On the fault-tolerant metric dimension of maximal outerplanar graphs

Abstract: Given a graph $G$, a set $W$ of vertices of $G$ is a resolving set if, for any pair of vertices $z_1$ and $z_2$ of $G$, there is a vertex $w\in W$ such that $d(w,z_1)\ne d(w,z_2)$, where $d(u,v)$ denotes the length of a shortest path between the vertices $u$ and $v$. The metric dimension of a graph $G$, $β(G)$, is the minimum cardinality of a resolving set of $G$. A resolving set $W$ is fault-tolerant if $W\setminus {w}$ is a resolving set for any $w\in W$. The fault-tolerant metric dimension of a graph $G$, denoted by $β'(G)$, is the minimum cardinality of a fault-tolerant resolving set of $G$. In this paper, we study the fault-tolerant metric dimension problem in maximal outerplanar graphs. In particular, we show that there are exactly two maximal outerplanar graphs with $β'(G)=3$. We also show that $4\le β'(G) \le 6$ for any maximal outerplanar graph $G$ of order at least 7 and metric dimension 2. Moreover, we establish the upper bound $β'(G) \le \left\lceil \frac{2n}{3} \right\rceil$ for maximal outerplanar graphs $G$ of order $n\ge 5$, by building in linear time a fault-tolerant resolving set of cardinality $\left\lceil \frac{2n}{3} \right\rceil$.