The Connected Detour Numbers of Special Classes of Connected Graphs

Simple finite connected graphs G � (V, E) of p≥ 2 vertices are considered in this paper. A connected detour set of G is defined as a subset S⊆V such that the induced subgraph G[S] is connected and every vertex of G lies on a u − v detour for some u, v ∈ S. 'e connected detour number cdn(G) of a graph G is the minimum order of the connected detour sets of G. In this paper, we determined cdn(G) for three special classes of graphs G, namely, unicyclic graphs, bicyclic graphs, and cog-graphs for Cp, Kp, and Km,n.


Introduction
For basic definitions of the concepts of graphs we refer to [1][2][3][4], and for detour distance and related terminologies in graphs, we refer to [5][6][7]. Let G � (V, E) be a connected simple graph of p vertices and q edges. We assume that p is finite and p(G) ≥ 2. For u, v ∈ V(G) the length of a maximum u − v path is called detour distance D (u, v).
e detour radius rad D G and the detour diameter diam D G (or D(G)) of G are defined as (2) A connected detour basis of G is a c.d.s. of order cdn(G) (see [8,9]).
A simple connected (p, q) graph G with p ≥ 3 is called unicyclic graph iff p � q. e graph G is called bicyclic iff q � p + 1.
e concept of connected detour number was introduced and studied by Santhakumaran and Athisayanathan in [9]. ey determined cdn for some special graphs such as K p , C p , K m,n , trees, and Hamilton graph. ere are many research papers on connected detour number and edge detour graphs (see [10][11][12][13][14]). Moreover, the concept of connected detour number and other related concepts have interesting applications in the channel assignment problem in radio technologies.
is motivated us to determine connected detour number for other classes of graphs. erefore, in this paper we determine the connected detour numbers for unicyclic graphs and bicyclic graphs. Moreover, the class of graphs called cog-graphs G c will be explained and determined the cnd(G c ) if G is a complete graph, tree, cycle graph, and complete bipartite graph.

The Connected Detour Number of Unicyclic Graphs
Let G be a connected graph of order p ≥ 3 and C the unique cycle in G, and let C be of length l ≥ 3. It is clear that C has no chords, and every vertex of G, which is not on C, is either a cut-vertex or an end-vertex. We shall determine the connected detour number of such graphs in terms of l and p. Let n be the number of vertices of C that are not cut-vertices. Denote If l � p, then G � C p so cdn(G) � 2. If p > l, then G contains at least one cyclic cut-vertex. If n � 0, that is every vertex of C is a cut-vertex, then by eorem 1.4 [Ref. 2] cdn(G) � p. From now on, we assume p > l.
Proof. If n � 1, then G contains exactly one vertex which is not a cut-vertex. It is clear that there is a detour joining the other two vertices of the triangle and v lies on it.
us, cdn(G) � p − 1. If n � 2, let u 1 and u 2 be cycle vertices which are not cut-vertices. Clearly, there is no path in G between two vertices of T(G) that contains u 1 or u 2 (see Figure 1). us, every c.d.b. B of G must contain either u 1 or u 2 . erefore, cdn(G) ≥ p − 1. If u 3 is the third vertex of the triangle and u 1 ∈ B, then u 2 lies on the u 1 − u 3 detour. erefore, cdn(G) ≤ p − 1. Hence, cdn(G) � p − 1.
To prove the converse, let cdn(G) � p − 1, and B(G) is a c.d.b. of G, then B(G) contains two vertices of the 3− cycle, one of them is a cut-vertex (∵p ≥ 4).
us, in view of eorem 1.4 [9], the 3− cycle has one or two vertices in T(G), that is, n � 1 or 2. Hence, the proof is completed.
From the hypothesis erefore, m � n. Hence, the proof of the theorem is completed. Proof. Let u be a vertex of C adjacent to v. en, there is a u − v detour consisting of all the vertices of C.
It is clear that there are no x − y detour containing vertices of T(G) for every pair x, y ∈ T(G). erefore, T(G) ∪ u { } is a connected detour basis of G, and hence cdn(G) � p − l + 2.
For connected unicyclic graphs having more than one cycle cut-vertex, we need the following definition. ese components divide the cycle vertices which are not cutvertices into m nonempty subsets A 1 , A 2 , . . . , A m , in successive order around C, as illustrated in Figure 3.
Example 2. Consider the unicyclic graph G shown in Figure 3. e set of cycle vertices which are not cut-vertices is W(G) � w 1 , w 2 , . . . , w n . It is clear that n � 11, m � 4, and l � 18. e set W(G) is partitioned into A 1 � w 1 , w 2 , A 2 � w 3 , w 4 , w 5 , w 6 , w 7 , A 3 � w 8 , and A 4 � w 9 , w 10 , w 11 . e c.d.n. for unicyclic graphs having more than one cutvertex is determined by the following theorem.
Proof. Let α be as defined in (7) for the graph G, and let S be a c.d.b. for G. By eorem 1.4 of Ref [9], S contains the set T(G). Since the induced subgraph G[S] is connected, then S must contain all the vertices of at least (m − 1) subsets from A 1 , A 2 , . . . , A m . Since S is of minimum order, then S does not contain the subset from A 1 , A 2 , . . . , A m that has maximum order, say A r . It is clear that there are two vertices x, y ∈ T(G) ∪ ∪ i≠r A i which are adjacent on C; hence, there is an x − y detour containing all the vertices of C. erefore, To prove the converse, let cdn(G) � p − β and let S ′ be a c.d.b. of G. If β is not equal to max |A i | : 1 ≤ i ≤ m , then either S ′ is not of minimum order or the induced subgraph G[S ′ ] is disconnected, contradicting the definition of connected detour basis.

