Publication

Abstract : {For vertices u and v in a connected graph G = (V, E), the monophonic detour distance dm(u,v) is the length of a longest u-v monophonic path in G. An u-v monophonie path of length dm (u,v) is an u-v monophonie detour or an u-v m-detour. The set /dm[u,n] consists of all those vertices lying on an u-v m-detour in G. Given a set S of vertices of G, the union of all sets Idm [u, v] for u, v 6 S, is denoted by Idm [S]. A set S is an m-detour convex set if Idm [S] = S. The m-detour convex hull [5]dm of 5 in G is the smallest m-detour convex set containing S. A set S of vertices of G is an m-detour set if Idm [S] = V and the minimum cardinality of an m-detour set is the m-detour number md(G) of G. A set 5 of vertices of G is an m-detour hull set if [S]dm = V and the minmimum cardinality of an m-detour hull set is the m-detour hull number md h(G) of G. Certain general properties of these concepts are studied. Bounds for the m-detour hull number of a graph are obtained. It is proved that every two integers a and b with 2 ≤ a ≤ b are realizable as the m-detour hull number and the m-detour number respectively, of some graph. Graphs G of order n for which mdj,(G) = n or mdh(G) = n - 1 are characterized. It is proved that for each triple a, b and k of positive integers with a lt; b and k ≥ 3, there exists a connected graph G with radmG = a