MMN-1950

# Minimal doubly resolving sets of antiprism and Möbius ladders

Consider a simple connected undirected graph $G=(V,E)$, where $V(G)$ represents the vertex set and $E(G)$ represents the edge set respectively. A subset $W$ of $V$ is called a resolving set for graph $G$ if for every two distinct vertices $x,y\in V$, there exist some vertex $w\in W$ such that $d(x,w)\neq d(y,w)$, 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 $u,v\in D$ such that $d(u,x)-d(u,y)\neq d(v,x)-d(v,y)$. 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 and for M\"{o}bius ladders.