On strong metric dimension of graphs and their complements

Abstract

A vertex x in a graph G strongly resolves a pair of vertices v,w if there exists a shortest x − w path containing v or a shortest x − v path containing w in G. A set of vertices S ⊆ V (G) is a strong resolving set of G if every pair of distinct vertices of G is strongly resolved by some vertex in S. The strong metric dimension of G, denoted by sdim(G), is the minimum cardinality over all strong resolving sets of G. For a connected graph G of order n ≥ 2, we characterize G such that sdim(G) equals 1, n − 1, or n − 2, respectively. We give a Nordhaus-Gaddum-type result for the strong metric dimension of a graph and its complement: for a graph G and its complement $\bar G$, each of order n ≥ 4 and connected, we show that $2 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 2} \right)$. It is readily seen that $sdim\left( G \right) + sdim\left( {\bar G} \right) = 2$ if and only if n = 4; we show that, when G is a tree or a unicyclic graph, $sdim\left( G \right) + sdim\left( {\bar G} \right) = 2\left( {n - 2} \right)$ if and only if n = 5 and $G \cong \bar G \cong C_5$, the cycle on five vertices. For connected graphs G and $\bar G$ of order n ≥ 5, we show that $3 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 3} \right)$ if G is a tree; we also show that $4 \leqslant sdim\left( G \right) + sdim\left( {\bar G} \right) \leqslant 2\left( {n - 3} \right)$ if G is a unicyclic graph of order n ≥ 6. Furthermore, we characterize graphs G satisfying $sdim\left( G \right) + sdim\left( {\bar G} \right) = 2\left( {n - 3} \right)$ when G is a tree or a unicyclic graph.