Weak edge triangle free detour number of a graph

For any two vertices u and v in a connected graph G = ( V, E ) , a u − v path P is called a u − v triangle free path if no three vertices of P induce a triangle. The triangle free detour distance D △ f ( u, v ) is the length of a longest u − v triangle free path in G. A u − v path of length D △ f ( u, v ) is called a u − v triangle free detour . A set S ⊆ V is called a weak edge triangle free detour set of G if every edge of G has both ends in S or it lies on a triangle free detour joining a pair of vertices of S. The weak edge triangle free detour number wdn △ f ( G ) of G is the minimum order of its weak edge triangle free detour sets and any weak edge triangle free detour set of order wdn △ f ( G ) is a weak edge triangle free detour basis of G. Certain properties of these concepts are studied. The weak edge triangle free detour numbers of certain classes of graphs are determined. Its relationship with the triangle free detour diameter is discussed and it is proved that for any three positive integers a, b and n of integers with 3 ≤ b ≤ n − a + 1 and a ≥ 4 , there exists a connected graph G of order n with triangle free detour diameter D △ f = a and wdn △ f ( G ) = b. It is also proved that for any three positive integers a, b and c with 3 ≤ a ≤ b and c ≥ b + 2 , there exists a connected graph G such that R △ f = a, D △ f = b and wdn △ f ( G ) = c.


Introduction
By a graph G = (V, E), we mean a finite undirected connected simple graph. For basic definitions and terminologies, we refer to Chartrand et al. [6]. The neighbourhood of a vertex v is the set N (v) consisting of all vertices u which are adjacent with v. A vertex v is an extreme vertex if the subgraph ⟨N (v)⟩ induced by its neighbourhood N (v) is complete.
The concept of geodetic number was introduced by Harary et al. [4,5,9]. For vertices u and v in a connected graph G, the distance d(u, v) is the length of a shortest u − v path in G. A u − v path of length d(u, v) is called a u − v geodesic. A set S ⊆ V is called geodetic set of G if every vertex of G lies on a geodesic joining a pair of vertices of S. The geodetic number g(G) of G is the minimum order of its geodetic sets and any geodetic set of order g(G) is called a geodetic basis of G.
The concept of detour number was introduced by Chartrand et al. [3]. 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 detour set of G if every vertex of 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 called a detour basis of G.
The concept of edge detour number was introduced by Santhakumaran an Athisayanathan [11,12]. A set S ⊆ V is called an edge detour set of G if every edge of G lies on a detour joining a pair of vertices of S. The edge detour number dn 1 (G) of G is the minimum order of its edge detour sets and any edge detour set of order dn 1 (G) is called an edge detour basis of G. A graph G is called an edge detour graph if it has an edge detour set.
The concept of weak edge detour number was introduced by Santhakumaran and Athisayanathan [13]. A set S ⊆ V is called a weak edge detour set of G if every edge of G has both ends in S or it lies on a detour joining a pair of vertices of S. The weak edge detour number dn w (G) of G is the minimum order of its weak edge detour sets and any weak edge detour set of order dn w (G) is a weak edge detour basis of G.
The concept of triangle free detour distance was introduced by Keerthi Asir and Athisayanathan [10]. The triangle free detour eccentricity e △f (v) of a vertex v in G is the maximum triangle free detour distance from v to a vertex of G. The triangle free detour radius, R △f of G is the minimum triangle free detour eccentricity among the vertices of G, while the triangle free detour diameter, D △f of G is the maximum triangle free detour eccentricity among the vertices of G.
The concept of triangle free detour number was introduced by Sethu Ramalingam and Athisayan athan [14]. A set S ⊆ V is called a triangle free detour set of G if every vertex of G lies on a triangle free detour joining a pair of vertices of S. The triangle free detour number dn △f (G) of G is the minimum order of its triangle free detour sets and any triangle free detour set of order dn △f (G) is called a triangle free detour basis of G.
In general, there are graphs G for which there exist edges which do not lie on a triangle free detour joining any pair of vertices of V. For the graph G given in Figure 1, the edge u 1 u 2 does not lie on a triangle free detour joining any pair of vertices of V. This motivates us to introduce the concept of weak edge triangle free detour set of a graph.
The following theorems will be used in the sequel.

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Weak edge triangle free detour number of a graph | S. Sethu Ramalingam and S. Athisayanathan Theorem 1.1. [14] Each extreme vertex of a graph G belongs to every triangle free detour set of G.

