Fault-Tolerant Metric and Partition Dimension of Graphs

Fault-Tolerant Metric and Partition Dimension of Graphs Muhammad Anwar Chaudhry, Imran Javaid? , Muhammad Salman Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University Multan, Pakistan

Abstract. A set W of vertices in a graph G is called a resolving set for G if for every pair of distinct vertices u and v of G there exists a vertex w ∈ W such that the distance between u and w is different from the distance between v and w. The cardinality of a minimum resolving set is called the metric dimension of G, denoted by β(G). A resolving set W 0 for G is fault-tolerant if W 0 \{w}, for each w in W 0 , is also a resolving set and the fault-tolerant metric dimension of G is the minimum cardinality of such a set, denoted by β 0 (G). We characterize all the graphs G such that β 0 (G) − β(G) = 1. A k-partition Π = {S1 , S2 , . . . , Sk } of V (G) is resolving if for every pair of distinct vertices u, v in G, there is a set Si in Π so that the minimum distance between u and a vertex of Si is different from the minimum distance between v and a vertex of Si . A resolving partition Π is fault-tolerant if for every pair of distinct vertices u and v in V (G), there are at least two sets Si , Sj in Π so that the minimum distance between u and a vertex of Si and a vertex of Sj is different from the minimum distance between v and a vertex of Si and a vertex of Sj . The cardinality of a minimum fault-tolerant resolving partition is called the fault-tolerant partition dimension, denoted by P(G). In this paper, we show that every pair a, b of positive integers with b ≥ 6 and d 2b e + 1 ≤ a ≤ b − 2 is realizable as the fault-tolerant metric dimension and the fault-tolerant partition dimension of some connected graphs. Also, we show that P(G) = n if and only if G = Kn or G = Kn − e.

Keywords: metric dimension, fault-tolerant metric dimension, partition dimension, fault-tolerant partition dimension. 2000 Mathematics Subject Classification: 05C12 ?

Corresponding author: [email protected]

2

1

M. A. Chaudhry, I. Javaid, M. Salman

Introduction

For a connected graph G with vertex set V (G) and edge set E(G), the distance between two vertices u and v in V (G) is the minimum number of edges in a u − v path and is denoted by d(u, v). The eccentricity ecc(v) of a vertex v in G is ecc(v) = maxu∈V d(v, u). The diameter of G, denoted by D, is the number maxv∈V ecc(v). Two vertices u and v in G are called the diametral vertices if d(u, v) = D. Given an ordered set W = {w1 , w2 , . . . , wk } ⊆ V (G), for each v ∈ V (G) the code of v with respect to W is (d(v, w1 ), d(v, w2 ), . . . , d(v, wk )), denoted by cW (v). The set W is called a resolving set for G if all vertices of G have distinct codes. The minimum cardinality of a resolving set of G is called the metric dimension of G and is denoted by β(G). A resolving set of order β(G) is called a metric basis of G [3]. Elements of bases were referred to as censors in an application given in [4]. If one of the censors does not work properly, we will not have enough information to deal with the intruder (fire, thief etc). In order to overcome this kind of problems, concept of fault-tolerant metric dimension was presented in [10]. Fault-tolerant resolving sets provide correct information even when one of the censors is not working. This concept is defined as follows: A resolving set W 0 of a graph G is fault-tolerant if W 0 \{w} is also a resolving set, for each w in W 0 . The fault-tolerant metric dimension of G is the minimum cardinality of a fault-tolerant resolving set, denoted by β 0 (G). A fault-tolerant resolving set of order β 0 (G) is called a fault-tolerant metric basis. For a vertex u of G and a subset S of V (G), the distance between u and S is d(u, S) = mins∈S d(u, s). Let Π = {S1 , S2 , . . . , Sk } be an ordered kpartition of V (G) and let v be a vertex of G. The k-tuple (d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )) is the code cΠ (v) of v with respect to the partition Π. A partition Π is called a resolving partition if for distinct vertices u and v of G, cΠ (u) 6= cΠ (v). The partition dimension of G is the cardinality of a minimum resolving partition, denoted by pd(G) [5, 6]. A resolving partition Π is f ault-tolerant if for every pair of distinct vertices u and v in V (G), the codes cΠ (v) and cΠ (u) differ by at least two coordinates. The cardinality of a minimum fault-tolerant resolving partition is called the fault-tolerant partition dimension, denoted by P(G) [11]. For an illustration of this concept, consider the graph G in Fig. 1. Let Π1 = {S1 , S2 , S3 , S4 }, where S1 = {v1 , v2 }, S2 = {v3 }, S3 = {v4 }, and S4 = {v5 }. Then the five codes with respect to Π1 are cΠ1 (v1 ) = (0, 1, 1, 1), cΠ1 (v2 ) = (0, 1, 2, 2), cΠ1 (v3 ) = (1, 0, 1, 2), cΠ1 (v4 ) = (1, 1, 0, 1), cΠ1 (v5 ) = (1, 2, 1, 0).

