Distance Connectivity in Graphs and Digraphs

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1 Distance Connectivity in Graphs and Digraphs M.C. Balbuena, A. Carmona Departament de Matemàtica Aplicada III M.A. Fiol Departament de Matemàtica Aplicada i Telemàtica Universitat Politècnica de Catalunya, Barcelona (Spain) Abstract Let G = (V, A) be a digraph with diameter D 1. For a given integer 2 t D, the t-distance connectivity κ(t) of G is the minimum cardinality of an x y separating set over all the pairs of vertices x, y which are at distance d(x, y) t. The t-distance edge connectivity λ(t) of G is defined similarly. The t-degree of G, δ(t), is the minimum among the out-degrees and in-degrees of all vertices with (out- or in-)eccentricity at least t. A digraph is said to be maximally distance connected if κ(t) = δ(t) for all values of t. In this paper we give a construction of a digraph having D 1 positive arbitrary integers c 2... c D, D > 3, as the values of its t-distance connectivities κ(2) = c 2,..., κ(d) = c D. Besides, a digraph that shows the independence of the parameters κ(t), λ(t), and δ(t) is constructed. Also we derive some results on the distance connectivities of digraphs, as well as sufficient conditions for a digraph to be maximally distance connected. Similar results for (undirected) graphs are presented. 1 Introduction The study of connectivity properties in graphs and digraphs has some applications to the design of reliable communication or interconnection networks. In particular, it is interesting to know sufficient conditions for a (di)graph to be maximally connected. See, for instance, the survey of Bermond, Homobono and Peyrat [2]. Apart from the standard Work supported in part by the Spanish Research Council (Comisión Interministerial de Ciencia y Tecnología, CICYT) under projects TIC and TIC E. e mail: carmona@etseccpb.upc.es 1

2 concepts of (vertex) connectivity and edge connectivity, a number of related connectivitymeasures have been considered in the literature. Thus, Harary defined in [11] the conditional connectivity as the minimum cardinality of a set of vertices, if any, whose deletion disconnects the (di)graph, and every remaining component satisfies a given condition. Another possible approach is to study the minimum sets whose deletion do not disconnect the (di)graph, but the resulting diameter does not exceed a given bound. This is, in fact, a particular instance of the so-called diameter vulnerability problem, which is closely related to the study of network fault-tolerance, and it has deserved special attention in the literature. See for instance the standard reference of Bollobás [3]. In this paper we study a generalization of the concepts of connectivity and edge connectivity which, to some extent, could also be considered as conditional connectivities. The difference is that, now, the condition it is not imposed on the remaining components, but on the original distance of the vertices which are disconnected. More precisely, we require that the disconnecting sets separate some vertices which, in the original (di)graph, were far apart enough from each other. This kind of connectivity was introduced by one of the authors and Fàbrega in [8], and it could be a useful tool to measure the reliability of the network as a function of the distance of the nodes which try to communicate to each other. The remaining of this section is devoted to recall the basic concepts and results used later, and to explain the new concepts. In Section 2 we construct some (di)graphs which have a given sequence of distance connectivities. In the last section, special attention is paid to derive sufficient conditions to get maximum distance connectivities. When we are dealing with graphs, the results are given in terms of the girth. To deal with the more general case of digraphs, the so-called parameter l and the condition of a digraph to be s-geodetic are used. This parameter was introduced in [5, 7] in the context of connectivity problems, and it is related to the number of short paths. Let us first give some of the notation used throughout the paper. Let G = (V, A) denote a digraph with (finite) set of vertices V = V (G) and set of (directed) edges A = A(G), which are ordered pairs of different vertices of V. So, neither loops nor multiple edges are allowed. If e = (x, y) A, we say that x is adjacent to y and that y is adjacent from x. Let Γ (x) and Γ + (x) denote respectively the set of vertices adjacent to and from x, that is the sets of in-neighbours and out-neighbours of x. Their cardinalities are the in-degree of x, δ (x) = Γ (x), and the out-degree of x, δ + (x) = Γ + (x). The minimum degree of 2