Theorem 1.2. [14]
If G is a connected graph of order n and triangle free detour diameter D △f , then dn △f (G) ≤ n − D △f + 1.
Throughout this paper G denotes a connected graph with at least two vertices.
2. Weak edge triangle free detour number of a graph Definition 2.1. Let G be a connected graph. A set S ⊆ V is called a weak edge triangle free detour set of G if every edge of G has both ends in S or it lies on a triangle free detour joining a pair of vertices of S. The weak edge triangle free detour number wdn △f (G) of G is the minimum order of its weak edge triangle free detour sets and any weak edge triangle free detour set of order wdn △f (G) is a weak edge triangle free detour basis of G.
Example 2.1. For the graph G given in Figure 2, it is clear that no two element subset of V is a weak edge triangle free detour set of G. It is easily seen that the set S 1 = {u, v, x} is a weak edge triangle free detour basis of G so that wdn △f (G) = 3. Also the set S 2 = {u, v, z} is another weak edge triangle free detour basis of G. Thus there can be more than one weak edge triangle free detour basis for a graph G. For the graph G given in Figure 3, S = {u, v} is a weak edge detour basis of G so that dn w (G) = 2 and S = {u, v, x, y, z} is a weak edge triangle free detour basis so that wdn △f (G) = 5. Hence the weak edge detour number and the weak edge triangle free detour number of a graph G are different.  Proof. A weak edge detour set needs at least two vertices so that dn w (G) ≥ 2. Since every weak edge triangle free detour set is also a weak edge detour set so that dn w (G) ≤ wdn △f (G). Also the set of all vertices of G is a weak edge triangle free detour set of G so that wdn △f (G) ≤ n. Thus 2 ≤ dn w (G) ≤ wdn △f (G) ≤ n.
Remark 2.1. The bounds in Theorem 2.1 are sharp. The set of two end-vertices of a path P n is its unique weak edge triangle free detour set so that wdn △f (G) = 2. For the complete graph K n , wdn △f (K n ) = n. Thus the path P n has the smallest weak edge triangle free detour number 2 and the complete graph K n has the largest possible weak edge triangle free detour number n.
Definition 2.2. A vertex v in a graph G is a weak edge triangle free detour vertex if v belongs to every weak edge triangle free detour basis of G. If G has a unique weak edge triangle free detour basis S, then every vertex of S is a weak edge triangle free detour vertex of G. Remark 2.2. A cut-vertex may or may not belong to a weak edge triangle free detour basis of a graph G. For the graph G given in Figure 4, x, y} are the only weak edge triangle free detour bases of G. The cut-vertex w belongs to every weak edge triangle free detour basis so that the cut-vertex w is the unique weak edge triangle free detour vertex of G.
For the graph G in Figure 5, S = {u, v, x, y} is a unique weak edge triangle free detour basis and the cut-vertex w is not a weak edge triangle free detour vertex of G.
In the following theorem we show that there are certain vertices in a connected graph G that are weak edge triangle free detour vertices of G.

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Weak edge triangle free detour number of a graph | S. Sethu Ramalingam and S. Athisayanathan  Theorem 2.2. Every extreme-vertex of a connected graph G belongs to every weak edge triangle free detour set of G. Also, if the set S of all extreme-vertices of G is a weak edge triangle free detour set, then S is the unique weak edge triangle free detour basis for G.
Proof. Let u be an extreme vertex of G and let S be a weak edge triangle free detour set of G.
Clearly u is an end-vertex and u does not lie on any triangle free detour joining a pair of vertices x, y ∈ S. So that S is not a triangle free detour set which is a contradiction.
Case 2. N [u] = K n (n ≥ 3). Since u / ∈ S, then u is an internal vertex of a x − y triangle free detour path say P, for some x, y ∈ S. Let v and w be the neighbours of u on P. Then v and w are not adjacent and so u is not an extreme vertex, which is a contradiction. If S is the set of all extreme vertices of G, then by the first part of this theorem, dn △f (G) ≥ |S|. If S is a triangle free detour set of G, then wdn △f (G) ≤ |S|. Hence wdn △f (G) = |S| and S is the unique weak edge triangle free detour basis for G.
Proof. This follows from Theorem 2.2.
In the following theorems we give the weak edge triangle free detour basis of certain graphs.  Proof. Let S = {u, v} be any set of two vertices of G. If u and v are adjacent, then D △f (u, v) = n − 1 and every edge e ̸ = uv of G lies on the u − v triangle free detour and the both ends of the edge uv belong to S. If u and v are antipodal, then D △f (u, v) = n 2 and every edge e of G lies on a u − v triangle free detour in G. Thus S is a weak edge triangle free detour set of G. Since |S| = 2, S is a weak edge triangle free detour basis of G.