Fault-Tolerant Metric and Partition Dimension of Graphs

3

v1

v G:

v

5

v 4

2

v3

Fig. 1. Illustrating fault-tolerant resolving partition.

Since these codes are distinct on at least two coordinates, Π1 is a faulttolerant resolving partition of G. Now, let Π2 = {S1 , S2 , S3 }, where S1 = {v1 , v2 }, S2 = {v3 }, and S3 = {v4 , v5 }. Then the corresponding codes are cΠ2 (v1 ) = (0, 1, 1), cΠ2 (v2 ) = (0, 1, 2), cΠ2 (v3 ) = (1, 0, 1), cΠ2 (v4 ) = (1, 1, 0), cΠ2 (v5 ) = (1, 2, 0). Π2 is also a resolving partition but not a fault-tolerant resolving partition for G since the codes of v1 and v2 differ by one coordinate only. Moreover, since no 3-partition is a fault-tolerant resolving partition for G, it follows that Π1 is a minimum fault-tolerant resolving partition of G, and P(G) = 4. Slater introduced the concept of resolving sets in 1975 when working with U. S. Sonar and Coast Guard Loran (Long range aids to navigation) stations [15]. Independently, Harary and Melter studied resolving sets in 1976 and used the term ‘metric dimension’ for it [9], the terminology which we have adopted. It was noted in [8], and an explicit construction was given in [13], showing that finding the metric dimension of a graph is NPhard. Their interest in this invariant was motivated by the navigation of robots in a graph space. A resolving set for a graph corresponds to the presence of distinctively labeled “landmark” vertices in the graph. It is assumed that a robot navigating a graph can sense its distance from each of the landmarks and hence uniquely determine its location in the graph. They also gave approximation algorithms for this invariant and established properties of graphs with metric dimension 2. Recently, these concepts have been reinvestigated by Johnson [12] of the Pharmaceutical Company while attempting to develop a capability of large data sets of chemical graphs.

