Graphs of order n with fault-tolerant partition dimension n-1

U.P.B. Sci. Bull., Series A, Vol. 76, Iss. 3, 2014 ISSN 1223-7027 GRAPHS OF ORDER n WITH FAULT-TOLERANT PARTITION DIMENSION n − 1 Imran Javaid1 , Muhammad Salman2 , Muhammad Anwar Chaudhry3 This paper gives the characterization of all the connected graphs G of order n ≥ 8 having fault-tolerant partition dimension n − 1. Keywords: resolving partition, fault-tolerant resolving partition, fault-tolerant partition dimension, diameter. MSC2010: 05C12. 1. Introduction The distance d(u, v) between two vertices u and v in a connected graph G with vertex set V (G) and edge set E(G) is the minimum number of edges in a u − v path. For a vertex v in G, the eccentricity ecc(v) is the maximum distance between v and any other vertex of G. The diameter of G, denoted by D, is the maximum eccentricity of a vertex v in G. Two vertices u and v in G are called the diametral vertices if d(u, v) = D. If two vertices u and v are adjacent (form an edge) in G, then we write as u ∼ v and if they are non-adjacent (do not form an edge), then we write as u ̸∼ v. We refer [1] for the general graph theoretic notations and terminology not described in this paper. Given an ordered set W “related to {w1 , w2 , . . . , wk } ⊆ V (G)”. For each v ∈ V (G), the representation of v with respect to W is the k-vector (d(v, w1 ), d(v, w2 ), . . . , d(v, wk )), denoted by r(v|W ). The set W is called a resolving set for G if all the vertices of G have distinct representations with respect to W . The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by dim(G). The metric dimension was first studied by Slater [2] and independently by Harary and Melter [3]. Slater described the usefulness of this notion when working with U.S. Sonar and Coast Guard Loran (Long range aids to navigation) stations. It was noted in [4] and an explicit construction was given in [5] showing that finding the metric dimension of a graph is NP-hard. For more results about the notion of metric dimension and its applications, we refer to a nice survey by Saenpholphat and Zhang [6] (see also [7, 8, 9, 10, 11, 12]). 1 Corresponding author: Center for Advanced Studies in Pure and Applied Mathematical, Bahauddin Zakariya University Multan 60800, Pakistan, E-mail:ijavaidbzu@gmail.com 2 Center for Advanced Studies in Pure and Applied Mathematical, Bahauddin Zakariya University Multan 60800, Pakistan 3 Center for Advanced Studies in Pure and Applied Mathematical, Bahauddin Zakariya University Multan 60800, Pakistan 159 160 Imran Javaid, Muhammad Salman, Muhammad Anwar Chaudhry Possibly to gain insight into the metric dimension, Chartrand et al. introduced the notion of a resolving partition and partition dimension [13, 14]. To define the partition dimension, the distance d(v, S) between a vertex v of G and S ⊆ V (G) is defined as min d(v, s). Let Π be an ordered k-partition “related to {S1 , S2 , . . . , Sk }” s∈S of V (G) and v be a vertex of G, then the k-vector (d(v, S1 ), d(v, S2 ), . . . , d(v, Sk )) is called the representation r(v|Π) of v with respect to the partitison Π. A partition Π is called a resolving partition if for distinct vertices u and v of G, r(u|Π) ̸= r(v|Π). The partition dimension of G