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For a connected graph G with at least two vertices and S a subset of vertices, the convex hull ${\left[S\right]}_G$ is the smallest convex set containing S. The hull number h(G) is the minimum cardinality among the subsets S of V(G) with ${\left[S\right]}_G=V\left(G\right)$. Upper bound for the hull number of strong product G ⊠ H of two graphs G and H is obtainted. Improved upper bounds are obtained for some class of strong product graphs. Exact values for the hull number of some special classes of strong product graphs are obtained. Graphs G and H...

For vertices x and y in a connected graph G, the detour distance D(x,y) is the length of a longest x - y path in G. An x - y path of length D(x,y) is an x - y detour. The closed detour interval ID[x,y] consists of x,y, and all vertices lying on some x -y detour of G; while for S ⊆ V(G), $I_D\left[S\right]={\bigcup }_{x,y\in S}{I}_D[x,y]$. A set S of vertices is a detour convex set if $I_D\left[S\right]=S$. The detour convex hull ${\left[S\right]}_D$ is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V(G) with ${\left[S\right]}_D=V\left(G\right)$....

For a nontrivial connected graph G = (V(G),E(G)), a set S⊆ V(G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g₁(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a special class of graphs. Exact values of the edge geodetic number of Cartesian product are obtained for several...

For two vertices u and v of a connected graph G, the set $I_G[u,v]$ consists of all those vertices lying on u-v geodesics in G. Given a set S of vertices of G, the union of all sets $I_G[u,v]$ for u,v ∈ S is denoted by $I_G\left[S\right]$. A set S ⊆ V(G) is a geodetic set if $I_G\left[S\right]=V\left(G\right)$ and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.