4

M. A. Chaudhry, I. Javaid, M. Salman

If d(x, u) 6= d(y, u), we shall say that vertex u separates vertices x and y. Let S(v) denotes the set of vertices of G which are separated by v and is called the separator of v. Since the distance of a vertex v with itself is zero so v contained in S(v). Likewise S(A) denotes the set of vertices of G which are separated by elements of A ⊆ V (G). If A = {v1 , v2 , . . . , vl } then S(A) and S(v1 , v2 , . . . , vl ) can be used interchangeably. So when building a minimum resolving set for G heuristically it makes sense to choose the vertices with separators as large as possible. We give the following algorithm to determine the metric dimension of a finite graph G. The Greedy Resolving Algorithm. Step 1. Find a vertex v1 ∈ V (G) with |S(v1 )| as large as possible and put this vertex into a set W . Step 2. Stop if S(W ) = V (G) otherwise find v2 ∈ V (G) with S(v1 , v2 ) as large as possible and put it into W . If S(W ) = V (G) then stop otherwise find another vertex v3 ∈ V (G) with S(v1 , v2 , v3 ) as large as possible and put it into W and continue this process until we find an integer k such that S(v1 , v2 , . . . , vk ) = V (G). If we put {v1 , v2 , . . . , vk } = W then, since v1 , v2 , . . . , vk are chosen with separator as large as possible, there is no l < k for which S(v1 , v2 , . . . , vl ) = V (G) which shows that W is a minimum resolving set for G and hence |W | is the metric dimension of G. The concept of metric dimension helps in studying another notion which can be used to identify the automorphism group of a graph, called the determining number of a graph: A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U . The determining number is the smallest size of a determining set, and is denoted by Det(G). This notion was introduces by Boutin in [2]. Independently, Erwin and Harary studied this notion and used the terms fixing set and fixing number [7]. The determining number can be obtained by using its connection with the metric dimension. The metric dimension of a graph provides an upper bound for the determining number of that graph. They proved that the determining number of a graph is bounded above by its metric dimension. Theorem 1. [7] For every connected graph G, Det(G) ≤ β(G). The remainder of the paper is divided into two sections. In the second section, we study the relationship between metric basis and fault-tolerant metric basis. Also, we characterize all the graphs G such that β 0 (G) − β(G) = 1. In the third section, we show that, for every pair a, b of positive integers with b ≥ 6 and d 2b e + 1 ≤ a ≤ b − 2, there exists a connected graph G such that P(G) = a and β 0 (G) = b. Also, we prove that fault-tolerant

Fault-Tolerant Metric and Partition Dimension of Graphs

5

partition dimension of a connected graph G is n if and only if G is one of the graphs Kn , Kn − e.

2

Fault-Tolerant Metric Dimension of Graphs

From the definition of metric dimension, it can be observed that the property of a given set W of vertices of a graph G to be a resolving set of G can be verified by investigating only the vertices of V (G)\W . This is because every vertex w ∈ W is the only vertex of G whose distance from w is 0. In the case of fault-tolerant metric dimension, one has to see whether every pair of vertices in G is being separated by at least two vertices in W 0 , a set which is candidate for being fault-tolerant resolving. A useful property in determining fault-tolerant metric dimension of graphs is the following lemma. Lemma 1. [11] A resolving set W 0 of a graph G is fault-tolerant if and only if every pair of vertices in G is resolved by at least two vertices of W 0 . From the definition of fault-tolerant metric dimension it can be seen that β 0 (G) ≥ β(G) + 1. (1) In [11], it was shown that the difference between the metric dimension and the fault-tolerant metric dimension can be arbitrarily large. Corollary 1. [11] For every natural number k, there exists a graph G such that β 0 (G) ≥ β(G) + k. Let Km,n denotes the complete bipartite graph with partite sets X and Y of cardinalities m ≥ 1 and n ≥ 1, respectively. In [9], it was shown that β(Km,n ) = m + n − 2. Any set of cardinality m + n − 1 of the vertices of Km,n does not form a fault-tolerant resolving set for Km,n . Hence we have the following result: Proposition 1. For any pair (m, n) of integers, β 0 (Km,n ) = m + n. Note that, if G is complete or complete bipartite graph then G has fault-tolerant metric dimension n and the only graph having fault-tolerant metric dimension 2 is a path. Thus one can see that 2 ≤ β 0 (G) ≤ n. Now we observe the relationship between the metric basis and the faulttolerant metric basis of a graph. The graph having the vertex set V = Z2 and the edge set E = {{u, v} : u − v ∈ {(0, ±1), (±1, 0)}} determined by d4 metric is called the square grid where d4 ((i, j), (i0 , j 0 )) = |i − i0 | + |j − j 0 | for any two vertices in Z2 . The index 4 is appropriate because it represents the number of points at

