Articles
Electronic Journal Of Graph Theory And Applications (23382287)13(1)pp. 217-230
Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if (Formula presented). A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for G. In this paper, using adjacency resolving sets and dominating sets of graphs, we study doubly resolving sets in the corona product of graphs G and H, G ☉ H. First, we obtain the upper and lower bounds for the doubly resolving number of the corona product G☉H in terms of the order of G and the adjacency dimension of H, then we present several conditions that make each of these bounds feasible for the doubly resolving number of G ☉ H. Also, for some important families of graphs, we obtain the exact value of the doubly resolving number of the corona product. © (2025), (Indonesian Combinatorics Society). All rights reserved.
Discrete Mathematics, Algorithms and Applications (17938317)
Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if d(v, x) − d(u, x) ≠ d(v, y) − d(u, y). A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for G. The aim of this paper is to investigate doubly resolving sets in the lexicographic product graphs. It is proved that if H ∈/ {P3, P3} or G does not have any vertex of degree 1, then ψ(G[H]) = dim(G[H]). Also ψ(G[H]) is computed in other cases. © World Scientific Publishing Company.
Discrete Applied Mathematics (0166218X)339pp. 178-183
Two vertices u,v in a connected graph G are doubly resolved by x,y∈G if d(v,x)−d(u,x)≠d(v,y)−d(u,y).A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for the graph G. In this paper all graphs G with ψ(G)=2 are characterized by using 2-connected subgraphs of G. © 2023 Elsevier B.V.
Kyungpook Mathematical Journal (12256951)63(1)pp. 123-129
For an ordered set W = {w1,w2,…,wk} of vertices and a vertex v in a connected graph G, the k-vector (Formula Presented) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, (Formula Presented) of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(Formula Presented), characterize all graphs G with dim(Formula Presented), and prove that dim(Formula Presented) if and only if G is the complete graph Kn or the star graph K1,n−1. © Kyungpook Mathematical Journal
Discrete Mathematics, Algorithms and Applications (17938317)14(4)
For a set W of vertices and a vertex v in a graph G, the k-vector r2(v|W) = (aG(v,w1),⋯,aG(v,wk)) is the adjacency representation of v with respect to W, where W = {w1,⋯,wk} and aG(x,y) is the minimum of 2 and the distance between the vertices x and y. The set W is an adjacency resolving set for G if distinct vertices of G have distinct adjacency representations with respect to W. The minimum cardinality of an adjacency resolving set for G is its adjacency dimension. It is clear that the adjacency dimension of an n-vertex graph G is between 1 and n - 1. The graphs with adjacency dimension 1 and n - 1 are known. All graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are studied in this paper. In terms of the diameter and order of G, a sharp upper bound is found for adjacency dimension of G. Also, a sharp lower bound for adjacency dimension of G is obtained in terms of order of G. Using these two bounds, all graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n - 2 are characterized. © 2022 World Scientific Publishing Company.