3 G, δ = δ(g), is the minimum over all the in-degrees and out-degrees of the vertices of G. For any pair of vertices x, y V, a path xx 1 x 2... x n 1 y from x to y, where the vertices are not necessarily distinct, is called an x y path. The distance from x to y, that is the length of a shortest path from x to y, is denoted by d G (x, y) or just d(x, y) if this does not lead to confusion, and D = D(G) = max x,y V {d(x, y)} stands for the diameter of G. The distance from x to F V, denoted by d(x, F ), is the minimum over all the distances d(x, f), f F. The distance from F to x, d(f, x), is defined analogously. A digraph G = (V, A) is said to be (strongly) connected when for any pair of vertices x, y V there always exists an x y path. Let G be a connected digraph. The positive eccentricity of a vertex x V is defined as e + (x) = max y V {d(x, y)}. The negative eccentricity e (x) is defined analogously. The positive radius of G is r + = min x V {e + (x)} and in a similar way it is defined the negative radius, r. The radius r of G is the minimum between the negative and the positive radius. Notice that the diameter is D = max x V {e + (x)} = max x V {e (x)}. Given x, y V such that (x, y) / A, a set S = S(x, y) V \{x, y} is called an x y separating set if there is no x y path in G\S. The (strong) local connectivity from x to y is κ(x, y; G) = κ(x, y) = min{ S : S is an x y separating set}. By the widely known Menger s theorem, κ(x, y) can also be defined as the maximum number of internally disjoint x y paths. Given t, 1 t D, the t distance connectivity of a digraph G, denoted by κ(t; G) or simply κ(t), is defined as κ(t) = min{κ(x, y) : x, y V, d(x, y) t}, if t 2, and κ(1) = κ(g) = κ being κ the standard connectivity (or vertex-connectivity) of G, that is the smallest number of vertices whose deletion results in a digraph that is either non-strongly connected or trivial, see [8]. From the above definition it is easily verified that κ = κ(1) = κ(2) κ(3) κ(d). (1) The concepts of edge x y separating set S = S (x, y), local edge connectivity λ(x, y) and t-distance edge connectivity λ(t) = λ(t; G) are defined analogously. In this case we have λ = λ(1) λ(2) λ(d), (2) 3

4 where λ = λ(g) denotes the (strong) edge connectivity of G, that is, the smallest number of edges whose deletion results in a non-strongly connected digraph. Throughout the paper, G stands for a connected digraph. Hence δ(g) 1. It is readily shown that κ(g) λ(g) δ(g), see Geller and Harary [10]. Then G is said to be maximally connected when κ = λ = δ. In fact, the first inequality holds for any t-distance connectivity, 1 t D, that is κ(t) λ(t). In general, it is no true that λ(t) δ, but the definition of λ(t) clearly verifies λ(t) δ + (x) for any x V such that e + (x) t, and λ(t) δ (x) for any x V with e (x) t. Then, for any t, 1 t D, it is useful to consider a new parameter, that we call the t-degree of G, which is defined as δ(t) = min{δ + (t), δ (t)}, where δ + (t) = min x V {δ + (x), e + (x) t} and δ (t) = min x V {δ (x), e (x) t}. As an immediate consequence of this definition we have δ = δ(1) = = δ(r) δ(r + 1) δ(d). (3) Now, for any t, 1 t D, κ(t) λ(t) δ(t). (4) A digraph G is said to be maximally t-distance connected if κ(t) = λ(t) = δ(t), and maximally t-distance edge connected if λ(t) = δ(t). Note that if G is maximally connected, then G is maximally t-distance connected for any 1 t r. We recall here that in the line digraph LG of a digraph G, each vertex represents an edge of G. Thus, V (LG) = {uv : (u, v) A(G)}; and a vertex uv is adjacent to a vertex wz iff v = w, that is, when the edge (u, v) is adjacent to the edge (w, z) in G. For any k > 1 the k-iterated line digraph, L k G, is defined recursively by L k G = LL k 1 G. From the definition it is evident that the order of LG equals the size of G, V (LG) = A(G), and that their minimum degrees coincide, δ(lg) = δ(g) = δ. Moreover, if G is d-regular (δ (x) = δ + (x) = d, for any x V ), d > 1, and has order n and diameter D, then L k G is also d-regular and has d k n vertices and diameter D(L k G) = D(G) + k. (5) See, for instance, Fiol, Yebra and Alegre [9] and Reddy, Kuhl, Hosseini and Lee [16]. In fact, (5) still holds for any strongly connected digraph other than a directed cycle, see Aigner [1]. Also, since the vertices of LG correspond to the edges of G and each path in LG of length t + 1, with 1 t + 1 D + 1, corresponds to a path in G of length t. Then 4