is the cardinality of a minimum resolving partition, denoted by pd(G). Based on the Chartrand et al. method of vertex-partitioning, Javaid et al. [15] partitioned the vertex set of a connected graph G into classes in such a way that any two distinct vertices in G have different distances from at least two classes of the partition. They referred this partition as a fault-tolerant resolving partition of V (G), defined as follows: Let Π be an ordered k-partition “related to {U1 , U2 , . . . , Uk }” of V (G), then Π is called a fault-tolerant resolving partition if for every pair of distinct vertices v, w in G, the representations r(v|Π) and r(w|Π) differ by at least two coordinates. The cardinality of a minimum fault-tolerant resolving partition is called the fault-tolerant partition dimension of G, denoted by P(G). We say that a class S distinguishes the vertices x and y of G if d(x, S) ̸= d(y, S). A partition Π distinguishes x and y if a class of Π distinguishes 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 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 Ui and Uj (i ̸= j) of a partition Π, d(x, Ui ) ̸= d(y, Ui ) and d(x, Uj ) ̸= d(y, Uj ) for all x, y ∈ Uk , k ̸= i, j. A useful property for finding the fault-tolerant partition dimension of a connected graph G is Lemma 3.1 placed in Annex-I. The join of two graphs G1 and G2 , denoted by G1 + G2 , is a graph with vertex set V (G1 )∪V (G2 ) and an edge set E(G1 )∪E(G2 )∪{uv | u ∈ V (G1 ) and v ∈ V (G2 )}. This paper is aim to characterize all the connected graphs G of order n ≥ 8 having fault-tolerant partition dimension n − 1. In the next section, we list all the connected graphs having fault-tolerant partition dimension one less than the order of the graph and prove that these are the only graphs having this property. 2. Classification of graphs of order n with fault-tolerant partition dimension n − 1 The graph G − e is a subgraph of G and can be obtained by deleting an edge e from G. The following is the list of graphs of order n having fault-tolerant partition dimension n − 1. It is worth mentioning that, in the list of graphs below, the graphs K with single subscript represent the complete graphs; and the graphs K with two subscripts separated by comma represent the complete bipartite graphs. G1 := K1,n−1 ; G2 := K1 + (K1 ∪ Kn−2 ); G3 := Kn − E(P3 ); G4 := Kn − E(P4 ); G5 := Kn − E(K3 ); G6 := K1,n−1 + e; G7 := Kn − E(2K2 ); G8 := Kn − E(3K2 ); G9 := Kn−1 − e and another vertex adjacent to end vertices of e; G10 := Kn−1 and a vertex adjacent to two vertices of Kn−1 ; G11 := K2 +Kn−3 with one edge deleted between K2 and Kn−3 and a vertex adjacent Graphs of order n with fault-tolerant partition dimension n − 1 161 to the vertices of K2 ; G12 := The same construction as G11 with K2 instead of K2 ; G13 := Kn−1 − e and a vertex adjacent to two vertices of Kn−1 , one of them being an end vertex of e; and the following four families of graphs: G1 := {Kn − E(K1,p + e), where 3 ≤ p ≤ n − 2}, G2 := {Kn − E(K1,p and a path P3 having one adge in common with K1,p ), where 3 ≤ p ≤ n − 3}, G3 := {Kn−1 − e and a vertex adjacent to p vertices of Kn−1 , where 2 ≤ p ≤ n − 3}, G4 := {Kn − E(K1,p ), where 2 ≤ p ≤ n − 3}. Figure 1, shown in