View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 3265–3296 www.elsevier.com/locate/disc Maximally edge-connected and vertex-connected graphs and digraphs: A survey Angelika Hellwiga, Lutz Volkmannb,∗ aInstitut für Medizinische Statistik, Universitätsklinikum der RWTH Aachen, PauwelsstraYe 30, 52074 Aachen, Germany bLehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany Received 8 December 2005; received in revised form 15 June 2007; accepted 24 June 2007 Available online 24 August 2007 Abstract Let D be a graph or a digraph. If (D) is the minimum degree, (D) the edge-connectivity and (D) the vertex-connectivity, then (D) (D) (D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge- connected if (D) = (D) and maximally vertex-connected if (D) = (D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively. © 2007 Elsevier B.V. All rights reserved. Keywords: Connectivity; Edge-connectivity; Vertex-connectivity; Restricted connectivity; Conditional connectivity; Super-edge-connectivity; Local-edge-connectivity; Minimum degree; Degree sequence; Diameter; Girth; Clique number; Line graph 1. Introduction Numerous networks as, for example, transport networks, road networks, electrical networks, telecommunication systems or networks of servers can be modeled by a graph or a digraph. Many attempts have been made to determine how well such a network is ‘connected’ or stated differently how much effort is required to break down communication in the system between some vertices. Clearly, it is desirable that a network stays connected as long as possible in case of faults should arise. Two classical measures that indicate how reliable a graph or a digraph D is are the edge-connectivity (D) and the vertex-connectivity or simply the connectivity (D) of D.If(D) is the minimum degree, then it is well known that (D)(D) and (D)(D). Thus, in order to construct reliable and fault-tolerant networks, sufficient conditions for graphs or digraphs D satisfying (D)=(D) (so-called maximally edge-connected) or (D)=(D) (so- called maximally connected) are of great interest. For the case that graphs or digraphs are maximally edge-connected or connected also further connectivity parameters are needed to investigate the fault tolerance, such as super-edge- connectivity, restricted (edge-)connectivity or local-edge-connectivity. Our objectives are to present results on (edge-)connectivity and maximally (edge)-connected graphs and digraphs. Furthermore, we take a look at the concept of conditional or restricted (edge-)connectivity. ∗ Corresponding author. E-mail addresses: [email protected] (A. Hellwig), [email protected] (L. Volkmann). 0012-365X/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2007.06.035 3266 A. Hellwig, L. Volkmann/ Discrete Mathematics 308 (2008) 3265–3296 For more information on connectivity in graphs we refer the reader to survey articles with different emphases by Fàbrega and Fiol [47], Mader [103], Oellermann [113] and Xu [177] and to the Ph.D. thesis by Hellwig [69]. 2. Some terminology and basic results We consider finite (di-)graphs without loops and multiple edges. For any (di-)graph D the vertex set is denoted by V(D)and the edge set (sometimes called oriented edge set or arc set in the case that D is a digraph) by E(D).We define the order of D by n = n(D) =|V(D)| and the size by m = m(D) =|E(D)|.IfG is a graph, then the vertex degree d(v) = dG(v) of a vertex v is the number of vertices of G adjacent to v. The minimum degree is denoted by = (G) and the maximum degree by = (G). The degree sequence of a graph G is defined as the nonincreasing sequence of the degrees of the vertices of G. For a vertex v ∈ V (G), the open neighborhood N(v)= NG(v) is the set of all vertices adjacent to v, and N[v]=NG[v]=N(v)∪{v} is the closed neighborhood of v.Ife = uv is an edge of a graph G, then let N(e)= NG(e) = (N(u) ∪ N(v))−{u, v}. For X ⊆ V (G) let G[X] be the subgraph of G induced by X. For a connected graph G, the distance d(u, v)=dG(u, v) between two vertices u and v is the length of a shortest path from u to v. The diameter dm(G) of a connected graph G is defined by dm(G) = maxu,v∈V (G) dG(u, v).IfX and Y