5 it can be shown that κ(t + 1; LG) = λ(t; G) if 1 t D 1, (6) and δ(t + 1; LG) = δ(t; G) if 1 t D. (7) To study the connectivity of graphs and digraphs, a new parameter related to the number of short paths was used in [5] (see also [7]), the definition of which is as follows. Definition 1.1 For a given digraph G = (V, A) with diameter D, let l = l(g), 1 l D, be the greatest integer such that, for any x, y V, (a) if d(x, y) < l, the shortest x y path is unique and there are no x y paths of length d(x, y) + 1; (b) if d(x, y) = l, there is only one shortest x y path. In [5] it is shown that for any digraph G without loops and different from a cycle this parameter satisfies an equality like (5). Namely, l(l k G) = l(g) + k. (8) Finally, we recall that a connected digraph G with diameter D is said to be s-geodetic, for some 1 s D, if any two vertices of G are joined by at most one path of length less than or equal to s. If s = D, the digraph is called strongly geodetic. See [6, 14]. Note that s min{l, g 1}, where g stands for the girth of G (that is the minimum length of a directed cycle). Moreover, if G is s-geodetic, then LG is s -geodetic with s = min{s + 1, g 1}. Similar notation and results apply for (undirected) graphs. For all definitions not given here we refer the reader to the books of Chartrand and Lesniak [4], Harary [13] and Harary, Norman and Cartwright [12]. 2 Constructions In this section, for a given sequence of D 1 positive integers c 2 c 3 c D, we construct a digraph G having these numbers as its t-distance connectivities. Likewise, we show that, for a given t, the parameters κ(t), λ(t) and δ(t) are independent. 5

6 The next lemma provides some upper bounds for the t-distance connectivities as a function of the order, the size and the diameter of a digraph, which will be useful to our study. The first bound was already given by Watkins [17] for the standard connectivity κ = κ(1). Lemma 2.1 Let G = (V, A) be a digraph with minimum degree δ > 1, diameter D, order n, size m and t-distance connectivities κ(t), λ(t). Then, (a) κ(t) n 2 D 1, 1 t D; (b) λ(t) m 2 D, 1 t D 1; λ(d) m D. Proof: (a) Let x, y V be two vertices such that d(x, y) t, and κ(x, y) their local connectivity. Since κ(x, y) is the maximum number of internally disjoint x y paths, the order of the digraph must satisfy κ(x, y)(d(x, y) 1) + 2 n. Then, κ(t)(t 1) + 2 n, and therefore κ(t) n 2 n 2 t 1. When t = D we have κ(d) D 1, and the result follows from (1). (b) From (5), (6) and case (a) we deduce that λ(t) = κ(t + 1; LG) m 2 D, 1 t D 1. Let x, y V such that d(x, y) = D and λ(x, y) their local edge connectivity. Since λ(x, y) is the maximum number of internally edge disjoint x y paths, the size of the digraph must satisfy: λ(x, y)d(x, y) m. Then λ(d)d m, and therefore λ(d) m D. When l = D, a similar reasoning allow us to obtain the following slightly better result. Lemma 2.2 Let G = (V, A) be a digraph with diameter D, order n, size m, parameter l = D, and t-distance connectivities κ(t), λ(t). Then, (a) κ(t) n 1 D, 1 t D; (b) λ(t) m 1 m+1 D+1, 1 t D 1; λ(d) D+1. Theorem 2.3 For any given D 1(> 2) positive integers c 2 c 3 c D, there exists a digraph G = (V, A) whose t-distance connectivities are κ(2) = c 2, κ(3) = c 3,, κ(d) = c D. 6