Annex-II, illustrates one graph of each family mentioned above for n = 8 and p = 3. We also list 7 graphs of order n with the fault-tolerant partition dimension n − 2 which will appear in the proofs of our lemmas. H1 := K2 + Kn−3 and a vertex adjacent to the vertices of Kn−3 ; H2 := Kn−2 and a path P4 joining two vertices of Kn−2 ; H3 := Kn−2 and a cycle C3 having a common vertex; H4 := Kn−2 and a path P3 having in common the central vertex of P3 ; H5 := Kn−2 and a path P3 having an end vertex common with Kn−2 ; H6 := K1,n−1 and a vertex adjacent to a diametral vertex of the star K1,n−1 ; H7 := Kn−2 and a path P4 having the central edge in common with Kn−2 . The relationship between the fault-tolerant partition dimension and the diameter of a connected graph was obtained by Javaid et al. in [8] (see Theorem 3.1 in Annex-I). Following is a consequence of Theorem 3.1, cited in Annex-I, will helps in proof of next lemmas. Corollary 2.1. Lat G be a connected graph of order n with P(G) = n − 1. Then diameter of G is at most three. The connected graphs having fault-tolerant partition dimension equal to the order of the graph have been characterized by Javaid et al (see Theorem 3.3 in Annex-I). Now, we show that the graphs listed above are the only graphs with fault-tolerant partition dimension n − 1. Let u be a diametral vertex in G with eccentricity 2. Denote Vi (u) = {v : v ∈ V (G), d(u, v) = i} for i = 1, 2. Then u ∼ u′ for each u′ ∈ V1 (u) and for each w ∈ V2 (u), w ∼ w′ for at least one w′ ∈ V1 (u). Now, we prove several lemmas which will help to prove our main result Theorem 2.1. Lemma 2.1. Let G be a connected graph of order n ≥ 8 with P(G) = n − 1 and diameter D = 2. If min(|V1 (u)|, |V2 (u)|) ≥ 3, then G belongs to G1 , or G3 , or G4 . Proof. With out loss of generality, we suppose that 3 ≤ r = |V1 (u)| ≤ |V2 (u)| = s = n − r − 1. Since n ≥ 8 and r ≥ 3, if there are three distinct vertices x, y, z in V1 (u) (or in V2 (u)) such that x ∼ y and x ̸∼ z, y ̸∼ z in G, then for two distinct vertices a, b in V2 (u) (or in V1 (u)), (u)(z)(a, x)(b, y)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, where π denotes a partition of V (G) \ {u, a, b, x, y, z} 162 Imran Javaid, Muhammad Salman, Muhammad Anwar Chaudhry having all the classes consisting a single vertex (which will be called a singleton sets partition). We deduce that P(G) ≤ n − 2, a contradiction. It follows that V1 (u) and V2 (u) induces Kr and Ks , or Kr − e and Ks − e, or Kr and Ks , respectively. In the case when V1 (u) induces Kr and V2 (u) induces Ks , we can chose distinct vertices u1 , v1 ∈ V1 (u) and u2 , v2 ∈ V2 (u) such that (u)(u1 , u2 )(v1 , v2 )π is a fault-tolerant resolving partition of V (G) having n − 2 classes, where π is a singleton sets partition of the remaining vertices, a contradiction. Now, we discuss the following two case: V1 (u) induces Kr and V2 (u) induces Ks or Ks − e (case 1), V1 (u) induces Kr − e and V2 (u) induces Ks or Ks − e (case 2). Case 1. If there are distinct vertices x, y ∈ V1 (u) and a, b ∈ V2 (u) such that a ̸∼ x and b ̸∼ y in G, then for a vertex z ∈ V1 (u) \ {x, y}, (a)(b)(u, z)(x, y)π is a faulttolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. We deduce that V1 (u) ∪ V2 (u) induces