Locating-dominating coalitions in graphs

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 $Π$ such that every set of $Π$ is not an LD-set in $G,$ but forms an LD-coalition with another set of $Π$. 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.

Figures (13)

• Figure 1: The maximal number of partners for the set $A$
• Figure 2: Cubic graph with the set $A$ having $2\Delta$ coalition partners
• Figure 3: Vertices $u$ and $v$ of $C_n$ with $V_1 \cap N(u) = V_1 \cap N(v)$.
• Figure 4: The blue vertex is $w_1=v_1$; $w_2$ is above and to the left of $w_1$; $A=\{ \text{non-white vertices }$ has gap configuration $[2,2,0,1,0,1,2]$.
• Figure 5: (a) The set $A'$ (b) If $A=\{\hbox{red vertices}\}$, the singleton is unique (blue).

Theorems & Definitions (36)

• Definition 1.1: Locating-dominating coalition
• Definition 1.2: Locating-dominating coalition partition
• Theorem 2.1: Slater slater88
• Theorem 2.2: Slater slater88
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Proposition 2.5
• proof