The vertex detour number of a graph (Q2469303)

Given an undirected graph \(G=(V,E)\), a detour between two vertices is a longest (simple) path between them. The vertex detour number \(d_x\) of \(G\) at \(x\) is the smallest cardinality of a \(x\)-detour set, i.e. the detours between \(x\) and any vertex of this set cover \(V\). The vertex detour number of \(G\) is the minimum vertex detour number. This is characterized for many common classes of graphs, and tight bounds are derived together connecting it with the earlier studied detour radius and diameter.