Fault-tolerant edge metric dimension of certain families of graphs

Abstract: Let WE = {w1,w2, . . . ,wk} be an ordered set of vertices of graph G and let e be an edge of G. Suppose d(x, e) denotes distance between edge e and vertex x of G, defined as d(e, x) = d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) , d(e2, x). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e,w1), d(e,w2), . . . , d(e,wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). In this paper, we initiate the study of fault-tolerant edge metric dimension. Let ẂE be edge metric generator of graph G, then ẂE is called fault-tolerant edge metric generator of G if ẂE \ {v} is also an edge metric generator of graph G for every v ∈ ẂE. A fault-tolerant edge metric generator of minimum cardinality is a fault-tolerant edge metric basis for graph G, and its cardinality is called fault-tolerant edge metric dimension of G. We also computed the fault-tolerant edge metric dimension of path, cycle, complete graph, cycle with chord graph, tadpole graph and kayak paddle graph.


Introduction and preliminaries
Suppose that G is connected, simple and undirected graph having edge set E(G) and vertex set V(G), respectively. The order of graph G is |V(G)| and size of graph G is |E(G)|. Moreover, ∆(G) and δ(G) represent the maximum and minimum degree of graph G respectively. Let W = {v 1 , v 2 , . . . , v k } be an ordered set of V(G) and let u be a vertex of G. The representation r(u | W) of u with respect to W is the k-tuple (d(u, v 1 ), d(u, v 2 ), . . . , d(u, v k )). If distinct vertices of G have distinct representation with respect to W, then W is called metric generator for G. A metric generator of minimum cardinality is Lemma 1.3. [3] For a connected graph G of order n, edim(G) ≥ 1 + log 2 δ(G) .
From the definition of fault-tolerant edge metric dimension, it can be seen that Lemma 1.4. For a connected graph G, The rest of paper is structured as follows: In the second section, we will study the fault-tolerant edge metric dimension of family of path, cycle and complete graphs. In third section, we will investigate the fault-tolerant edge metric dimension of family of cycle with chord graphs C m n . In fourth section, fault-tolerant edge metric dimension of family of tadpole graphs G l n will be determined. In last section, we will compute the fault-tolerant edge metric dimension of family of kayak paddle graphs G l n,m .

Fault-tolerant edge metric dimension of family of path, cycle and complete graphs
In this section, we will investigate the fault-tolerant edge metric dimension of family of paths, cycles and complete graphs. The family P n have V(P n ) = {u 1 , u 2 , . . . , u n } and E(P n ) = {u i u i+1 : 1 ≤ i ≤ n − 1}. The family P n for n = 10 is shown in Figure 1. The following theorem tells us the edge metric dimension of P n .
Proof. In order to compute fault-tolerant edge metric dimension of P n , we havé W E = {u 1 , u n } ⊂ V(P n ), we have to show thatẂ E is a fault-tolerant edge metric generator of P n . For this, we give representations of each edge of P n .
We see that there are no two tuples having the same representations. This shows that fault-tolerant edge metric dimension of P n is less than or equal to 2. Since by Lemma 1.4, P n has fault-tolerant edge metric dimension greater than or equal to 2. Hence fault-tolerant edge metric dimension is equal to 2.
The family C n have V(C n ) = {u 1 , u 2 , . . . , u n } and E(C n ) = {u i u i+1 : 1 ≤ i ≤ n − 1} ∪ {u n u 1 }. The family C n for n = 15 is shown in Figure 2. The following theorem tells us the edge metric dimension of C n . Theorem 2.3. [3] For any integer n ≥ 3, edim(C n ) = 2. Proof. In order to compute fault-tolerant edge metric dimension of C n , we have the following cases.
, we have to show thatẂ E is a fault-tolerant edge metric generator of C n . For this, we give representations of each edge of C n .
(1, 0, 0), if i = 2; Case (ii). n is even. TakeẂ E = {u 1 , u 2 , u 3 } ⊂ V(C n ), we have to show thatẂ E is a fault-tolerant edge metric generator of C n . For this, we give representations of each edge of C n .
We see that there are no two tuples having the same representations. This shows that fault-tolerant edge metric dimension of C n is less than or equal to 3. Since by Lemma 1.4, C n has fault-tolerant edge metric dimension greater than or equal to 3. Hence fault-tolerant edge metric dimension of C n is equal to 3.
The family C m n for n = 20 and m = 9 is shown in Figure 3. The following theorem tells us the edge metric dimension of C m n . Now, we will compute the fault-tolerant edge metric dimension of C m n .
Proof. In order to compute fault-tolerant edge metric dimension of C m n , we have the following cases. Case (i). Both n and m are even.
, we have to show thatẂ E is a fault-tolerant edge metric generator of C m n . For this, we give representations of each edge of C m n .
Case (ii). n is odd and m is even.
For this, we give representations of each edge of C m n .
Case (iii). n is even and m is odd.
, we have to show thatẂ E is a fault-tolerant edge metric generator of C m n . For this, we give representations of each edge of C m n .
Case (iv). Both n and m are odd.
, we have to show thatẂ E is a fault-tolerant edge metric generator of C m n . For this, we give representations of each edge of C m n .
We see that there are no two tuples having the same representations in all the four cases. This shows that fault-tolerant edge metric dimension of C m n is less than or equal to 3. Since by Lemma 1.4, C m n is not a path so fault-tolerant edge metric dimension of C m n is greater than or equal to 3. Hence fault-tolerant edge metric dimension of C m n is 3.

