Edge fault tolerance analysis of super k-restricted connected networks C. Balbuena a, * , P. García-Vázquez b a Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain b Departamento de Matemática Aplicada I, Universidad de Sevilla, Sevilla, Spain article info Keywords: Connectivity Fault-tolerance k-Restricted edge connectivity Super-k k abstract An edge cut X of a connected graph G is a k-restricted edge cut if G  X is disconnected and every component of G  X has at least k vertices. Additionally, if the deletion of a minimum k-restricted edge cut isolates a connected component of k vertices, then the graph is said to be super-k k . In this paper, several sufficient conditions yielding super-k k graphs are given in terms of the girth and the diameter. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction The topology of a multiprocessor system can be modeled as an undirected graph G ¼ðV ; EÞ, where V represents the set of all processors and E represents the set of all connecting links between the processors. Among all fundamental properties for interconnection networks, the connectivity and edge connectivity are major parameters widely used for measures of func- tionality of the system. A basic definition of the connectivity of a graph G is defined as the minimum number of edges (or vertices) whose removal from G produces a disconnected graph. For example, the minimum number of faulty links in an n-cube that results in the remaining nodes being disconnected is its edge connectivity n. But the only case that n faulty links can disconnect an n-cube is that all these n links are neighboring to a same node [16]. However, the probability that all n faulty links are neighbors of the same node is very small. The use of forbidden faulty set [17] is motivated by the fact that the traditional graph connectivity model cannot correctly reflect network resilience of large systems. The vertices or edges in a forbidden faulty set cannot fail at the same time. By restricting the forbidden fault set to be the sets of neighboring edges of any spanning subgraph with not more than k-vertices in the faulty networks, Fàbrega and Fiol [12,13] introduced the k-extra-edge-connectivity of interconnection networks (where k is a non-negative integer) as follows. Given a graph G and a non-negative integer k, the k-extra-edge-connectivity k k ðGÞ of G is the minimum cardinality of a set of edges of G, if any, whose deletion disconnects G and every remaining com- ponent contains at least k vertices. The notion of k-extra-edge-connectivity was later extended to the directed case in [1]. The concept of the extraconnectivity was inspired by the definition of conditional connectivity introduced by Hararay [15] who asked for the minimum cardinality of a set of edges of G, if any, whose deletion disconnects G such that every remaining component satisfies some prescribed property. In this paper we adopt the following definition. Definition 1.1 ([12,13]). Given a graph G and a positive integer k P 1, the k-restricted edge connectivity k k ðGÞ of G is the minimum cardinality of a set of edges of G, if any, whose deletion disconnects G and every remaining component contains at least k vertices. From the definition, we immediately have that if k k ðGÞ exists, then k i ðGÞ exists for any i < k and k i ðGÞ 6 k k ðGÞ. Observe that k 1 ðGÞ is just the standard edge-connectivity kðGÞ (the minimum number of edges whose deletion disconnects the graph). Fur- thermore, the restricted edge connectivity k 0 ðGÞ defined by Esfahanian and Hakimi [11] is k 0 ðGÞ¼ k 2 ðGÞ. The restricted edge 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.01.060 * Corresponding author. E-mail addresses: m.camino.balbuena@upc.edu (C. Balbuena), pgvazquez@us.es (P. García-Vázquez). Applied Mathematics and Computation 216 (2010) 506–513 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc