Metric dimension of Andrásfai graphs

Abstract

A set $W\subseteq V(G)$ is called a resolving set, if for each pair of distinct vertices $u,v \in V(G)$ there exists $t \in W$ such that $d(u,t)\neq d(v,t)$, where $d(x,y)$ is the distance between vertices $x$ and $y$. The cardinality of a minimum resolving set for $G$ is called the metric dimension of $G$ and is denoted by $dim_{M}(G)$. This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and $And(k)\square P_n$. Also, we provide upper and lower bounds for $dim_M(And(k)\square C_n)$.