The fault-tolerant metric dimension of the king's graph

The concept of resolving the set within a graph is related to the optimal placement problem of access points in an indoor positioning system. A vertex $w$ of the undirected connected graph $G$ resolves the vertices $u$ and $v$ of $G$ if the distance between vertices $w$ and $u$ differs from the distance between vertices $w$ and $v$. A subset $W$ of vertices of $G$ is called a resolving set, if every two distinct vertices of $G$ are resolved by some vertex of $w \in W$. The metric dimension of $G$ is a minimum cardinality of its resolving set.The set of access points of the indoor positioning system corresponds to the resolving set of vertices in the graph.The minimum number of access points required to locate each of the vertices corresponds to the metric dimension of graph. A resolving set $W$ of the graph $G$ is fault-tolerant if $W \setminus \{w\}$ is also a resolving set of $G$, for each $w \in W$. The fault-tolerant metric dimension of the graph $G$ is a minimum cardinality of the fault-tolerant resolving set. In the indoor positioning system the fault-tolerant resolving set provides correct information even when one of the access points is not working. The article describes a special case of a graph called the king's graph, or the strong product of two paths.The king's graph is a building model in some indoor positioning systems. In this article we give an upper bound for the fault-tolerant metric of the king's graph and a formula for a particular case of the king's graph. Refs 20. Figs 2.