Ye, Yongsheng; Lv, Changhong; Liu, Qingmin The geodetic numbers of Cartesian products of graphs. (English) Zbl 1125.05058 Math. Appl. 20, No. 1, 158-163 (2007). Summary: For any two vertices \(u\) and \(v\) in a graph \(G\), a \(u\)-\(v\) geodesic is the shortest path between \(u\) and \(v\). Let \(I(u, v)\) denote the set of all vertices lying on a \(u\)-\(v\) geodesic. For a vertex subset \(S\), let \(I(S)\) denote the union of all \(I(u, v)\) for \(u, v\in S\). The geodetic number \(g(G)\) of a graph \(G\) is the minimum cardinality of a set \(S\) with \(I(S)=V(G)\). In this paper, a sufficient and necessary condition for the equality of \(g(G)\) and \(g(G\times K_3)\) is presented, and for a tree \(T\), we give the geodetic number of \(T\times K_m\) and \(C_n\times K_m\). Cited in 2 Documents MSC: 05C38 Paths and cycles 05C12 Distance in graphs Keywords:convex set; Cartesian product; geodesic; geodetic number PDFBibTeX XMLCite \textit{Y. Ye} et al., Math. Appl. 20, No. 1, 158--163 (2007; Zbl 1125.05058)