6

M. A. Chaudhry, I. Javaid, M. Salman

distance one from a given point with respect to d4 metric. The set of vertices (i, j) ∈ Z2 with |i| ≤ n and |j| ≤ m in square grid is called a rectangle, denoted by Rn,m . In [11], it was shown that there is an infinite class of rectangles Rn,m such that some fault-tolerant metric bases contain a basis of Rn,m and others contain no basis of Rn,m . We make the following three observations about the relationship between metric basis and fault-tolerant metric basis in some well known classes of graphs. Proofs are straightforward and so omitted. Observation 1. If G is a complete graph of order at least 3 or a complete bipartite graph, then every fault-tolerant metric basis W 0 contains a basis W of G as a proper subset. Let Cn denotes the cycle on n ≥ 3 vertices. Two vertices u and v of Cn are antipodal if d(u, v) = n2 otherwise, they are non-antipodal. Note that every two vertices are non-antipodal in an odd cycle. In [13], it was shown that β(Cn ) = 2 and two vertices forms a resolving set for Cn if and only if they are non-antipodal. In [11], it was shown that β 0 (Cn ) = 3 and a set W 0 of three vertices forms a fault-tolerant resolving set for Cn if and only if no two vertices of W 0 are antipodal. Thus we have the following observation: Observation 2. If G is a cycle of order at least 3, then every fault-tolerant metric basis W 0 contains a basis W of G as a proper subset. A branch of a tree T at a vertex v is the subgraph induced by v and one of the component of T \ {v}. A branch of T at v which is a path is called a branch path when degree of v is at least 3. Hernando et al. [10] studied the fault-tolerant metric dimension of trees. The following theorem was proved for the metric dimension of trees in [15]. Theorem 2. [15] Let T be a tree with set L of endpoints with |L| ≥ 3. Let L1 , L2 , . . . , Lk be the components of the subtree induced by the set of all branch paths, and let ei be the number of branch paths in T that are in Li then β(T ) = |L| − k, and S is a metric basis iff it consists of exactly ei − 1 of the branch paths of Li , for each Li ,1 ≤ i ≤ k. Following theorem for fault-tolerant metric basis of trees was proved in [10]. Theorem 3. [10] Let T ba a tree with set L of endpoints with |L| ≥ 3. Let L1 , L2 , . . . Lk be the components of the subtree induced by the set of all branch paths, and let ei be the number of branch paths in T that are in Li . Let Ei be the set of endpoints corresponding to branch paths where ei = 1 then β 0 (T ) = |L \ E1 | and L \ E1 is fault-tolerant metric basis.

Fault-Tolerant Metric and Partition Dimension of Graphs

7

From Theorem 2 and Theorem 3 we have the following observation: Observation 3. If T is a tree, then every fault-tolerant metric basis contains a basis of T as a proper subset. Now we characterize all the graphs G such that the cardinality of a fault-tolerant metric basis of G is one more than the cardinality of a metric basis of G. For this purpose, we need the following definition given in [1]. Definition 1. Bases are said to have the exchange property in a graph G if whenever U and W are bases of G and w ∈ W then there is u ∈ U such that U − {u} ∪ {w} is a basis of G. The following theorem characterizes all the graphs G such that β 0 (G) − β(G) = 1. Theorem 4. Let G be a graph. β 0 (G) − β(G) = 1 if and only if exchange property holds for bases of G. Proof. Let G be a graph. Suppose U and W are bases of G and w ∈ W then there is u ∈ U so that U − {u} ∪ {w} is a basis of G. Then W 0 = U ∪ {w} is a fault-tolerant basis of G which implies that β 0 (G) − β(G) = 1. Conversely, let W be a basis of cardinality β(G) and W 0 be a faulttolerant basis of cardinality β 0 (G) of G. Since β 0 (G) − β(G) = 1 then U = W 0 − {w0 }, for any w0 ∈ W 0 , is a basis of G. We assume contrarily that for some u ∈ U there exist no w ∈ W so that W − {w} ∪ {u} = S is a basis of G but then β 0 (G) − β(G) 6= 1, a contradiction.  By [11] and [14], β 0 (Rn,m ) − β(Rn,m ) = 2 which implies that the exchange property of bases does not hold in an infinite family of rectangles Rn,m . Next, we give examples of regular graphs (graphs in which all the vertices have same degree) having the exchange property of bases. Example 1. The graph having the vertex set V = Z2 and the edge set 0 0 determined by the Euclidean metric, p i, e., two vertices (i, j), (i , j ) ∈ V 0 0 0 2 0 2 forms an edge if d((i, j), (i , j )) = (i − i ) + (j − j ) , is referred to as digital plane, denoted by (Z2 , ε). By the digital plane we mean the set of all points which have integral coordinates. It was shown in [14] that β(Z2 , ε)=3 and a set of any three points which are not collinear form a metric basis for (Z2 , ε). It easily follows that fault-tolerant metric dimension of digital plane is 4 and fault-tolerant metric bases consist of a set of four points such that any three points are not collinear.

8

M. A. Chaudhry, I. Javaid, M. Salman

Example 2. The graph the having vertex set V = Z2 and the edge set E = {{u, v} : u−v ∈ {(0, ±1), (±1, 0), (1, ±1), (−1, ±1)}} determined by d8 metric is referred to as king grid, denoted by (Z2 , ε8 ) where d8 ((i, j), (i0 , j 0 )) = max(|i − i0 |, |j − j 0 |). The index 8 is appropriate because it represents the number of points at distance one from a given point with respect to d8 metric. The set of vertices (i, j) ∈ Z2 with |i| ≤ n and |j| ≤ n in king grid is called a square, denoted by Sn . It was shown in [14] that the metric dimension of Sn is three. It easily follows that the fault-tolerant metric dimension of Sn is four and fault-tolerant metric basis consists of the following points (n, n), (−n, n), (−n, −n), (n, −n) of Sn . In both examples, by Theorem 4, the exchange property of bases holds since the fault-tolerant metric dimension of these graphs is one more than their metric dimension.

3

Fault-Tolerant Partition Dimension of Graphs

We shall say that a class S separates the vertices x and y of G if d(x, S) 6= d(y, S). A partition Π separates x and y if a class of Π separates x and y. From these definitions it can be observed that the property of a given partition Π of a graph G to be a resolving partition of G can be verified by investigating the pairs of vertices in the same class. Indeed, d(x, Si ) = 0 for every vertex x ∈ Si but d(x, Sj ) 6= 0 with j 6= i. It follows that x ∈ Si and y ∈ Sj are separated either by Si or Sj for every i 6= j. Likewise, it can be observed that the property of a given partition Π of a graph G to be a fault-tolerant resolving partition of G can be verified by investigating that every pair of vertices in the same class is separated by at least two classes of Π. That is, for two classes Si and Sj (i 6= j) of a partition Π, d(x, Si ) 6= d(y, Si ) and d(x, Sj ) 6= d(y, Sj ) for all x, y ∈ Sk , k 6= i, j. Thus, it can be observed that pd(G) ≤ P(G).

(2)

The fault-tolerant metric dimension and the fault-tolerant partition dimension are related as shown in [11]. Theorem 5. [11] If G is a non trivial connected graph, then P(G) ≤ β 0 (G) + 1. Following [5], we provide an upper and lower bound for the fault-tolerant partition dimension of a graph in terms of its order and diameter. For an integer n and D with 2 ≤ D < n, we define η(n, D) as the least positive integer ν for which n ≤ (D + 1)ν .

