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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), ID[S] = ⋃ x,y∈S ID[x, y]. A set S of vertices is a detour convex set if ID[S] = S. The detour convex hull [S]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 [S]D = V (G). Let x be any vertex in a connected graph G. For a vertex y in G, denote by IG[y] , the set of all vertices distinct from x that lie on some x − y detour of G; while for S ⊆ V (G), ID[S] x = ⋃ y∈S ID[y] . For x / ∈ S, S is an x-detour set of G if ID[S] x = V (G) − {x} and an x-detour set of minimum cardinality is the x-detour number dx(G) of G. For x / ∈ S, S is an x-detour convex set if ID[S] x = S. The x-detour convex hull of S, [S] D is the smallest x-detour convex set containing S. The x-detour hull number dhx(G) is the minimum cardinality among the subsets S of V (G) − {x} with [S]xD = V (G) − {x}. In this paper, we investigate how the detour hull number and the vertex detour hull number of a connected graph are affected by adding a pendant edge.

ON THE DETOUR AND VERTEX DETOUR HULL NUMBERS OF A GRAPH