7 Proof: We know that κ(d) n 2 D 1. Then, the order of the digraph must satisfy n = V c D (D 1) + 2. Therefore, we construct a digraph G with n = c D (D 1) + 2, and we denote its vertices as v 0, v ij, 1 i D 1, 1 j c D and v D. We consider a partition of V into the subsets B 0 = {v 0 }, B i = {v i1,, v icd }, 1 i D 1 and B D = {v D }. The adjacencies of the digraph are the following: Γ + (v 0 ) = B 1 ; Let w B i, 1 i D 2. Its out-neighbours are Γ + (w) = B i+1, w v i1, and Γ + (v i1 ) = {v i+1,1,, v i+1,cd i } B i+1 ; Γ + (w) = V {w}, for any w B D 1 ; Γ + (v D ) = B D 1. Let us show that the above digraph verifies the required conditions. First of all, we compute the diameter of G. For any w B D 1, e + (w) = 1, since these vertices are adjacent to any vertex of G except themselves, and e + (v D ) = 2 since Γ + (v D ) = B D 1. For any w B i, 1 i D 2, d(w, v D 1,1 ) = D 1 i, and hence, d(w, y) D i, for any y V. Therefore, e + (w) = D i D 1. Finally, e + (v 0 ) = D, because d(v 0, v D ) = D and if v V, v v D, d(v 0, v) < D. Thus, D(G) = max v V {e + (v)} = e + (v 0 ) = D. Since there are only two vertices at distance D, namely, v 0 and v D and there are exactly c D internally disjoint v 0 v D paths in G, κ(d) = κ(v 0, v D ) = c D. To study the other t-distance connectivities we first calculate the local connectivities κ(x, y) for any x, y V. First, notice that for any y V, Γ + (v D ) Γ (y), and hence κ(v D, y) = δ + (v D ) = c D. If x = v i1, 1 i D 2, we recall that Γ + (v i1 ) = {v i+1,1,..., v i+1,cd i } B i+1. (By way of illustration consider for instance x = v D 3,1, see Fig. 1.) We distinguish three different cases according to vertex y V, keeping in mind that for any y B j+1, 1 j D 1, either B j Γ (y) or B j {v j1 } Γ (y). (i) When y = v j1, 1 j D 1, we have that B j 1 Γ (v j1 ), and therefore we can find the following internally disjoint v i1 v j1 paths: v i1 v i+1r v j 1r v j1 if i + 1 < j D 1, or v i1 v i+1r v D 1r v j1 if 1 j i + 1, where 1 r c D i. Hence κ(v i1, v j1 ) = δ + (v i1 ) = c D i. (ii) When y B j, y v j1, 1 j D 2, in the worst case we have that v j 1,1 / Γ (y). In this case we can find the following internally disjoint v i1 y paths: 7

8 v i1 v i+1r v j 1,r y if i + 1 < j D 1, or v i1 v i+1r v D 1,r y if 1 j i + 1, where in both cases 2 r c D i. Besides, we have the following path from v i1 to y, which is disjoint with the above ones: v i1 v i+1,1 v D 1,1 y. Therefore κ(v i1, y) = δ + (v i1 ) = c D i, for any y B j, y v j1, 1 j D 2. (iii) When y = v 0 or y = v D, we have that Γ (y) = B D 1 and therefore we can find the following internally disjoint v i1 y paths: v i1 v i+1,r v D 1,r y for 1 r c D i, and hence κ(v i1, y) = c D i. Figure 1: Finally, if x = v 0 or x B i, x v i1, 1 i D 2, then Γ + (x) = B i+1 and it is easy to verify that κ(x, y) = c D for any y V. From the above results we get κ(2) = min{κ(x, y) : d(x, y) 2} = c 2, κ(3) = min{κ(x, y) : d(x, y) 3} = c 3,..., κ(d) = min{κ(x, y) : d(x, y) D} = c D. We now proceed to show that the three parameters κ(t), λ(t), and δ(t) of a digraph are independent. Theorem 2.4 For any given positive integer satisfying a b c and 3 t, there exists a digraph G = (V, A) with diameter D t, having these numbers as parameters, that is, κ(t) = a, λ(t) = b, and δ(t) = c. Proof: We construct a digraph with order n = V = s(d 1) + 2, s > c, and denote its vertices as in the proof of Theorem 3.1, that is, V = B 0... B D, being B 0 = {v 0 }, B i = {v i1,..., v is }, 1 i D 1, B D = {v D }. We next describe the adjacencies: Γ + (v) = V \{v}, for any v B D 1 ; Γ + (v D ) = B D 1 ; For any v B i, 0 i D t 1, or D t + 2 i D 2, Γ + (v) = B i+1 ; 8

