$. Our purpose in this paper is to initiate the study of locating-dominating coalitions in graphs. We first investigate the existence of LDC-partitions. We also obtain lower and upper bounds on $C_{L}(G)$. We characterize connected graphs $G$ of order $n\ge 3$ satisfying $C_L(G) = n,$ as well as those trees $T$ such that $C_L(T)=n-1$. In addition, we determine the exact values of $C_L(G)$ for some classes of graphs. Moreover, we investigate the computational complexity of the decision problem associated with locating-dominating coalition partitions. To the best of our knowledge, this is the first work that addresses the algorithmic complexity of a decision problem related to coalition partitions, not only for this locating-dominating model but for coalition partitions in general.