Abstract
A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then $$\mathrm{sg}(G)$$ is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair $$\{x,y\}\subseteq S$$ so that these $${|S|\atopwithdelims ()2}$$ geodesics cover all the vertices of G. In this paper, we first give some bounds for strong geodetic number in terms of diameter, connectivity, respectively. Next, we show that $$2\le \mathrm{sg}(G)\le n$$ for a connected graph G of order n, and graphs with $$\mathrm{sg}(G)=2,n-1,n$$ are characterized, respectively. In the end, we investigate the Nordhaus–Gaddum-type problem and extremal problems for strong geodetic number.
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