Fault-Tolerant Metric and Partition Dimension of Graphs

9

Theorem 6. Let G be a graph of order n ≥ 3 and diameter D. Then η(n, D) ≤ P(G) ≤ n − D + 2. Proof. For upper bound, let u and v be vertices of G for which d(u, v) = D and let u = v1 , v2 , . . . , vD+1 be a path of length D. Let V (G) = {v1 , v2 , . . . , vD , . . . , vn }, then it is straightforward to show that the partition Π = {S1 , S2 , . . . , Sn−D+2 } of V (G) is a fault-tolerant resolving (n − D + 2)-partition, where S1 = {v1 , v2 , . . . , vD−1 } and Si = {vi+D−2 } for 2 ≤ i ≤ n − D + 2. Now for lower bound, let P(G) = ν and Π be a fault-tolerant resolving ν-partition of V (G). Since each code of a vertex of G is a ν-vector, every coordinate of which is a nonnegative integer not exceeding D and all n codes are distinct, it follows that n ≤ (D + 1)ν . Hence P(G) = ν ≥ η(n, D).  As a consequence of Theorem 6, we have the following corollary: Corollary 2. If G is a graph of order n ≥ 3 and P(G) = n then D ≤ 2. For each integer n ≥ 3, P(G) = 3 if G is a path Pn but converse of this statement is not true since there exists an other family of graphs namely cycles Cn , for each integer n ≥ 3, having fault-tolerant partition dimension three. For each integer n ≥ 3, if G is a complete graph of order n then the fault-tolerant partition dimension of G and G − e is n as shown in the next lemma. Lemma 2. Let Kn (n ≥ 3) be the complete graph, then P(Kn ) = n = P(Kn − e). Proof. It is straightforward to see that P(Kn ) = n. Now we prove that P(Kn − e) = n. We prove this by double inequality. By Theorem 6, P(Kn − e) ≤ n since the diameter of Kn − e is two. For other inequality, we assume contrarily that P(Kn − e) = n − 1 and Π = {S1 , S2 , . . . , Sn−1 } be a fault-tolerant resolving (n − 1)-partition of V (Kn − e). Then only one class of Π, say S1 , contains two vertices of Kn − e. Since D = 2, there are only two diametral vertices in Kn − e. Let us call the diametral vertices x and y, then we have the following three cases: Case 1: If S1 = {x, y} then cΠ (x) = cΠ (y). Case 2: If S1 = {x, u} where u ∈ V (Kn − e) − {y}. Then the codes cΠ (x) and cΠ (u) differ by one coordinate only. Case 3: If S1 = {u, v} where u, v ∈ V (Kn −e)−{x, y} then cΠ (u) = cΠ (v).

10

M. A. Chaudhry, I. Javaid, M. Salman

All three cases give the contradiction to the fact that Π is a faulttolerant resolving partition. Hence P(Kn − e) ≥ n.  Thus we can see that if G is a connected graph of order n ≥ 3, then certainly 3 ≤ P(G) ≤ n. Theorem 7. If Km,n be the complete bipartite  if m+1 P(Km,n ) = max(m, n) + 1 if  max(m, n) if

graph for m, n ≥ 1. Then m − n = 0, |m − n| = 1, |m − n| ≥ 2.

Proof. Let Km,n be the complete bipartite graph with partite sets V = {v1 , v2 , . . . , vm } and U = {u1 , u2 , . . . , un }. We have the following three cases: Case 1: (m − n = 0) It is not difficult to see that the partition Π = {S1 , S2 , . . . , Sm+1 }, where Si = {vi , ui } for 1 ≤ i ≤ m − 1, Sm = {vm } and Sm+1 = {um }, is a fault-tolerant resolving (m + 1)-partition of V (Km,n ) which implies that P(Km,n ) ≤ m + 1. Conversely, suppose that P(Km,n ) = n and Π = {S1 , S2 , . . . , Sn } be a fault-tolerant resolving partition of V (Km,n ). Since each vertex in V is adjacent with all the vertices in U and vice versa, so every class of Π contains exactly one vertex from V and one vertex from U . Without loss of generality, we assume that Si = {vi , ui } for all 1 ≤ i ≤ n then cΠ (vi ) = (1, . . . , 0, . . . , 1) = cΠ (ui ) where 0 is at ith place, a contradiction. Hence P(Km,n ) = m + 1. Case 2: (|m − n| = 1) Without loss of generality, assume that max(m, n) = n. The partition Π = {S1 , S2 , . . . , Sm+2 }, where Si = {vi , ui } for 1 ≤ i ≤ m−1, Sm = {vm }, Sm+1 = {um } and Sm+2 = {vm+1 }, is a fault-tolerant resolving (m + 2)partition of V (Km,n ) which shows that P(Km,n ) ≤ m + 2. Conversely, we assume contrarily that Π = {S1 , S2 , . . . , Sm+1 } is a fault-tolerant resolving partition of V (Km,n ). Since each vertex in V is adjacent to all the vertices in U and vice versa, so we can put exactly one vertex from V and one vertex from U into each class of Π. Thus, without loss of generality, we assume that Si = {vi , ui } for 1 ≤ i ≤ m and Sm+1 = {vm+1 }. Then cΠ (vi ) = (1, . . . , 0, . . . , 1, 2), cΠ (ui ) = (1, . . . , 0, . . . , 1, 1)