9 We construct a complete symmetric digraph Kc a+1 with the first c a+1 vertices of B D t and all of these vertices have also the following out-neighbours: Γ + (v D t,i ) = {v D t+1,1,..., v D t+1,a }, 1 i c a + 1. Besides, for any v B D t different from such vertices, Γ + (v) = B D t+1 ; For any v B D t+1, v / Γ + (v D t,1 ), Γ + (v) = B D t+2 ; For any v Γ + (v D t,1 ) B D t+1, Γ + (v D t+1,i ) = v D t+2,i, 1 i a 1, and Γ + (v D t+1,a ) = {v D t+2,a,..., v D t+2,b }. Figure 2: κ(t) = a, λ(t) = b, δ(t) = c. The obtained digraph is shown in Fig. 2. Let us see now that it satisfies the theorem. Notice that any x V, x v D t,i, 1 i c a+1, such that e + (x) t, satisfies δ + (x) = s > c and δ (x) = 2s > c, and also it is easy to prove that κ(x, y) = λ(x, y) = s for any y V. Therefore, δ(t) = δ + (v D t,i ) = c, 1 i c a + 1, since e + (v D t,i ) = t. Furthermore, for any y V, we have κ(v D t,i, y) = a, since there exist exactly a internally disjoint paths from v D t,i to y. For instance, for i = 1 such paths are v D t,1 v D t+1,i v D 1,i y, 1 i a. Hence, κ(t) = a. On the other hand, we have the following b edge-disjoint paths from v D t,1 to any y V : v D t,1 v D t+1,i v D t+2,i v D 1,i y, 1 i a; and v D t,1 v D t,j v D t+1,a v D t+2,j v D 1,j y, a + 1 j b. Hence, λ(t) = b. We observe that if t = 1, κ(t) = κ, λ(t) = λ, δ(t) = δ. In this case a construction of a digraph verifying Theorem 2.4 was given by Geller and Harary in [8]. In the case of (undirected) graphs we have a result similar to Theorem 2.3. Now, recall that the radius and diameter of a graph G are always related by the following inequalities: r D 2r. Besides, if κ = δ we have κ = κ(1) = κ(2) = = κ(ρ) = = κ(r) where ρ = D 2. Theorem 2.5 For any given D ρ + 1 positive integers c ρ c ρ+1 c D, with c ρ 2 and D > 4 there exists a digraph G = (V, A) whose t-distance connectivities are κ(2) = = κ(ρ) = c ρ, κ(ρ + 1) = c ρ+1,, κ(d) = c D. 9

