The detour hull number of a graph

Abstract

For vertices \(u\) and \(v\) in a connected graph \(G=(V, E)\), the set \(I_D[u,v]\) consists of all those vertices lying on a \(u-v\) longest path in \(G\). Given a set \(S\) of vertices of \(G\), the union of all sets \(I_D[u,v]\) for \(u,v\in S\), is denoted by \(I_D[S]\). A set \(S\) is a detour convex set if \(I_D[S]=S\). The detour convex hull \([S]_D\) of \(S\) in \(G\) is the smallest detour convex set containing \(S\). The detour hull number \(d_h(G)\) is the minimum cardinality among the subsets \(S\) of \(V\) with \([S]_D=V\). A set \(S\) of vertices is called a detour set if \(I_D[S]=V\). The minimum cardinality of a detour set is the detour number \(dn(G)\) of \(G\). A vertex \(x\) in \(G\) is a detour extreme vertex if it is an initial or terminal vertex of any detour containing \(x\). Certain general properties of these concepts are studied. It is shown that for each pair of positive integers \(r\) and \(s\), there is a connected graph \(G\) with \(r\) detour extreme vertices, each of degree \(s\). Also, it is proved that every two integers \(a\) and \(b\) with \(2\leq a\leq b\) are realizable as the detour hull number and the detour number respectively, of some graph.  For each triple \(D,k\) and \(n\) of positive integers with \(2\leq k\leq n-D+1\) and \(D\geq 2\), there is a connected graph of order \(n\), detour diameter \(D\) and detour hull number \(k\).  Bounds for the detour hull number of a graph are obtained. It is proved that \(dn(G)=dh(G)\) for a connected graph \(G\) with detour diameter at most 4. Also, it is proved that for positive integers \(a,b\) and \(k\geq 2\) with \(a< b\leq 2a\), there exists a connected graph \(G\) with detour radius \(a\), detour diameter \(b\) and detour hull number \(k\). Graphs \(G\)  for which \({d}_{h}(G)=n-1\) or \(d_h(G)=n-2\) are characterized.