Kr + Ks (subcase 1.1) or (Kr + Ks ) − e (subcase 1.2) or Kr + (Ks − e) (subcase 1.3) or (Kr + (Ks − e)) − e (subcase 1.4). Subcase 1.1. In this case, G ∈ G4 . Subcase 1.2. In this case, G ∈ G3 . Subcase 1.3. In this case, G ∈ G1 . Subcase 1.4. Let a ̸∼ b and c ̸∼ x in G for x ∈ V1 (u) and a, b, c ∈ V2 (u). Then we can chose a vertex y ∈ V1 (u) \ {x} and a vertex d ∈ V2 (u) \ {a, b, c} such that (u)(a)(c)(b, y)(x, d)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, where π is a singleton sets partition of the remaining vertices, a contradiction. Case 2. By the similar arguments as Case 1, V1 (u) ∪ V2 (u) induces (Kr − e) + Ks (subcase 2.1) or ((Kr − e) + Ks ) − e (subcase 2.2) or (Kr − e) + (Ks − e) (subcase 2.3) or ((Kr − e) + (Ks − e)) − e (subcase 2.4). Subcase 2.1. In this case, G ∈ G3 . Subcase 2.2. Let a ̸∼ b for a, b ∈ V1 (u) and c ̸∼ x for c ∈ V1 (u), x ∈ V2 (u). Then (i) For c ̸= a, b, (b)(u, c)(a, x)π is a fault-tolerant resolving (n−2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. (ii) For c = a or b, (x)(u, d)(c, y)π, where d ∈ V1 (u) \ {a, b, c} and y ∈ V2 (u) \ {x}, is a fault-tolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. Subcase 2.3. Let a ̸∼ b and x ̸∼ y in G for a, b ∈ V1 (u) and x, y ∈ V2 (u). Then for c ∈ V1 (u) \ {a, b}, (a)(x)(u, c)(b, y)π is a fault-tolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. Subcase 2.4. Let a ̸∼ b, x ̸∼ y and c ̸∼ z in G for a, b, c ∈ V1 (u) and x, y, z ∈ V2 (u). Then (i) For c ̸= a, b and z ̸= x, y, (a)(b)(y)(u, c)(z, x)π is a fault-tolerant resolving (n−2)partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. (ii) For c ̸= a, b and z = x or y, (a)(b)(u, c)(z, w)π, where w ∈ V2 (u) \ {x, y, z}, is a fault-tolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. (iii) For c = a or b and z ̸= x, y, (y)(u, c)(z, x)π is a fault-tolerant resolving (n − 2)partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. Graphs of order n with fault-tolerant partition dimension n − 1 163 (iv) For c = a or b and z = x or y, (u, d)(c, z)π, where d ∈ V1 (u) \ {a, b, c}, is a faulttolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. Lemma 2.2. Let G be a connected graph of order n ≥ 8 with P(G) = n−1 and diameter D = 2. If min(|V1 (u)|, |V2 (u)|) ≤ 2, then G belongs to G = {G1 , G2 , . . . , G13 }, or G1 , or G2 , or G3 , or G4 . Proof. We shall consider the following cases: Case 1. |V1 (u)| = 2, |V2 (u)| = n − 3, Case 2. |V1 (u)| = n − 3, |V2 (u)| = 2, Case 3. |V1 (u)| = 1, |V2 (u)| = n − 2, Case 4. |V1 (u)| = n − 2, |V2 (u)| = 1. Case 1. Suppose that V1 (u) = {v, w}. If V2 (u) contains three distinct vertices x, y, z such that x ̸∼ y and x ∼ z in G, then the pair {y, z} distinguished by x. Since n ≥ 8, we can find another vertex u′ ∈ V2 (u) such that (u, u′ )(y, z)π is a faulttolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, which contradicts the hypothesis. It follows that V2 (u) induces Kn−3 (subcase 1.1), or Kn−3 (subcase 1.2). Subcase 1.1. If one of the vertices of V1 (u), say v, has the property that there exist x, y ∈ V2 (u) such that v ̸∼ x and v ∼ y in G, then either v ∼ w or v ̸∼ w in G, (x)(u, v)(y, w)π is a fault-tolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. One deduce that v and w are