$$\varDelta $$-Convexity Number and $$\varDelta $$-Number of Graphs and Graph Products

Abstract

The \(\varDelta \)-interval of \(u,v \in V (G)\), \(I_{\varDelta }(u,v)\), is the set formed by u, v and every w in V(G) such that \(\{u, v, w\}\) is a triangle \((K_3)\) of G. A set S of vertices such that \(I_{\varDelta } (S)=V(G)\) is called a \(\varDelta \)-set. \(\varDelta \)-number is the minimum cardinality of a \(\varDelta \)-set. \(\varDelta \)-graph is a graph with all the vertices lie on some triangles. If a block graph is a \(\varDelta \)-graph, then we say that it is a block \(\varDelta \)-graph. A set \(S\subseteq V (G)\) is \(\varDelta \)-convex if there is no vertex \(u \in V(G)\setminus S\) forming a triangle with two vertices of S. The convexity number of a graph G with respect to the \(\varDelta \)-convexity is the maximum cardinality of a proper convex subset of G. We have given an exact value for the convexity number of block \(\varDelta \)-graphs with diameter \({\le }3\), block \(\varDelta \)-graphs with diameter \({>}3\) and the two standard graph products (Strong, Lexicographic products), a bound for Cartesian product. Also discussed some bounds for \(\varDelta \)-number and a realization is done for the \(\varDelta \)-number and the hull number.