Fault-tolerant edge metric dimension of family of tadpole graphs G l n
In this section, we will compute the fault-tolerant edge metric dimension of family of tadpole graphs G l n . The family G l n have V(G l n ) = {v 1 , v 2 , . . . , v n , u 1 , u 2 , . . . , u l } and E(G l n ) = {v i v i+1 : 1 ≤ i ≤ n − 1} ∪ {u s u s+1 , : 1 ≤ s ≤ l − 1} ∪ {v n u 1 , u 1 v 1 }. The graph G l n for n = 9 and l = 7 is shown in Figure 4. The following theorem tells us the edge metric dimension of G l n .
Theorem 4.1. [29] For all n ≥ 2, l ≥ 3, edim(G l n ) = 2. Proof. In order to compute fault-tolerant edge metric dimension of G l n , we have the following cases. Case (i). n is odd. LetẂ E = {v 1 , v n , u m } ⊂ V(G l n ), we have to show thatẂ E is a fault-tolerant edge metric generator of G l n . For this, we give representations of each edge of G l n .
Case (ii). n is even. LetẂ E = {v 1 , v n , u m } ⊂ V(G l n ), we have to show thatẂ E is a fault-tolerant edge metric generator of G l n . For this, we give representations of each edge of G l n .
We see that there are no two tuples having the same representations. This shows that fault-tolerant edge metric dimension of G l n is less than or equal to 3 and now we try to show that fault-tolerant edge metric dimension of G l n is greater than or equal to 3. Since by Lemma 1.4, G l n is not a path so faulttolerant edge metric dimension of G l n is greater than or equal to 3. Hence fault-tolerant edge metric dimension of G l n is equal to 3.

Fault-tolerant edge metric dimension of family of kayak paddle graphs G l n,m
In this section, we will compute the edge metric dimension of family of kayak paddle graphs G l n,m . The family G l n,m have V(G l n,m ) = {u 1 , u 2 , . . . , u m , v 1 , v 2 , . . . , v n , w 1 , w 2 , . . . , w l } and E(G l n,m ) = {v i v i+1 : The family G l n,m for n = 8, m = 5 and l = 4 is shown in Figure 5. The following theorem tells us the edge metric dimension of G l n,m . Theorem 5.1. [29] For every n ≥ 2, m ≥ 2 and l ≥ 4, edim(G l n,m ) = 2. Figure 5. Kayak Paddle graph G 4 8,5 . Now, we will compute the fault-tolerant edge metric dimension of G l n,m . Theorem 5.2. For n ≥ 2, m ≥ 2 and l ≥ 4, f edim(G l n,m ) = 4. Proof. In order to compute fault-tolerant edge metric dimension of G l n,m , we have the following cases. Case (i). n is odd and m is even. LetẂ E = {v 1 , v 2 , u 1 , u 2 } ⊂ V(G l n,m ), we have to show thatẂ E is a fault-tolerant edge metric generator of G l n,m . For this, we give representations of each edge of G l n,m .
Case (ii). Both n and m are even. LetẂ E = {v 1 , v 2 , u 1 , u 2 } ⊂ V(G l n,m ), we have to show thatẂ E is a fault-tolerant edge metric generator of G l n,m . For this, we give representations of each edge of G l n,m .
Case (iii). Both n and m are odd. LetẂ E = {v 1 , v 2 , u 1 , u 2 } ⊂ V(G l n,m ), we have to show thatẂ E is a fault-tolerant edge metric generator of G l n,m . For this, we give representations of each edge of G l n,m .
We see that there are no two tuples having the same representations. This shows that fault-tolerant edge metric dimension of G l n,m is less than or equal to 4 and now we try to show that fault-tolerant edge metric dimension of G l n,m is grater than or equal to 4. For this purpose, we have to show that there is no fault-tolerant edge metric generator having cardinality 3, we suppose on contrary that fault-tolerant edge metric dimension of G l n,m is 3 and let W E = {v i , v j , v k }. Then the Table 1 shows all order pairs of edges (e, f ) for which r(e|Ẃ E ) = r( f |Ẃ E ). Table 1. (e, f ) for which r(e|Ẃ E ) = r( f |Ẃ E ). Conditions on i, j and k (e, f ) 1 ≤ i, j, k ≤ n (u 1 w l , u m w l ) 1 ≤ i, j ≤ n, 1 ≤ k ≤ l (u 1 w l , u m w l ) 1 ≤ i ≤ n and 1 ≤ j, k ≤ l (u 1 w l , u m w l ) 1 ≤ i, j, k ≤ l (u 1 w l , u m w l ) 1 ≤ i ≤ n, 1 ≤ j ≤ l and 1 ≤ k ≤ m If we takeẂ E \ {v k } (u 1 w l , u m w l ) 1 ≤ i, j ≤ n, and 1 ≤ k ≤ m If we takeẂ E \ {v k } (w 1 w 2 , w 1 v 1 ) or (w 1 w 2 , w 1 v n ) In all possibilities, we conclude that there is no fault-tolerant edge metric generator of 3 vertices. Hence fault-tolerant edge metric dimension of G l n,m is 4.

Conclusions
In this paper, we have computed the fault-tolerant edge metric dimension of some planar graphs path, cycle, complete, cycle with chord, tadpole and kayak paddle. It is observed that the fault-tolerant edge metric dimension of these graphs is constant and does not depend on the number of vertices. It is concluded that the fault-tolerant edge metric dimension of families of path graphs is two, the fault-tolerant edge metric dimension of families of cycle graphs, cycle with chord graphs, tadpole graphs is three and the fault-tolerant edge metric dimension of kayak paddle graphs is found to be four. Here we end with an open problem.

Open Problem
Characterize all families of graphs for which difference of fault-tolerant metric dimension and edge metric dimension is one.