are two subsets of V (G), then we denote by (X, Y ) the set of edges with one endpoint in X and the other one in Y and by d(X,Y) = dG(X, Y ) = minx∈X, y∈Y dG(x, y) the distance between X and Y. The girth g of a graph is the length of its shortest cycle. A (,g)-cage is a -regular graph with 2 and girth g 3 having the least possible number of vertices. Let p be a positive integer. A (di-)graph D is said to be p-partite if it is possible to partition V(D)into p subsets V1,V2,...,Vp (called partite sets) such that every edge joins a vertex of Vi to a vertex of Vj for i = j.IfG is a p-partite graph such that uv ∈ E(G) for every vertex u ∈ Vi and every v ∈ Vj for i = j, then we speak of a complete p-partite or complete multipartite graph.Forp = 2, such graphs are called bipartite graphs. The complete bipartite graph Kp,q has the bipartition A, B with |A|=p and |B|=q such that for every pair of vertices u ∈ A and v ∈ B there exists the edge uv. A vertex-cut in a graph G is a set X of vertices of G such that G − X is disconnected. The vertex-connectivity or simply the connectivity = (G) of a graph G is the minimum cardinality of a vertex-cut of G if G is not complete, and (G) = n − 1ifG is the complete graph Kn of order n.Anedge-cut in a graph G is a set S of edges of G such that G − S is disconnected. If S is a minimal edge-cut of a connected graph G, then, necessarily, G − S consists of exactly two components. Every nontrivial graph has an edge-cut. The edge-connectivity = (G) of a graph G is the minimum ¯ cardinality of an edge-cut of G if G is nontrivial and (K1) = 0. We denote by G the complementary graph of a graph G. A graph G is self-complementary if G = G¯ . For other graph theory terminology we follow Bondy and Murty [21] or Chartrand and Lesniak [29]. Connectivity, edge-connectivity and minimum degree of a graph are related by a basic inequality due to Whitney [171] in 1932. Theorem 2.1 (Whitney [171] 1932). For any graph G, (G)(G)(G). (1) Chartrand and Harary [27] showed that this result is best possible in the sense that if a,b and c are positive integers such that a bc, then there exists a graph G with (G) = a, (G) = b and (G) = c. Because of (1), it is reasonable to say that a graph G is maximally connected if (G) = (G) and maximally edge-connected if (G) = (G). The following result on the diameter of a graph and its complement will be used later, and can be found in the book by Bondy and Murty [21] on p. 14 as an exercise (for a proof cf. [148, p. 19]). Theorem 2.2 (Bondy and Murty [21] 1976). If G is a graph of diameter dm(G)4, then dm(G)¯ 2. Note that Theorem 2.2 generalizes the earlier result by Ringel [133] that a nontrivial self-complementary graph has diameter 2 or 3 as well as the observation that dm(G)3 implies dm(G)¯ 3 by Harary and Robinson [68]. A. Hellwig, L. Volkmann/ Discrete Mathematics 308 (2008) 3265–3296 3267 3. Maximally edge-connected graphs Sufficient conditions for equality of edge-connectivity and minimum degree for graphs were given by several authors. The starting point was an article by Chartrand [26] in 1966. He observed that if is large enough, then the second inequality in (1) becomes an equality. Theorem 3.1 (Chartrand [26] 1966). If G is a graph of order n(G)2(G) + 1, then (G) = (G). In 1974, Lesniak [91] proved the following strengthening of this result. Theorem 3.2 (Lesniak [91] 1974). If G is a graph with dG(u) + dG(v)n(G) − 1 for all distinct non-adjacent vertices u and v, then (G) = (G). One year later, Plesník [127] presented a nice condition based on the diameter that implies Theorems 3.1 and 3.2. Theorem 3.3 (Plesník [127] 1975). If G is a graph of diameter dm(G)2, then (G) = (G). In 1989, Plesník and Znám [128] could relax the condition in Theorem 3.3 in the sense that some distances can be greater than 2. Theorem 3.4 (Plesník and Znám [128] 1989). If in a connected graph G no four vertices u1,v1,u2,v2 with d(u1,u2), d(u1,v2), d(v1,u2), d(v1,v2)3 exist, then (G) = (G). A pair of sets X, Y ⊂ V (G) with dG(X, Y ) = k, where k is a positive integer, is called k-distance maximal, if there exist no sets X1 ⊇ X and Y1 ⊇ Y with X1 = X or Y1 = Y such that dG(X1,Y1) = k.