Nonlocal-Adjacency Metric Dimension of Graphs

Abstract

Let $ T = \{t_1, t_2, \ldots, t_k\} \subseteq V(G)$ be an ordered subset of the vertex set of a graph G, and let $u \in V(G)$ be a vertex in G. The adjacency metric representation of vertex u with respect to the set T is the k-vector $r_A(u \mid T ) = (d_A(u, t_1), d_A(u, t_2), \ldots, d_A(u, t_k))$. The set T is called a nonlocal-adjacency metric resolving set of the graph G if $r_A(u \mid T ) \ne r_A(w \mid T )$ for every pair of vertices $u, v \in G$ with u not adjacent to v. The minimum cardinality of a nonlocal-adjacency metric resolving set of G is called the nonlocal-adjacency metric dimension of G, denoted by $dim_{Anl}(G)$. In this paper, we present graphs obtained from the degree corona product of two graphs. The degree corona product of graphs G and H, denoted by $G \odot_{\deg} H$, is the graph constructed by taking a graph G and $\sum_{i=1}^{|V(G)|} \deg(v_i)$ copies $H_{ij}$ of graph H, and then connecting every vertex $v_i \in V(G)$ to all vertices in $H_{ij}$ for every $j \in \{1, 2, \ldots, \deg(v_i)\}$ and $i \in \{1, 2, \ldots, |V(G)|\}$. Furthermore, we determine and analyze the nonlocal-adjacency metric dimension of basic graphs $G_b \in \{P_n, C_n\}$, centered graphs $G_c \in \{K_n, S_n, K_1 + P_n, K_1 + C_n, K_m + \overline{K_n}\}$, and the degree corona product graphs $G_c \odot_{\deg} K_1$. In addition, we provide upper bounds, characterizations of the nonlocal-adjacency metric dimension of graphs, and examples of applications of this concept.