On the local metric dimension of dipyramidal graph and king graph
Javas Alfreda Belva Yoga Pratama, Tri Atmojo Kusmayadı · AIP conference proceedings · 2021
Let G be a simple connected graph with a set of vertices V(G) and set of edges E(G). The distance from vertex u to vertex v on graph G is the length of the shortest path from vertex u to vertex v, which is denoted by d(u, v). Suppose there is an ordered set W = {w1, w2, w3,…,wn} and v is a vertex on graph G, then the representation of vertex v to W is an n - tuple ordered pair, namely r(v|W) = (d(v, w1),d(v, w2),d(v,w3),…, d(v, wn)). A set W is called a local resolving set if r(u|W) ≠ r(v|W) for each pair of vertex u and vertex v adjacent to each other on graph G. The number of vertices in the local metric base of graph G is called the local metric dimension and is denoted by lmd(G). Dipyramidal graph DPn is the skeleton of an n-sided dipyramid with n ≥ 3. King graph Kmn is a graph with m × n vertices in which each vertex represents a square in an m × n chessboard, and each edge corresponds to a legal move by a king with m, n ≥ 2. In this paper, the local metric dimension of a dipyramidal graph and a king graph were explained.