10 Proof: We construct a graph G = (V, A) with order n = V = c D (D 1)+2, and denote the vertices as in the proof of Theorem 2.3. The adjacencies are the following (see Fig. 3): Γ(v 0 ) = B 1, and Γ(v D ) = B D 1 ; All the edges between the partite sets: B i and B i+1, 1 i ρ 2; B ρ 1 and (B ρ {v ρ1 }) and (B j {v j1 }) and (B j+1 {v j+1,1 }), ρ j D 1; Γ(v i1 ) = {v i+1,1 } {(c i 1) first vertices of B i 1 }, for any ρ i D 1. To prove that the above graph satisfies the required conditions we follow the same lines of reasoning as in the above-mentioned theorem. The details are omitted. Figure 3: 3 Maximally distance connected digraphs In this section we give some conditions for a digraph to be maximally t-distance connected. From now on we suppose δ(t) > 1, since if δ(t) = 1 the digraph is obviously maximally t-distance connected. The following result shows that for a s-geodetic digraph with parameter l, the distance connectivity κ(2l) uniquely determine the connectivities κ(t) for any t 2s, and analogously for the t-distance edge connectivity. Theorem 3.1 Let G = (V, A) be a s-geodetic digraph with diameter D, parameter l, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, (a) κ(t) = min{δ(t), κ(2l)}, for any t 2s; (b) λ(t) = min{δ(t), λ(2l + 1)}, for any t 2s + 1. Proof: We will first prove (a). Let F be a minimum t-distance disconnecting set of G, that is F = κ(t). Then, there exist two vertices u, v V \F such that d(u, v) t 10

11 and there are no paths from u to v in G\F. Then, the set V \F can be partitioned into two disjoint nonempty sets V, V + such that G\F has no edges from V to V +. Let the vertices of V and V + be respectively partitioned into subsets V i and V j, where 1 i k, 1 j k, according to their distance to and from F, that is, V i = {x V : d(x, F ) = i} and V j = {x V + : d(f, x) = j}. As any path from V to V + goes through F, the distance from a vertex in V k to one in V k is at least k + k and hence D k + k. Without loss of generality, suppose k k (if not, use the converse digraph of G.) We know that κ(t) δ(t). Since t 2s 2l it follows that κ(t) κ(2l), therefore κ(t) min{δ(t), κ(2l)}. To show the converse inequality we distinguish two cases: (a.1) k l. Then, if x V k, y V k, we have d(x, y) k + k 2l. Hence, κ(t) = F κ(2l). (a.2) k l 1. We now consider two subcases: (i) k s 1. Let us consider the afore-mentioned vertex u V i, i k, and let u 1,..., u δ(t) be δ(t) of its out-neighbours. For each u i, let f i be a vertex in F at minimum distance from u i. If f i = f j for some i j, then there would be two disjoint u f i paths of length at most k + 1 s, contradicting that G is s-geodetic. Hence, κ(t) δ(t). (ii) s k. Let x V k, we have that e + (x) t since, for any y V k, d(x, y) k+k 2s t. Let x 1,..., x δ(t) be δ(t) of its out-neighbours. As in case (i), for each x i, let f i be a vertex in F at minimum distance from x i. Now, if f i = f j, for some i j, then there would be two disjoint x f i paths of length k or k+1 contradicting the definition of l, since k d(x, f i ) 1+d(x i, f i ) 1+k l. So κ(t) δ(t). (b) In this case, let E be a minimum t-distance disconnecting set and consider the two disjoint vertex sets F = {f : (f, f ) E} and F = {f : (f, f ) E}. Then we define V i = {x : d(x, F ) = i} V and V j = {x : d(f, x) = j} V + as above. Now the distance from a vertex in V k to one in V k is at least k + k + 1 and hence D k + k + 1. Furthermore, we have λ(t) {δ(t), λ(2l + 1)}. To complete the proof we again consider two cases: (b.1) k l. So, if x V k, y V k, we get d(x, y) k + k + 1 2l + 1. Then, λ(t) = E λ(2l + 1). 11