adjacent to all the vertices in V2 (u) or one of them is not adjacent to any vertex of V2 (u). But in the last case, we get D = 3 unless v ∼ w, which contradicts the hypothesis. If v ∼ w in G and for example v ̸∼ v ′ for any vertex v ′ ∈ V2 (u), then it follows that w ∼ w′ for all w′ ∈ V2 (u). In this case G ∼ = G6 . If v ∼ z and w ∼ z in G for all z ∈ V2 (u), then (i) G ∼ = K2,n−2 if v ̸∼ w in G, but P(G) = n − 2, by Theorem 3.2, a contradiction. (ii) G ∼ = K2 + Kn−2 if v ∼ w in G, but P(G) ≤ n − 2, Since there exists a vertex x ∈ V2 (u) such that (u, v)(w, x)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, where π is a singleton partition of V (G) \ {u, v, w, x}, a contradiction. Subcase 1.2. If one of the vertices of V1 (u), say w, has the property that there exist three vertices x, y, z ∈ V2 (u) such that w ̸∼ x, w ̸∼ y and w ∼ z in G, then either v ∼ w or v ̸∼ w in G, (x)(y)(u, z)(v, w)π is a fault-tolerant resolving (n − 2)partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. It follows that if v ∼ c or w ∼ c for at least one vertex c ∈ V2 (u), then it is adjacent to at least n − 4 vertices in V2 (u). If v ̸∼ w one obtains that both v and w adjacent to at least n − 4 vertices in V2 (u) since otherwise D = 3. Consider now the case when both v and w are adjacent to at least n − 4 vertices in V2 (u). If v and w are adjacent to all n − 3 vertices of V2 (u), then G ∼ = G10 = G9 if v ̸∼ w and G ∼ if v ∼ w. If one of v and w is adjacent to n − 4 vertices in V2 (u) and other one is adjacent to all n − 3 vertices of V2 (u), then G ∼ = G11 if v ̸∼ w in G and G ∼ = G12 if v ∼ w in G. It is not possible that both v and w are adjacent to exactly n − 4 vertices of V2 (u). Indeed, if there exist distinct vertices x, y ∈ V2 (u) such that v ̸∼ x, w ̸∼ y 164 Imran Javaid, Muhammad Salman, Muhammad Anwar Chaudhry and both v and w are adjacent to n − 4 vertices of V2 (u), then either v ∼ w in G or v ̸∼ w in G, there exists z ∈ V2 (u) \ {x, y} such that (x)(y)(u, z)(v, w)π is a faulttolerant resolving (n − 2)-partition of V (G), where π is a singleton sets partition of the remaining vertices, a contradiction. Consider now the case when v ∼ w in G and v ̸∼ v ′ for each vertex v ′ ∈ V2 (u). If w ∼ w′ for all w′ ∈ V2 (u), then G ∼ = H3 . In this case, there exist distinct vertices x, y ∈ V2 (u) such that (u, y)(v, x)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. If w ̸∼ t for one vertex t ∈ V2 (u), then d(u, t) = 3 which contradicts the equality D = 2. Case 2. In this case, let V2 (u) = {s, t}. If V1 (u) contains three distinct vertices v, w, x such that v ∼ w and v ̸∼ x in G, then we can find another vertex y ∈ V1 (u)\{v, w, x} such that (v)(u, y)(w, x)π is a fault-tolerant resolving (n−2)-partition of V (G) which implies that P(G) ≤ n − 2, a contradiction. It follows that V1 (u) induces Kn−3 (subcase 2.1) or Kn−3 (subcase 2.2). Subcase 2.1. If s ̸∼ t in G, since D = 2 we obtain that V1 (u) ∪ V2 (u) induces a subgraph isomorphic to K2,n−3 . By considering two distinct vertices x, y ∈ V1 (u), we deduce that (t)(u, x)(s, y)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, contradiction. It follows that s ∼ t in G. Suppose that one of the vertices s, t, say t, has the property that t ̸∼ t1 for one vertex t1 ∈ V1 (u), but t ∼ t2 for at least one vertex t2 of V1 (u). Then (t)(s, t1 )(u, t2 )π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. Hence s ∼ v and t ∼ v in G for all v ∈ V1 (u) and G ∼ = H1 , but P(G) ≤ n − 2 in this case. Since we can find two distinct vertices x, y ∈ V1 (u) such that (u)(s, x)(y, t)π, where π is a singleton sets partition of V (G) \ {s, t, u, x, y}, is a fault-tolerant resolving partition having n − 2 classes, a contradiction. Subcase 2.2. Both the vertices s and t must adjacent to at least one vertex of V1 (u), otherwise D = 3. If s ∼ y and t ∼ y in G for all y ∈ V1 (u), then G ∼ = G3 if s ∼ t in G and G ∼ = G5 if s ̸∼ t in G. Consider now the case when, for all v ∈ V1 (u), v ∼ t in G. When s ̸∼ t and s ̸∼ v1 , s ̸∼ v2 , . . . , s ̸∼ vi in G for v1 , v2 , . . . , vi ∈ V1 (u), where i ∈ {1, 2, . . . , n − 4}, then G ∈ G1 . When s ∼ t; if s ̸∼ s′ in G for a single vertex s′ ∈ V1 (u), then G ∼ = G4 , if s ̸∼ v1 , s ̸∼ v2 , . . . s ̸∼ vi in G for v1 , v2 , . . . , vi ∈ V1 (u), where i ∈ {2, 3, . . . , n − 4}, then G ∈ G2 . For i = n − 4, G ∼ = G13 . A similar situation occurs when v ∼ s for all v ∈ V1 (u). The remaining subcase is that when both s and t are not adjacent to at least one vertex of V1 (u). If there exist four distinct vertices a, b, c, d ∈ V1 (u) such that a ̸∼ s, d ̸∼ t and c ∼ s, b ∼ t in G, let v ∈ V1 (u) \ {a, b, c, d}. If v ∼ s or v ∼ t in G, then either s ̸∼ t or s ∼ t, (u)(s, c)(t, b)π is a fault-tolerant resolving (n − 2)-partition of V (G), which implies that c ∼ s, b ∼ t are only edges joining s and t to vertices in V1 (u). In this case, G ∼ = H2 if s ∼ t, but P(G) ≤ n − 2 since (u)(c, s)(b, t)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. If s ̸∼ t in G, then d(s, t) = 3, a contradiction. If a and d coincide or b and c coincide, then it remains to consider only two subcases: s ̸∼ x and t ̸∼ x in G for a single vertex x ∈ V1 (u) (subcase 2.2.1), s ∼ x′ and t ∼ x′ for a single vertex x′ ∈ V1 (u) (subcase 2.2.2). Subcase 2.2.1. Either s ̸∼ t or s ∼ t (in this case G ∼ = Kn − E(C4 )), there are two distinct vertices v, w ∈ V1 (u) such that (u)(x)(s, v)(t, w)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, a contradiction. Graphs of order n with fault-tolerant partition dimension n − 1 165 Subcase 2.2.2. If s ∼ t ∈ E(G), then G ∼ = H3 and if s ̸∼ t, then G ∼ = H4 . In both the cases, (u, t)(s, x)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, a contradiction. Case 3. Let V1 (u) = {x}, it follows that x ∼ x′ for all x′ ∈ V2 (u). If V2 (u) induces Kn−2 or Kn−2 , then G is isomorphic to G1 or G2 , respectively. Otherwise, there exists a diametral vertex y ∈ V2 (u) such that 2 ≤ |V1 (y)| ≤ n − 3, hence |V2 (y)| ∈ {2, n − 3} and we are again in the Case 1, or in the Case 2, relatively to y. Case 4. Let V2 (u) = {v}. If degree of v is one, then v is a diametral vertex and |V1 (v)| = 1, hence Case 3 occurs again. Otherwise, let degree of v, d(v), is grater than or equal to two. If there exist six distinct vertices a, b, c, d, e, f ∈ V1 (u) (since n ≥ 8) such that a ̸∼ d, b ̸∼ e, c ̸∼ f and a ∼ b, a ∼ c, a ∼ e, a ∼ f, b ∼ c, b ∼ d, b ∼ f, c ∼ d, c ∼ e, d ∼ e, d ∼ f, e ∼ f in G, then (u)(a, b)(c, v)π is a faut-tolerant