The Connected Detour Numbers of Connected Bicyclic Graphs
A (p, q) graph is bicyclic if and only if q � p + 1. us, if G is a connected bicyclic graph, then G contains either three cycles having some edges in common or contains exactly two cycles having no edges in common. e connected detour number for a block bicyclic graph is determined by the following result.

Proposition 3.
Let G be a 2-connected bicyclic graph of order p ≥ 5 as shown in Figure 4. en, Proof.
Conversely, if m ≠ n, say m > n and k ≠ m, n, then G does not contain adjacent vertices u, v such that u, v { } is a detour set. Hence, the proof of Part (i) is completed.
(ii) If m, n, and k are different, say m > n > k, then it is clear that Hence, the proof of the proposition is completed.
□ Remark 2. If G is a 2-connected bicyclic graph of order p ≥ 4 with a cycle C and with exactly one chord, that is, an edge joining nonadjacent vertices of C, then cdn(G) � 2. is section is divided into two subsections according to the types of the bicyclic graphs.

e Connected Detour Numbers of Bicyclic Graphs of
ree Cycles. Now assume that G is a connected bicyclic graph of order p ≥ 9 with one or more cut-vertices and with three cycles, that is, three x − y paths which are internally vertex disjoints denoted by Q 1 , Q 2 , and Q 3 as shown in Figure 5. We assume without loss of generality that  We shall determine the connected detour number for three kinds of bicyclic graphs of three cycles.
Case 1. Assume that each Q i , 1 ≤ i ≤ 3, contains at least one cut-vertex other than x and y. Moreover, let T ′ be the set of all cycle cut-vertices in G.
en, we have the following proposition which determines the c.d.b. of such kind of bicyclic graph G.

Proposition 4.
Let G be a connected bicyclic graph of three cycles and with one or more cut-vertices on each Q i , i � 1, 2, 3, other than x and y as explained above and shown in Figure 5. en, where S is a subset of T of minimum order such that the erefore, cdn(G) � |T| + |S|. Hence, the proof is completed. e following example illustrates Proposition 3.2. □ Example 3. Consider the bicyclic graph G shown in Figure 6.
Case 2. Assume that G contains exactly one x − y path that does not contain cut-vertices, other than x and y. So we have three possibilities for such bicyclic graph G: (i) Let Q 1 and Q 3 each contains at least one internal cut-vertex, and Q 2 does not contain an internal cutvertex.

Example 4.
Consider the graphs G i , 1 ≤ i ≤ 4, as shown in Figure 7.
It is easy to verify that: Case 3. Assume that the connected bicyclic graph G consists of two x − y paths, and each path does not contain cut-vertices but only one x − y path contains internal cutvertices. If Q 1 contains at least two internal cut-vertices, and Q 2 and Q 3 have no cut-vertices, n � k, then Q 1 ∪ Q 2 is a unicyclic graph, denoted H 1,2 ′ . It is clear that where α 1,2 ′ is given for H 1,2 ′ as defined in Definition 1. us, Similar results we have if Q i (i � 2, 3) has at least two internal cut-vertices and the other x − y paths have no cutvertices. erefore, cdn(G) � p(G) − m − α 2,3 ′ , for i � 2 and m � k, where α 2,3 ′ is for the unicyclic graph H 2,3 ′ and α 1,3 ′ is for H 1,3 ′ . Figure 5 has exactly one cycle cut-vertex which is a vertex of the x − y path Q i (1 ≤ i ≤ 3) including x and y, and the other two x − y paths have equal lengths, then cdn(G) � 1 + |T|.