12 (b.2) k l 1. If k 1, we can reason as in case (a.2). When k = 0, we have V = F. Then, let u F with e + (u) t, and let u 1, u 2,..., u δ(t) be δ(t) of its out-neighbours. If Γ + (u) F it is clear that δ(t) F E = λ(t). Otherwise, suppose that u 1, u 2,..., u j F and u j+1, u j+2,..., u δ(t) F. So, E contains the following edges (u i, w), 1 i j, w F, and (u, u i ), j + 1 i δ(t). Since G has no loops, all these edges are different and, therefore, δ(t) E = λ(t). Corollary 3.2 Let G = (V, A) be a s-geodetic digraph with diameter D, parameter l, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, (a) κ(t) = δ(t) for any t 2s, if D 2l 1; (b) λ(t) = δ(t) for any t 2s + 1, if D 2l. Proof: (a) As D 2l 1, using the same notation as in Theorem 3.1, we have 2k k + k D 2l 1. Hence k l 1 and we are in case (a.2) of the above proof, then κ(t) = δ(t). (b) Use the same reasoning as in (a). Since s 1, for t = 1 the above corollary gives a sufficient condition for a digraph G to have maximum connectivity, which was already formulated in [5, 7, 8]. A digraph is said to be maximally distance connected if κ(t) = λ(t) = δ(t) for any 1 t D, and maximally edge distance connected if λ(t) = δ(t) for any 1 t D. The next corollary gives a sufficient condition for a digraph to be maximally distance connected. Corollary 3.3 Let G = (V, A) be a s-geodetic digraph with diameter D, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, (a) κ(t) = δ(t) for any t D, if D 2s 1; (b) λ(t) = δ(t) for any t D, if D 2s. 12

13 If k is large enough, the k-iterated line digraph L k G satisfies the conditions on the diameter of Corollary 3.2 since, from (5) and (8), D(L k G) 2l(L k G) 1 k D(G) 2l(G) + 1; D(L k G) 2l(L k G) k D(G) 2l(G). Moreover, if G is s-geodetic then L k G is s -geodetic with s = min{s + k, g 1}, where g denotes the girth of G. Corollary 3.4 Let G = (V, A) be a s-geodetic digraph with diameter D, parameter l, girth g, and t-degree δ(t). Let s = min{s + k, g 1}. Then, (a) κ(t; L k G) = δ(t; L k G) for any t 2s, if k D 2l + 1; (b) λ(t; L k G) = δ(t; L k G) for any t 2s + 1, if k D 2l. Let now G = (V, E) be a graph and consider its associated symmetric digraph G, that is the digraph obtained from G by replacing each edge xy E by the directed edges (x, y) and (y, x) forming a digon. Then it turns out that the parameters l = l(g ) and s = s(g ), the maximum value for which G is s-geodetic, are equal and closely related with the girth g of G. Indeed, it is easy to realize that l = s = g 1 2. Furthermore, as a minimum t-distance disconnecting set cannot contain digons, we have that κ(t; G ) = κ(t; G), λ(t; G ) = λ(t; G), and the next corollary is now clear. Corollary 3.5 Let G = (V, E) be a graph with diameter D, girth g, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, (a) κ(t) = min{δ(t), κ(g 1)} for any t g 1, g odd, κ(t) = min{δ(t), κ(g 2)} for any t g 2, g even; (b) λ(t) = min{δ(t), λ(g)} for any t g, g odd, λ(t) = min{δ(t), λ(g 1)} for any t g 1, g even. Hence, we get the following sufficient conditions involving the diameter and the girth for a graph to be maximally distance connected. 13

14 Corollary 3.6 Let G = (V, E) be a graph with diameter D, girth g, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, for any 1 t D, (a) κ(t) = δ(t) if (b) λ(t) = δ(t) if { D g 2, g odd D g 3, g even; { D g 1, g odd D g 2, g even. A similar study can be carried out in the case of bipartite digraphs. Note that, between any two vertices of a bipartite digraph there are no two paths whose lengths differ by one. Hence, in this case the parameter l can be redefined as follows. Definition 3.7 For a given bipartite digraph G, let l = l(g), 1 l D, be the greatest integer such that, for any two vertices x, y V at distance d(x, y) l, the shortest x y path is unique. The following result is the analogue of Theorem 3.1. Theorem 3.8 Let G = (V, A), V = U 1 U 2, be a s-geodetic bipartite digraph with diameter D, parameter l, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, (a) κ(t) = min{δ(t), κ(2l + 1)}, for any t 2s; (b) λ(t) = min{δ(t), λ(2l + 2)}, for any t 2s + 1. Proof: We use the same notation as in the proof of Theorem 3.1. Thus, as before, in proving (a) we distinguish two cases: (a.1) k l. If k l + 1 we are done since d(x, y) d(x, F ) + d(f, y) 2l + 2. Then, κ(t) = F κ(2l + 2) κ(2l + 1). On the other hand, if k = l we need to consider two subcases: (i) V k U i for each i=1,2. Then there exist two vertices x V k U 1, x V k U 2 such that d(x, y) k + k 2l and, similarly, d(x, y) k + k 2l. Hence, since x and x belong to different partite sets, at least one of the above distances must be not smaller than 2l + 1. Hence, κ(t) = F κ(2l + 1). 14

