Journal of Computer and System Sciences 115 (2021) 64–72 Contents lists available at ScienceDirect Journal of Computer and System Sciences www.elsevier.com/locate/jcss Edge fault-tolerance analysis of maximally edge-connected ✩ graphs and super edge-connected graphs ∗ Shuang Zhao a, Zongqing Chen b, Weihua Yang c, Jixiang Meng a, a College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China b School of Science, Tianjin Chengjian University, Tianjin 300384, China c Department of Mathematics, Taiyuan University of Techology, Taiyuan 030024, China a r t i c l e i n f o a b s t r a c t Article history: Edge fault-tolerance of interconnection network is of significant important to the design Received 8 October 2019 and maintenance of multiprocessor systems. A connected graph G is maximally edge- Received in revised form 20 May 2020 connected (maximally-λ for short) if its edge-connectivity attains its minimum degree. G is Accepted 8 July 2020 super edge-connected (super-λ for short) if every minimum edge-cut isolates one vertex. Available online 23 July 2020 The edge fault-tolerance of the maximally-λ (resp. super-λ) graph G with respect to the Keywords: maximally-λ (resp. super-λ) property, denoted by mλ(G) (resp. Sλ(G)), is the maximum − Edge fault-tolerance integer m for which G S is still maximally-λ (resp. super-λ) for any edge subset S Maximally edge-connected with |S| ≤ m. In this paper, we give upper and lower bounds on mλ(G). Furthermore, we Super edge-connected completely determine the exact values of mλ(G) and Sλ(G) for vertex transitive graphs. Vertex transitive graph © 2020 Elsevier Inc. All rights reserved. 1. Introduction It is well-known that a network can be modeled by a graph G = (V (G), E(G)), where the vertex set V (G) represents the node-set of the network and the edge set E(G) represents the link-set of the network. Then requirements in the design of the network correspond to graph measures. In [33], several measurements for building networks, such as diameter, hamiltonicity, connectivity and symmetry are discussed. Since link faults may happen when a network is put in use, apart from such network requirements, it is practically meaningful and important to consider faulty networks [17]. The problems of diameter and hamiltonicity have been solved on a variety of faulty networks, see [16–18,25,30,35]. Comparatively, there are few results about connectivity, concerning with the faulty networks. As a traditional measurement for reliability and fault-tolerance of the network, the edge-connectivity of a connected graph G, denoted by λ(G), is the minimum cardinality of a set of edges (named as edge-cut) of G, whose removal from G makes the remaining graph no longer connected. It is well-known that λ(G) ≤ δ(G), where δ(G) is the minimum degree of G. Hence, a graph G with λ(G) = δ(G) is said to be maximally edge-connected, maximally-λ for short. To design more reliable networks, besides the requirement of maximal edge-connectivity, it is also desirable that the number of minimum edge-cuts is as small as possible. For this purpose, Bauer et al. [3]defined the super edge-connected graphs. A connected graph G is called super edge-connected, or simply super-λ, if every minimum edge-cut isolates one vertex. A more refined measurement is the restricted edge-connectivity, proposed by Esfahanian and Hakimi [9]. The restricted edge-connectivity λ (G), is the minimum cardinality of a set of edges (named as restricted edge-cut) of G, if any, whose dele- ✩ The research is supported by NSFC (No. 11531011). * Corresponding author. E-mail address: [email protected] (J. Meng). https://doi.org/10.1016/j.jcss.2020.07.002 0022-0000/© 2020 Elsevier Inc. All rights reserved. S. Zhao et al. / Journal of Computer and System Sciences 115 (2021) 64–72 65 tion makes G not connected and the remaining graph contains no isolated vertices. A graph G is said to be λ -connected, if λ (G) exists. A λ -connected graph G is called super restricted edge-connected, or simply super-λ , if every minimum restricted edge-cut isolates an edge. Known results about λ(G) and λ (G) are presented in [6,12,13,20–24,27,28,32,34,35]. In [14], Hong, Meng and Zhang proposed a parameter, denoted by Sλ(G), to consider super edge-connectivity in graphs with faulty edges. The index Sλ(G) is the maximum integer m for which G − S