MINIMAL DOUBLY RESOLVING SETS OF ANTIPRISM GRAPHS AND MOBIUS LADDERS

摘要

Consider a simple connected graph G = (V(G),E(G)), where V(G) represents the vertex set and E(G) represents the edge set respectively. A subset W of V (G) is called a resolving set for a graph G if for every two distinct vertices x, y is an element of V (G), there exist some vertex w is an element of W such that d(x, w) not equal d(y, w), where d(u, v) denotes the distance between vertices u and v. A resolving set of minimal cardinality is called a metric basis for G and its cardinality is called the metric dimension of G, which is denoted by beta(G). A subset D of V(G) is called a doubly resolving set of G if for every two distinct vertices x,y of G, there are two vertices u, v is an element of D such that d(u,x) - d(u,y) not equal d(v,x) - d(v,y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by psi(G). In this paper, we determine the minimal doubly resolving sets for antiprism graphs denoted by A(n) with n >= 3 and for Mobius ladders denoted by M-n, for every even positive integer n >= 8. It has been proved that psi(A(n)) = 3 for n >= 3 and psi(M-n) = {3, if n 0 or 4 (mod 8) 4, if n 2 or 6 (mod 8) for every even positive integer n >= 8.