15 (ii) V k U i = for i=2, say. Then all the out-neighbours of x V k must be in V k 1. Since e + (x) 2l t, we can consider δ(t) of its out-neighbours and, as before, if f i = f j for some i j, there would be two disjoint x f i paths of length l, a contradiction. Hence, κ(t) δ(t). (a.2) k l 1. This case is proved as in Theorem 3.1. Case (b) can be proved by using a similar reasoning. As a consequence of the above theorem we can now obtain results which are similar to Corollaries As could be expected, the obtained upper bounds on the diameter are now one greater than before. For instance, we get the following conditions for a bipartite graph to be maximally distance connected. Corollary 3.9 Let G = (V, E) be a bipartite graph with diameter D, (even) girth g, t-distance connectivities κ(t) and λ(t), and t-degree δ(t). Then, for any 1 t D, (a) κ(t) = δ(t) if D g 2; (b) λ(t) = δ(t) if D g 1. Therefore, any bipartite graph with diameter three is maximally distance edge connected (From Theorem 3.8, the result also applies for digraphs.) A similar result for the standard edge connectivity of a graph was given by Plesník and Znám [15] and, in the case of digraphs, by Fiol and Fàbrega [8]. References [1] M. Aigner, On the linegraph of a directed graph. Math. Z. 102 (1967) [2] J.-C. Bermond, N. Homobono, and C. Peyrat, Large fault-tolerant interconnection networks. Graphs and Combinatorics 5 (1989) [3] B. Bollobás, Extremal Graph Theory. Academic Press, London (1978). 15

16 [4] G. Chartrand and L. Lesniak, Graphs and Digraphs. Wadsworth, Monterey, CA (1986). [5] J. Fàbrega and M. A. Fiol, Maximally connected digraphs. J. Graph Theory 13 (1989), [6] J. Fàbrega and M.A. Fiol, Extraconnectivity of graphs with large girth. Discrete Math., to appear. [7] M.A. Fiol, J. Fàbrega and M. Escudero, Short paths and connectivity in graphs and digraphs. Ars Combin. 29B (1990) [8] M.A. Fiol and J. Fàbrega, On the distance connectivity of graphs and digraphs. Discrete Math. 125 (1994) [9] M.A. Fiol, J.L.A. Yebra and I. Alegre, Line digraph iterations and the (d, k) digraph problem. IEEE Trans. Comput. C-33 (1984) [10] D. Geller and F. Harary, Connectivity in digraphs. Lec. Not. Math. 186, Springer, Berlin (1970) [11] F. Harary, Conditional connectivity, Networks 13 (1983) [12] F. Harary, R.Z. Norman, D. Cartwright, Introduction à la Théorie des Graphes Orientés. Dunod, Paris (1968). [13] F. Harary, Graph Theory. Addison-Wesley, Reading (1971). [14] J. Plesník and S. Znám, Strongly geodetic directed graphs, Acta Fac. Rerum Natur. Univ. Comenian., Math. Publ. 29 (1974) [15] J. Plesník and S. Znám, On equality of edge-connectivity and minimum degree of a graph, Arch. Math. (Brno) 25 (1989) [16] S.M. Reddy, J.G. Kuhl, S.H. Hosseini and H. Lee, On digraphs with minimum diameter and maximum connectivity. Proceedings of the 20th Annual Allerton Conference (1982) [17] M.E. Watkins, A lower bound for the number of vertices of a graph. Amer. Math. Monthly 74 (1967)