Fault-Tolerant Metric and Partition Dimension of Graphs

11

for all 1 ≤ i ≤ m, where 0 is at ith place. That is, the codes cΠ (vi ) and cΠ (ui ) differ by one coordinate which is a contradiction to the fact that Π is a fault-tolerant resolving partition. Hence P(Km,n ) = m + 2 = max(m, n) + 1. Case 3: (|m − n| ≥ 2) Without loss of generality, assume that max(m, n) = m. It is easy to see that the partition Π = {S1 , S2 , . . . , Sm }, where Si = {vi , ui } for 1 ≤ i ≤ n and Sj = {vj } for n + 1 ≤ j ≤ m, is a fault-tolerant resolving m-partition of V (Km,n ) which shows that P(Km,n ) ≤ m. On the other hand, since each vertex in V is adjacent to all the vertices in U and vice versa, no two vertices of V contained in the same class of a fault-tolerant resolving (m − 1)-partition of V (Km,n ) which shows that P(Km,n ) ≥ m.  As a consequence of Theorem 7, we have the following corollary for stars. Corollary 3. Let K1,n−1 be the star for n ≥ 3, then P(K1,2 ) = 3 and P(K1,n−1 ) = n − 1 for all n ≥ 4. Next we show that every pair a, b of positive integers with b ≥ 6 and d 2b e + 1 ≤ a ≤ b − 2 is realizable as the fault-tolerant metric dimension and the fault-tolerant partition dimension of some connected graphs. Theorem 8. For every pair a, b of positive integers with b ≥ 6 and d 2b e + 1 ≤ a ≤ b − 2, there exists a connected graph G such that P(G) = a and β 0 (G) = b. Proof. Let G = Km,n with m = a and n = b − a. It is a routine exercise to verify that β 0 (G) = m + n = a + b − a = b. Since d 2b e + 1 ≤ a ≤ b − 2, it follows that m − n > 1. By Theorem 7, P(G) = a.  We have already seen, by Lemma 2, that the graphs Kn and Kn − e (n ≥ 3) have fault-tolerant partition dimension n. We now show that these are the only graphs of order n with the fault-tolerant partition dimension n. Theorem 9. Let G be a connected graph of order n ≥ 3. Then P(G) = n if and only if G is one of the graphs Kn , Kn − e. Proof. P(Kn ) = n = P(Kn −e), by Lemma 2. Assume that G is a connected graph of order n ≥ 3 having P(G) = n. By Corollary 2, we deduce that D ≤ 2. The only graph having diameter 1 is the complete graph. Thus G = Kn when D = 1.