Conversely, assume that S is a weak edge triangle free detour basis of G. Let S ′ be any set of two adjacent vertices or two antipodal vertices of G. Then as in the first part of this theorem, S ′ is a weak edge triangle free detour basis of G. Hence |S| = |S ′ | = 2. Let S = {u, v} ⊆ V. If u and v are not adjacent and u and v are not antipodal, then the edges of the u − v geodesic that do not lie on the u − v triangle free detour in G so that S is not a weak edge triangle free detour set of G, which is a contradiction. Proof. Let G be an odd cycle C n (n ≥ 5). If {u, v} is any set of two adjacent vertices of G. It is clear that D △f (u, v) = n − 1. Then every edge e ̸ = uv of G lies on the u − v triangle free detour and the both ends of the edge uv belong to S so that S is a weak edge triangle free detour set of G.
Since |S| = 2, S is a weak edge triangle free detour basis of G.
Conversely, assume that S is a weak edge triangle free detour basis of G. Let S ′ be any set of two adjacent vertices of G. Then as in the first part of this theorem S ′ is a weak edge triangle free detour basis of G. Hence |S| = |S ′ | = 2. Let S = {u, v} ⊆ V. If u and v are not adjacent, then the edges of u − v geodesic do not lie on the u − v triangle free detour in G so that S is not a weak edge triangle free detour set of G, which is a contradiction. Thus S consists of any two adjacent vertices of G.
Theorem 2.6. Let G be a complete bipartite graph K n,m (2 ≤ n ≤ m). Then a set S ⊆ V is a weak edge triangle free detour basis of G if and only if S consists of any two vertices of G.
Proof. Let G be the complete bipartite graph K n,m . Let X and Y be bipartite sets of G with |X| = n and |Y | = m. Let S = {u, v} be any set of two vertices of G.
Let xy ∈ E such that x ∈ X and y ∈ Y. If x ̸ = u, then the edge xy lies on the u − v triangle free detour P : u, y, x, . . . , v of length 2n − 2. If x = u, then the edge xy lies on the u − v triangle free detour P : u = x, y, . . . , v of length 2n − 2. Hence S is a weak edge triangle free detour basis of G.
Case 2. Let u, v ∈ Y. It is clear that D △f (u, v) = 2n. Let xy ∈ E such that x ∈ X and y ∈ Y. If y ̸ = v, then the edge xy lies on the u − v triangle free detour P : u, x, y, . . . , v of length 2n. If y = v, then the edge xy lies on the v − u triangle free detour P : v = y, x, . . . , u of length 2n. Hence S is a weak edge triangle free detour set of G. In all cases, since |S| = 2, S is a weak edge triangle free detour basis of G.
Case 3. u ∈ X and v ∈ Y. It is clear that D △f (u, v) = 2n − 1. Let xy ∈ E. If xy = uv, then both of its ends are in S. Let xy ̸ = uv be such that x ∈ X and y ∈ Y. If x ̸ = u and y ̸ = v, then the edge xy lies on the u − v triangle free detour P : u, y, x, . . . , v of length 2n − 1. If x = u and y ̸ = v, then the edge xy lies on the u − v triangle free detour P : u = x, y, . . . , v of length 2n − 1. Hence S is a weak edge triangle free detour set of G.
Conversely, let S be a weak edge triangle free detour basis of G. Let S ′ be any set consisting of two vertices of G. Then as in the first part of this theorem, S ′ is a weak edge triangle free detour basis of G. Hence |S| = |S ′ | = 2 and it follows that S consists of any two vertices of G.
Theorem 2.7. Let G be the wheel W n = K 1 + C n−1 (n ≥ 6). Then a set S ⊆ V is a weak edge triangle free detour basis of G if and only if S consists of every vertex of G.
Proof. Let K 1 = {w} and C n−1 : v 1 , v 2 , . . . , v n−1 , v 1 be the cycle of length n − 1. Let W n = K 1 + C n−1 (n ≥ 6) be a wheel and S be a weak edge triangle free detour basis consists of every vertex of G. Then all the edges of C n−1 lie on the triangle free detour joining any pair of two adjacent vertices of C n−1 . Also D △f (w, v i ) = 1(1 ≤ i ≤ n − 1), then every edge of W n lies on a triangle free detour joining a pair of vertices of S. Hence S is a weak edge triangle free detour set of G. Since |S| = n, S is a weak edge triangle free detour basis of G.