Remark 3. If the bicyclic graph G depicted in
From now on, we assume that m > n > k ≥ 1 (see Figure 5). If Q 1 contains internal cut-vertices and Q 2 and Q 3 contain no cut-vertices, then we may assume that the distance from x to the first cut-vertex along the x − y path Q 1 is not more than the distance from y to the last cut-vertex along Q 1 . Let H ″ 1,3 be the unicyclic graph constructed from Q 1 ∪ Q 3 ∪ w 1 , z , where w 1 z is an end-edge incident to vertex w 1 of Q 3 . It is clear H ″ 1,3 contains vertices x and w 1 in addition to vertices from Q 1 , and so erefore, If the distance from y to the last cut-vertex along Q 1 is less than the distance from x to the first cut-vertex along Q 1 , then we have the unicyclic graph where α ‴ 1,3 is the number defined for H ‴ 1,3 (Definition 1). We have results similar to (20a) and (20b) for the cases where Q i (i � 2, 3) has internal cut-vertices, the other two x − y paths have no internal cut-vertices and m > n > k ≥ 1. Namely, cdn(G) � p(G) − m − α ″ 2,3 or cdn(G) � p(G) − m − α ‴ 2,3 for i � 2 or 3 and the unicyclic graphs H ″ 2,3 or H ‴ 2,3 .
Remark 4. If the vertex x or the vertex y is the only cycle cut-vertex of the bicyclic graph G shown in Figure 5 and m > n > k ≥ 1, then

e Connected Detour Numbers of Bicyclic Graphs of Two
Cycles. Let G be a bicyclic graph containing exactly two cycles C 1 and C 2 , either having one vertex in common or there is a path joining a vertex of C 1 to a vertex of C 2 . us, G is considered to consist of two unicyclic graphs G 1 and G 2 having exactly one vertex v in common. Let G i ′ (i � 1, 2) be a uncyclic graph obtained from G i by adding to it an end-edge vw i . e connected detour number of G is determined by the following theorem.

The Connected Detour Numbers of Cog-Graphs
Let G be a connected (p, q)-graph, then G (c) is the graph constructed from the graph G with q additional vertices u 1 , u 2 , . . . , u q corresponding to the edges e 1 , e 2 , . . . , e q of G and 2q additional edges obtained from joining u i to the two vertices of e i for all i � 1, 2, . . . , q. Such class of graphs are called cog-graphs of G. For example, let G be a star of order five, then G (c) is cog-star of order nine shown in Figure 8. Clearly if G is (p, q)-graph then G (c) is (p + q, 3q)-graph. e proofs of the following elementary results are obvious.

