The forcing connected detour number of a graph

For two vertices u and v in a graph G = (V,E), the detour distance D(u, v) is the length of a longest u—v path in G. A u—v path of length D(u, v) is called a u—v detour. A set S ⊆ V is called a detour set of G if every vertex in G lies on a detour joining a pair of vertices of S. The detour number dn(G) of G is the minimum order of its detour sets and any detour set of order dn(G) is a detour basis of G. A set S ⊆ V is called a connected detour set of G if S is detour set of G and the subgraph G[S] induced by S is connected. The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G. A subset T of a connected detour basis S is called a forcing subset for S if S is the unique connected detour basis containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G), is fcdn(G) = min{fcdn(S)}, where the minimum is taken over all connected detour bases S in G. The forcing connected detour numbers of certain standard graphs are obtained. It is shown that for each pair a, b of integers with 0 ≤ a < b and b ≥ 3, there is a connected graph G with fcdn(G) = a and cdn(G) = b.


Introduction
By a graph G = (V, E) we mean a finite undirected graph without loops or multiple edges.The order and size of G are denoted by p and q respectively.We consider connected graphs with at least two vertices.For basic definitions and terminologies we refer to [1,5].For vertices u and v in a connected graph G, the detour distance D(u, v) is the length of a longest u-v path in G.A u-v path of length D(u, v) is called a u-v detour.It is known that the detour distance is a metric on the vertex set V .Detour distance and detour center of a graph were studied in [2,4].
A vertex x is said to lie on a u-v detour P if x is a vertex of P including the vertices u and v.A set S ⊆ V is called a detour set if every vertex v in G lies on a detour joining a pair of vertices of S. The detour number dn(G) of G is the minimum order of a detour set and any detour set of order dn(G) is called a detour basis of G.A vertex v that belongs to every detour basis of G is a detour vertex in G.If G has a unique detour basis S, then every vertex in S is a detour vertex in G.These concepts were studied in [3].A set S ⊆ V is called a connected detour set of G if S is a detour set of G and the subgraph G[S] induced by S is connected.The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G.A vertex v in G is a connected detour vertex if v belongs to every connected detour basis of G.If G has a unique connected detour basis S, then every vertex in S a connected detour vertex of G.The connected detour number of a graph was introduced and studied in [6].
For the graph G given in Figure 1.1, the sets S 1 = {v 1 , v 3 }, S 2 = {v 1 , v 5 } and S 3 = {v 1 , v 4 } are the three detour bases of G so that dn(G) = 2.It is clear that no two element subset of V is a connected detour set of G. However the set S 4 = {v 1 , v 2 , v 3 } is a connected detour basis of G so that cdn(G) = 3.Also the set S 5 = {v 1 , v 2 , v 5 } is another connected detour basis of G. Thus there can be more than one connected detour basis for a graph G. Graphs are often used to model network of real life problems and some definite part is always present in a minimum possible spanning set in a particular problem.For each connected detour basis S in a connected graph G, there is always some subset T of S that uniquely determines S as the connected detour basis containing T .Such subsets are called forcing subsets for S, and in this paper we briefly describe the properties satisfied by these sets in a graph.
The following theorem is used in the sequel.
Theorem 1.1.[6] All the end vertices and all the cut vertices of a connected graph G belong to every connected detour set of G.
Throughout this paper G denotes a connected graph with at least two vertices.

The Forcing Connected Detour Number
Definition 2.1.Let G be a connected graph and S a connected detour basis of G.A subset T ⊆ S is called a forcing subset for S if S is the unique connected detour basis containing T .A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G), is fcdn(G) = min{fcdn(S)}, where the minimum is taken over all connected detour bases S in G.      Then there is a vertex x j (1 ≤ j ≤ a) such that x j / ∈ T .Let y j be a vertex of F j distinct from x j .Then S 0 = (S − {x j }) ∪ {y j } is also a connected detour basis such that it contains T .Thus S is not the unique connected detour basis containing

Example 2 . 2 .
For the graph G given in Figure2.1, S 1 = {u, s, w, t, v} is the unique connected detour basis of G so that fcdn(G) = 0 and for the graph G given in Figure1.1, S 2 = {v 1 , v 2 , v 3 } and S 3 = {v 1 , v 2 , v 5 } are the only connected detour bases of G so that fcdn(G) = 1.