Conversely, let S be a weak edge triangle free detour basis of G. Let S 1 be any set consisting of every vertex of G. Then as in the first part of this Theorem, S 1 is a weak edge triangle free detour basis of G. Hence |S| = |S 1 | = n and it follows that S consists of every vertex of G.  The following theorems give realization results.
Theorem 2.8. For any two positive integers k and n with 2 ≤ k ≤ n, there exists a connected graph G of order n with wdn △f (G) = k.
Proof. Case 1. 2 ≤ k = n. Any complete graph G has the desired property. Case 2. 2 ≤ k < n, let P be a path of order n − k + 2. Then the graph G obtained from P by adding k − 2 new vertices to P and joining them to any cut-vertex of P is a tree of order n and so by Corollary 2.1, wdn △f (G) = k. Theorem 2.9. For each positive integer k ≥ 3, there exists a connected graph G with a vertex v of degree k in G such that v does not belongs to a weak edge triangle free detour basis of G and wdn △f (G) = k.
Proof. For k ≥ 2, let G be the graph obtained from the complete graph is a weak edge triangle free detour set of G. However, S ∪ {v 2 , v 3 } is a weak edge triangle free detour set of G and hence by Theorem 2.2, S ∪ {v 2 , v 3 } is a weak edge triangle free detour basis of G so that wdn △f (G) = k.  Case 2. a = b − 1. Consider the graph G given in Figure 6. Let S = {u 1 , u 2 , . . . , u a−1 , u a } be the set of all end vertices of G. By Theorems 1.1 and 2.2, S is contained in every triangle free detour set and every weak edge triangle free detour set of G. It is easily seen that S is a triangle free detour set of G and so dn △f (G) = a, but S is not a weak edge triangle free detour set of G. Let T = S ∪ {v 3 }. Then T is a weak edge triangle free detour set of G and so wdn △f (G) = b = a + 1.  Proof. Case 1. For 4 ≤ a = b, any tree with a end vertices has the desired properties, By Corollary 2.1.
Case 2. For 4 ≤ a < b. Let P 4 : x, u 1 , u 2 , u 3 be path of order 4. Let G 1 be the graph obtained from the path P 4 by adding a − 3 new vertices w 1 , w 2 , . . . , w a−3 and joining each vertex Let G 2 be the graph obtained from the graph G 1 by adding b − a + 2 new vertices v 1 , v 2 , . . . , v b−a+2 and joining each vertex v j (1 ≤ j ≤ b−a+2) to u 1 and u 2 in P 4 . Let G = G 2 be the required graph of order b+3 is shown in Figure 8. Since S = {x, w 1 , w 2 , . . . , w a−3 } is the set of all end vertices of G, then every weak edge detour set of G contains S and S is not a weak edge detour set of G. Let S 1 = S ∪ {u 1 , u 2 }. It is easily verified that S 1 is a weak edge detour set of G and so that dn w (G) = |S 1 | = a. Next, we show that wdn △f (G) = b. By Theorem 2.2, every weak edge triangle free detour set of G contains S. Clearly, S is not a weak edge triangle free detour set of G. It is easily verified that each v i (1 ≤ i ≤ b − a + 2) must belong to every weak edge triangle free detour set of G. Thus T = S ∪ {v 1 , v 2 , . . . , v b−a+2 } is a weak edge triangle free detour set of G, it follows from Theorem 2.2 that T is a weak edge triangle free detour basis of G and so wdn △f (G) = b.

Weak edge triangle free detour number and triangle free detour diameter of a graph
We have seen that by Theorem 1.2, dn △f (G) ≤ n − D △f + 1. However, in the case of weak edge triangle free detour number of a graph, this is not true.