Proposition 5
(1) e cog-graph G (c) does not contain end-vertices. (2) If the graph G has n end-vertices, then G (c) contains exactly (q + n) vertices of degree 2.
Let V(G) � v 1 , v 2 , . . . , v p and V(G (c) ) � V(G) ∪ u 1 , u 2 , . . . , u q }. If (x 1 , x 2 , . . . , x k− 1 , x k ) is an x 1 − x k detour in G, then (x 1 , y 1 , x 2 , y 2 , . . . , x k− 1 , y k− 1 , x k ) is an x 1 − x k detour in G (c) , in which y i ∈ U � u 1 , u 2 , . . . , u q for 1 ≤ i ≤ k − 1 and y i is the vertex that corresponds to edge Moreover, if Q is an y − y ′ detour in G (c) y, y ′ ∈ U, (as shown in Figure 9), then Any way, if S is a detour set of G, then S may not be a detour set of G (c) . Also for some graphs G, cdn (G) ≠ cdn(G (c) ). For example, if G is an odd cycle graph C p with exactly one chord, then cdn(G) � 2 and cdn(G (c) ) � 3. But there are special graphs G such that cdn(G) � cdn(G (c) ), as given in the following proposition. Proposition 6. Let G be a connected graph. If G is a tree or a cycle graph, then Proof. It is obvious. e following concepts were introduced by Santhakumaran and Athisayanathan in [12]. e edge detour number dn 1 (G) of G is the minimum order of its edge detour set. Any edge detour set of order dn 1 (G) is called an edge detour basis of G. A graph which has an edge detour set is called a edge detour graph (denoted E.D. graph)." ere are graphs which are not E.D. graphs because they do not have edge detour sets [12]. For E.D. graphs we give the following definition. e connected detour number cdn 1 (G) of G is defined by Proof. One can easily check that the statement is true for p 3, 4, and 5. Now assume that the statement is true for p r ≥ 5, and consider K r+2 . Let x, y be any pair of vertices of K r+2 , and let K r K r+2 − x, y and V(K r ) v 1 , v 2 , . . . , v r− 1 , v r } as shown in Figure 10.
By induction hypothesis for every pair v i , v j of vertices of K r , every edge other than v i v j of K r lies on a v i − v j detour Q in K r .
It is clear that the two edges xv i , yv j or (xv j , yv i ) with Q produce an x − y detour in K r+2 .
is is true for all i, j 1, 2, . . . , r, i ≠ j. us, every edge of K r+2 other than xy lies on some x − y detour in K r+2 . erefore, the proposition is true for K r+2 . Hence, by induction on p the proposition is true for K p , p ≥ 3. □ Corollary 1. For each complete graph K p with p ≥ 3, Proof. Let u, v, and w be any three vertices in K p . By Proposition 8 every edge of K p other than uv (resp., uw) lies on an u − v detour (resp., u − w detour). us, u, v, w { } is a c.e.d.s. of K p . Clearly, cdn 1 (K p ) > 2, and hence cdn 1 (K p ) 3.

Journal of Mathematics
K r,s , r, s ≥ 3, and consider K r+1,s+1 . Let xy be any edge of K r+1,s+1 , and let K r,s � K r+1,s+1 − x, y as shown in Figure 11 in which its vertex set is X ∪ Y, X � x 1 , x 2 , . . . , x r , and Y � y 1 , y 2 , . . . , y s . By induction hypothesis, every edge of K r,s other than x i y j (1 ≤ i ≤ r, 1 ≤ j ≤ s) lies on x i − y j detour Q in K r,s . Clearly, each x i − y j detour Q in K r,s implies x − y detour Q ′ (namely, x, y j − x i detour, and y) in K r+1,s+1 . Moreover, each edge of K r,s with edges xy j and yx i lie on Q ′ . Since this holds for i � 1, 2, . . . , r and j � 1, 2, . . . , s, then every edge of K r+1,s+1 (other than xy) lies on an x − y detour in K r+1,s+1 .
us, by induction the proposition holds for every K m,n , m, n ≥ 2.
Proof. Consider the vertices x 1 , x 2 , and y 1 of K m,n where x 1 x 2 ∉ E(K m,n ) and x 1 y 1 , x 2 y 1 ∈ E(K m,n ). en, by Proposition 9 every edge of K m,n (other than x 1 y 1 ) lies on an x 1 − y 1 detour, and x 1 y 1 lies on an x 2 − y 1 detour in K m,n . erefore, x 1 , x 2 , y 1 is a c.e.d.s. of K m,n , and hence cdn 1 (K m,n ) � 3. Proof. Let xy be an edge of K m,n , then by Proposition 9 every edge other than xy lies on an x − y detour in K m,n . From the definition of cog-graphs, every vertex other than z lies on an x − y detour in K (c) m,n , where z is the vertex that corresponds to the edge xy in K (c) m,n . Adding the edge yz to every such x − y detour in K (c) m,n we obtain x − z detours, and hence every K (c) m,n lies on an x − z detour in K (c) m,n . Hence, cdn(K (c) m,n ) � 2.

Conclusions
e connected detour numbers for three classes of connected simple graphs are determined in this research paper. e three classes are unicyclic graphs, bicyclic graphs, and cog-graphs for C c p , K c p , and K c m,n . We think that the methods used in proving the results in Section 3 can be used to determine the connected detour numbers for bridge graphs and chain graphs (defined in [16]) that are constructed from finite pairwise disjoint unicyclic graphs.
It is shown that cdn(G c ) is related to cdn 1 (G), and in view of Proposition 10 we suggest the following problem: characterize edge detour graphs G such that cdn(G (c) ) � cdn 1 (G).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Figure 11: K r+1,s+1 , r, s ≥ 3.