12

M. A. Chaudhry, I. Javaid, M. Salman

Let D = 2 and u be a vertex having ecc(u) = 2. Denote Vi (u) = {v : v ∈ V (G), d(u, v) = i} f or 1 ≤ i ≤ 2. Vertex u is adjacent to all the vertices in V1 (u) and each vertex in V2 (u) is adjacent to at least one vertex in V1 (u). Consider first n ≥ 5. If min(|V1 (u)|, |V2 (u)|) ≥ 2, then we can choose v1 ∈ V1 (u) and v2 ∈ V2 (u) such that (u)(v1 , v2 )π is a fault-tolerant resolving (n − 1)-partition of V (G) where π denotes a partition of V (G) \ {u, v1 , v2 } having all the classes consisting of a single vertex (which will be called a singleton partition). We deduce that P(G) ≤ n − 1, a contradiction. So either |V1 (u)| = 1, |V2 (u)| = n − 2 or |V1 (u)| = n − 2, |V2 (u)| = 1. A. If |V1 (u)| = 1, |V2 (u)| = n − 2 and suppose that V1 (u) = {v} then there exist two vertices w, x ∈ V2 (u) such that wx ∈ E(G) or wx 6∈ E(G). In both cases (u, w)π is a fault-tolerant resolving (n − 1)-partition of V (G) where π is a singleton partition of the remaining vertices, a contradiction of the hypothesis. It follows that V2 (u) induces Kn−2 or Kn−2 . In the first case G = K1,n−1 but P(G) = n − 1, by Corollary 3, a contradiction. In the second case G = K1 + (K1 ∪ Kn−2 ) but P(G) ≤ n − 1 since (u, v)π is a fault-tolerant resolving partition of V (G) having n − 1 classes, a contradiction. B. If |V1 (u)| = n − 2, |V2 (u)| = 1. Suppose that V2 (u) = {v}. If d(v) = 1 (d(v) denotes the degree of a vertex v) then v is a diametral vertex and |V1 (v)| = 1, hence case A occurs again. Otherwise let d(v) ≥ 2. If there exist two vertices x, y ∈ V1 (u) such that xy ∈ E(G) or xy 6∈ E(G), we can find w ∈ V1 (u) and (u)(w, v)π is a fault-tolerant resolving (n − 1)-partition of V (G) where π is a singleton partition of the remaining vertices, a contradiction. It follows that V1 (u) induces Kn−2 or Kn−2 . In the first case, since D = 2, v is adjacent to all the vertices of Kn−2 which implies that G = K2,n−2 but P(K2,n−2 ) ≤ n − 1, by Theorem 7, which contradicts the hypothesis. We obtain that V1 (u) induces Kn−2 . If v is not adjacent to a vertex w ∈ V1 (u), since d(v) ≥ 2 it follows that there exist distinct vertices x, y ∈ V1 (u) such that xv, yv ∈ E(G). In this case (u)(x, v)π is a faulttolerant resolving (n − 1)-partition of V (G), a contradiction. We deduce that v is adjacent to all the vertices of Kn−2 and in this case G = Kn − e. The proof is complete for n ≥ 5. It remains to consider the cases n = 3, 4. For n = 3 the unique connected graph of order three having diameter two is K3 − e. For n = 4, if d(v) ≥ 2 then following B, we obtain that G is either K4 − e or C4 . But, it is not difficult to see that P(C4 ) = 3. It completes the proof. 

Fault-Tolerant Metric and Partition Dimension of Graphs

13

Acknowledgements The authors are grateful to the anonymous referee whose valuable suggestions resulted in producing an improved paper. This research is supported by the Higher Education Commission of Pakistan.

References 1. D. L. Boutin, Determining sets, resolving sets and the exchange property, Graphs and Combin., 2009. 2. D. L. Boutin, Identifying graphs automorphisms using determining sets, The Electron. J. Combin., 13, 2006. 3. P. S. Buczkowski, G. Chartrand, C. Poisson, P. Zhang, On k-dimensional graphs and their bases, Periodica Math. Hung. 46(1)(2003) 9-15. 4. G. Chartrand, P. Zhang, The theory and applicatons of resolvability in graphs, A survey, In Proc. 34 Southeastern International Conf. on Combinatorics,Graph Theory and Computing, Congr. Numer. 160(2003) 47-68. 5. G. Chartrand, E. Salehi, P. Zhang, The partition dimension of a graph, Aequationes Math. 59(2000) 45-54. 6. G. Chartrand, E. Salehi, P. Zhang, On the partition dimension of a graph, Congr. Numer. 130(1998) 157-168. 7. D. Erwin, F. Harary, Destroying automorphisms by fixing nodes, Disc. Math. 306(2006) 3244-3252. 8. M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, New York 1979. 9. F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combin. 2(1976) 191-195. 10. C. Hernando, M. Mora, P. J. Slater, David R. Wood, Fault-Tolerant Metric Dimension of Graphs, Proc. Internat. Conf. Convexity in Discrete Structures, Ramanujan Math. Society Lecture Notes 5(2008) 81-85. 11. I. Javaid, M. Salman, M. A. Chaudhry, S. Shokat, Fault-Tolerance in Resolvability, Util. Math. 80(2009) 263-275. 12. M. A. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Statist 3(1993) 203-236. 13. S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Disc. Appl. Math. 70(1996) 217-229. 14. R. A. Melter, I. Tomescu, Metric bases in digital geometry, Computer Vision, Graphics, and Image Processing 25(1984) 113-121. 15. P. J. Slater, Leaves of trees, Congr. Numer. 14(1975) 549-559.