Remark 3.1. In the case of weak edge triangle free detour number wdn △f (G) of a graph G, there are graphs for which wdn △f (G) = n − D △f + 1, wdn △f (G) > n − D △f + 1 and wdn △f (G) < n − D △f + 1. For any cycle C n of order n ≥ 4, D △f = n − 1 and wdn △f (C n ) = 2 so that wdn △f (G) = n − D △f + 1. For any wheel W n of order n ≥ 6, D △f = n − 2 and wdn △f (W n ) = n so that wdn △f (W n ) > n − D △f + 1. For the graph G in Figure 9, n = 6, D △f = 4 and wdn △f (G) = 2 so that wdn △f (W n ) < n − D △f + 1.    T is a weak edge triangle free detour set of G and hence it follows from Theorem 2.2 that T is a weak edge triangle free detour basis of G so that wdn △f (G) = c. Subcase 1.2. Let a be an even integer. Construct the graph H as above Subcase(1.1) in Figure  10. Then G is obtained from H by adding c − 3 new vertices w 1 , w 2 , w 3 , . . . , w c−3 to H and joining each w i (1 ≤ i ≤ c − 3) to the vertex u b−a−1 and is shown in Figure 11. It is easily verified that a ≤ e △f (x) ≤ b for any vertex x in G and e △f (v 0 ) = a and e △f (v 1 ) = b. Thus R △f = a and D △f = b. Now, we show that wdn △f (G) = c. Let S = {w 1 , w 2 , . . . , w c−2 , u b−a } be the set of all end-vertices of G. As in Case 1, S is not a weak edge triangle free detour set of G. Let It is easy to see that S 1 is not a weak edge triangle free detour set of G. Clearly the set T = S ∪ {v 1 , v a } is a weak edge triangle free detour set of G. Hence it follows from Theorem 2.2 that T is a weak edge triangle free detour basis of G and wdn △f (G) = c.
. . , v 2a be a cycle of order 2a and W b+2 = K 1 + C b+1 be the wheel with V (C b+1 ) = {u 1 , u 2 , . . . , u b+1 },K 1 = {u 0 }. Let F be the graph obtained from W b+2 and P 2a by identifying u 0 of W b+2 with v 1 of P 2a as shown in Figure 12. It is easily verified that a ≤ e △f (x) ≤ b for any vertex x in G and e △f (v a ) = a and e △f (u 1 ) = b. Thus R △f = a and D △f = b. Let S = {u 1 , u 2 , . . . , u b+1 , v 2a } is a weak edge triangle free detour set of G so that wdn △f (G) = b + 2 = c.  Proof. Case 1. When a is even, let G be the graph obtained from the cycle C a : u 1 , u 2 , . . . , u a , u 1 of order a by adding b − 1 new vertices v 1 , v 2 , . . . , v b−1 and joining each vertex v i (1 ≤ i ≤ b − 1) to u 1 and adding n − a − b + 1 new vertices w 1 , w 2 , . . . , w n−a−b+1 and joining each vertex w i (1 ≤ i ≤ n − a − b + 1) to both u 1 and u 3 and is shown in Figure 13. It is easily verified that the order of the graph G is n and triangle free detour diameter D △f = a. Now, we show that wdn △f (G) = b. Let S = {v 1 , v 2 , . . . , v b−1 } be the set of all end-vertices of G. Since no edge of G other than the edges u 1 v i (1 ≤ i ≤ b−1) lies on a triangle free detour joining a pair of vertices of S, S is not a weak edge triangle free detour set of G. Let T = S ∪ {v}, where v is the antipodal vertex of u 1 in C a . Then every edge of G lies on a triangle free detour joining a vertex v i (1 ≤ i ≤ b − 1) and v so that T is a weak edge triangle free detour set of G. Now, it follows from Theorem 2.2 that T is a weak edge triangle free detour basis of G and so wdn △f (G) = b. Case 2. When a is odd, let G be the graph obtained from the cycle C a : u 1 , u 2 , . . . , u a , u 1 of order a by adding b − 2 new vertices v 1 , v 2 , . . . , v b−2 and joining each vertex v i (1 ≤ i ≤ b − 2) to u 1 and adding n − a − b + 2 new vertices w 1 , w 2 , . . . , w n−a−b+2 and joining each vertex w i (1 ≤ i ≤ n − a − b + 2) to both u 1 and u 3 . It is easily verified that the order of the graph G is n, triangle free detour diameter D △f = a and is shown in Figure 14. Now, we show that wdn △f (G) = b. Let S = {v 1 , v 2 , . . . , v b−2 } be the set of all end-vertices of G. As in Case 1, S is not a weak edge triangle free detour set of G. Let S 1 = S ∪ {v}, where v is the any vertex of G such that v ̸ = v i (1 ≤ i ≤ b − 2). It is easy to see that S 1 is not a weak edge triangle free detour set of G. Now, the set T = S ∪ {u 2 , u a } is a weak edge triangle free detour set of G. Hence it follows from Theorem 2.2 that T is a weak edge triangle free detour basis of G and so wdn △f (G) = b.