resolving partition of V (G) having n − 2 classes, where π is a singleton sets partition of the remaining vertices, a contradiction. It follows that V1 (u) induces Kn−2 − e (subcase 4.1), or Kn−2 − E(2P2 ) (subcase 4.2), or Kn−2 (subcase 4.3), or Kn−2 (subcase 4.4). Subcase 4.1. Since d(v) ≥ 2, if v ∼ v1 , v ∼ v2 , . . . , v ∼ vi in G for v1 , v2 , . . . , vi ∈ V1 (u), where i ∈ {2, 3, . . . , n − 3}, then G ∈ G3 . For i = 2, if v1 , v2 ∈ V1 (u) are end vertices of e then G ∼ = G9 and if v1 is end vertex of e and v2 is not, then G ∼ = G13 . ′ ′ If v ∼ v for all v ∈ V1 (u), then G ∼ G . = 7 Subcase 4.2. Let a, b, c, d ∈ V1 (u) such that a ̸∼ c and b ̸∼ d in G. If there exist two distinct vertices x, y ∈ V1 (u) \ {a, b, c, d} such that v ∼ a, v ∼ y but v ̸∼ x in G, then (u)(v)(c)(d)(a, x)(b, y)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. We deduce that v ∼ v ′ for all v ′ ∈ V1 (u). In this case G ∼ = G8 . ′ ′ Subcase 4.3. Since D = 2 it follows that v ∼ v for all v ∈ V1 (u) and G ∼ = K2,n−2 , but P(G) ≤ n − 2 since for every two distinct vertices s, t of V1 (u), (t, u)(s, v)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. Subcase 4.4. If v ̸∼ v1 , v ̸∼ v2 , . . . , v ̸∼ vi in G for v1 , v2 , . . . , vi ∈ V1 (u), where i ∈ {1, 2, . . . , n − 4}, then G ∈ G4 . The case for i = n − 3 is not occur since d(v) ≥ 2. If v ∼ v ′ for all v ′ ∈ V1 (u), then G ∼ = Kn − e, but P(Kn − e) = n, by Theorem 3.3. Lemma 2.3. Let G be a connected graph of order n ≥ 8 with P(G) = n − 1 and diameter D = 3. Then G ∼ = G2 . Proof. Let s be a diametral vertex having ecc(s) = 3. Denote Vi (s) = {s′ : s′ ∈ V (G), d(s′ , s) = i} for i = 1, 2, 3. Let t ∈ V1 (s), u ∈ V2 (s) and v ∈ V3 (s). If there are w, x ∈ V (G)\{s, t, u, v} belonging to different sets from V1 (s), V2 (s), V3 (s), then (s)(x)(t, w)(u, v)π is a fault-tolerant partition of V (G) having n − 2 classes, a contradiction. It follows that we can consider only three cases. Case 1. |V1 (s)| = |V2 (s)| = 1, |V3 (s)| = n − 3, Case 2. |V1 (s)| = |V3 (s)| = 1, |V2 (s)| = n − 3, Case 3. |V2 (s)| = |V3 (s)| = 1, |V1 (s)| = n − 3. Case 1. Suppose that V1 (s) = {t}, V2 (s) = {u}, then u ∼ u′ for all u′ ∈ V3 (s) otherwise, there exists a vertex v ∈ V3 (s) such that d(s, v) = 4, contradiction. If there exist three distinct vertices x, y, z ∈ V3 (s) such that x ̸∼ y and x ∼ z in 166 Imran Javaid, Muhammad Salman, Muhammad Anwar Chaudhry G, then (z)(s)(t, x)(u, y)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. Hence V3 (s) induces Kn−3 or Kn−3 . In the first case G ∼ = H5 but P(G) ≤ n−2 since there is a vertex v ∈ V3 (s) such that (s, v)(t, u)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. In the second case G ∼ = H6 and we get a contradiction by the same argument as previous case. Case 2. Let V1 (s) = {u} and V3 (s) = {v} then u ∼ u′ for all u′ ∈ V2 (s). As above, V2 (s) induces Kn−3 (subcase 2.1) or Kn−3 (subcase 2.2). v ∼ w for at least one vertex w ∈ V2 (s). If there exist two vertices x, y ∈ V2 (s) \ {w} such that v ̸∼ x and v ∼ y in G, then (x)(w)(u, s)(v, y)π ia s fault-tolerant resolving partition of V (G) having n − 2 classes, a contradiction. It follows that v ∼ v ′ for all v ′ ∈ V2 (s) or v ̸∼ v ′ for any vertex in v ′ ∈ V2 (s) \ {w}. Subcase 2.1. If v ∼ v ′ for all v ′ ∈ V2 (s), then G ∼ = G2 . Otherwise, G ∼ = H7 but P(G) ≤ n − 2 since for every t ∈ V2 (s) \ {w}, (w)(s, t)(u, v)π is a fault-tolerant resolving (n − 2)-partition of V (G), a contradiction. Subcase 2.2. If v ∼ v ′ for all v ′ ∈ V2 (s), then G ∼ = K2,n−2 − e. Otherwise, G ∼ = H6 . In both the cases, P(G) ≤ n − 2 since for every t ∈ V2 (s) \ {w}, (w)(u, s)(v, t)π is a fault-tolerant resolving partition of V (G) having n − 2 classes, a contradiction. Case 3. Suppose that V2 (s) = {x} and V3 (s) = {y}. In this case y is a diametral vertex and |V1 (y)| = 1. Hence y instead of s we have Cases 1 and 2, which completes the proof. Now, our main result is the following: Theorem 2.1. Let G be a connected graph of order n ≥ 8. Then P(G) = n − 1 if and only if G belongs to G = {G1 , G2 , . . . , G13 }, or G1 , or G2 , or G3 , or G4 . Proof. By using Lemma 3.1, it is a routine exercise to verify that all the graphs enumerated in the statement have the fault-tolerant partition dimension n − 1. Conversely, let G be a connected graph of order n ≥ 8 having P(G) = n − 1. Then D ≤ 3, by Corollary 2.1. When D = 1, then G is isomorphic to the complete graph Kn and P(G) = n, by Theorem 3.3. When D = 2, 3, then Lemmas 2.1, 2.2 and 2.3 conclude the proof. 3. Conclusion In this paper, we considered the generalization of the fault-tolerant metric dimension “the fault-tolerant partition dimension”. Inspired by the works, done by Chartrand et al. in [13, 14] and by Javaid et al. in [8], on the characterization of all the connected graphs of order n having partition dimension (the generalization of the metric dimension) n, n − 1, n − 2, and the fault-tolerant partition dimension n, we classified all the connected graphs with fault-tolerant partition dimension one less than the order of the graphs. Acknowledgements The authors would like to express their deep gratitude to the referee(s) for careful reading of the earlier version of the manuscript and several insightful comments. Graphs of order n with fault-tolerant partition dimension n − 1 167 Annex-I Lemma 3.1. [15] Let Π be a fault-tolerant resolving partition of V (G) and u, v ∈ V (G). If d(u, w) = d(v, w) for all w ∈ V (G) \ {u, v}, then u and v belong to distinct classes of Π. Theorem 3.1. [8] Let G be a connected graph of order n ≥ 3 and diameter D. Then η(n, D) ≤ P(G) ≤ n − D + 2, where η(n, D) is the least positive integer ν for which n ≤ (D + 1)ν . Theorem 3.2. [8] If Km,n be the complete bipartite  if  m+1 max(m, n) + 1 if P(Km,n ) =  max(m, n) if graph for m, n ≥ 1, then m − n = 0, |m − n| = 1, |m − n| ≥ 2. Theorem 3.3. [8] Let G be a connected graph of order n. Then P(G) = n if and only if G is one of the graphs Kn and Kn − e. Annex-II G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 A1 G13 G12 G11 A2 A3 A4 Figure 1. Illustration of the graphs for n = 8. Ai ∈ Gi for i = 1, 2, 3, 4 and p = 3, n = 8. Deleted edges colored by red (dotted edges) and new edges colored by blue (thick edges). 168 Imran Javaid, Muhammad Salman, Muhammad Anwar Chaudhry REFERENCES [1] G. Chartrand and L. Lesniak, Graphs and Digraphs, 3rd ed., Chapman and Hall, London, 1996. [2] P. J. 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