is still super-λ for any edge subset S with |S| ≤ m. In [14], the authors gave upper and lower bounds on Sλ(G) and presented more refined bounds on Sλ(G) for some special classes of graphs. Motivated by these, a quite natural problem can be introduced as follows. For a graph G with some connectivity property P, how many faulty edges can be tolerated such that the remaining graph still has property P? We can define a parameter with respect to property P, denoted by mP (G), as the maximum integer m for which G − S still has property P for any edge subset S with |S| ≤ m. The parameter mP (G) can be used to evaluate the reliability and fault-tolerance more reasonably. If one seeks to destroy a network in such a way that the damaged network no longer has property P, then we have to destroy at least mP (G) + 1edges. Thus, the value of mP (G) will provide a beneficial reference for engineers when designing or selecting networks to build parallel systems. Wang and Lu [29]studied three families of super-λ networks with faulty edges. In [15], Hong and Xu investigated mP (G) when P denotes the property of super-λ , and determined the exact value of mP (G) for two families of networks. Wang et al. [31] considered super-λ Cartesian product graphs with faulty edges. In [8], Cheng and Hsieh gave upper and lower (k) bounds on mP (G), where P denotes the property of super-λ (see [8]for definition) for k ≥ 3. This paper is concerned with this parameter, denoted by mλ(G), when P denotes the property of maximally-λ. In [26], Sun, Zhao and Meng studied mλ(G) for two families of networks. The rest of this paper is organized as follows. In Section 2, we introduce some definitions and related results which are to be used in this paper. In Section 3, we present upper and lower bounds on mλ(G). In Section 4, we focus on giving values of mλ(G) under some conditions. In Section 5, we completely determine how many edges can be deleted such that the remaining graph is still super-λ for super-λ vertex transitive graphs. 2. Definitions and related results In this section, we introduce some definitions and related results which will be used in the following sections. For graph-theoretical terminologies and notations not given here, we follow [5,11]. = | | Let G (V (G), E(G)) be a graph. The order of G is the size of V (G) . We use NG (v) to denote the neighbor set of the vertex v ∈ V (G), and NG (A) = ( ∈ NG (v)) \ A for A ⊆ V (G). We write dA (v) =|NG (v) ∩ A|. For X, X ⊆ V (G), notations v A [X, X ]G and G[X] are used to denote the set of edges between X and X in G, and the subgraph of G induced by X, respectively. When the graph under consideration is obvious, we use ω(X) instead of [X, X], where X = V (G) \ X. A clique of G is a subset X of V (G) such that G[X] is complete. If, furthermore, |X| = k, then X is a k-clique. We use Kn, Cn and K1,n−1 to denote the complete graph, the cycle and the star of order n. In the design of network topology, highly symmetric graphs, for instance vertex transitive graphs or edge transitive graphs, are popular due to their desirable properties [2,19,33]. If for any two vertices u, v ∈ V (G), there is an automorphism φ ∈ Aut(G) such that φ(u) = v, then G is vertex transitive, where Aut(G) denotes the automorphism group of G. A vertex transitive graph is always regular. It is known that vertex transitive graphs are maximally-λ [22]. Circulant graphs are ≤ ≤ n examples of vertex transitive graphs. Let Zn be the ring of integers modulo n and 1 a1 < a2 <...<ak 2 be k integers. The circulant graph G(n; a1, a2, ..., ak) is a graph with vertex set Zn, and for any i ∈ Zn, i is adjacent to i ± a1, i ± a2, ..., i ± ak (mod n). And G(n; a1, a2, ..., ak) is connected if and only if gcd(n, a1, a2, ..., ak) = 1. G is edge transitive if for any two edges e, f ∈ E(G), there is an automorphism φ ∈ Aut(G) such that φ(e) = f . Since edge transitive graph G satisfies κ(G) = δ(G) [32], where κ(G) is the vertex-connectivity of G, combining with κ(G) ≤ λ(G) ≤ δ(G), we see that edge transitive graph is also maximally-λ. Now, we introduce the definition of the edge fault-tolerance of a maximally-λ graph G with respect to the maximally-λ property. Definition 2.1. A maximally-λ graph G is m-maximally edge-connected (m-maximally-λ for short) if G − S is still maximally-λ for any edge subset S ⊆ E(G) with |S| ≤ m.