On the conditional diagnosability of Cayley graphs generated by 2-trees and related networks Eddie Cheng Department of Mathematics & Statistics Oakland University Rochester, MI 48309, USA. Email: echeng@oakland.edu L´ aszl´ o Lipt´ ak Department of Mathematics & Statistics Oakland University Rochester, MI 48309, USA. Email: liptak@oakland.edu Ke Qiu Department of Computer Science Brock University St. Catharines, ON, Canada L2S 3A1 Email:kqiu@brocku.ca Zhizhang Shen Department of Computer Science & Technology Plymouth State University Plymouth, NH 03264, USA Email:zshen@plymouth.edu Abstract—In this note, we utilize existing results to derive the exact value of the conditional diagnosability for Cayley graphs generated by 2-trees, which generalize the alternating group graphs. In addition, the corresponding problem for arrangement graphs and hyper Petersen networks will also be discussed. Keywords-Fault diagnosis; self-diagnosable system; compari- son diagnosis model; conditional diagnosability; Cayley graph; k-trees; arrangement graphs; hyper Petersen networks I. I NTRODUCTION Thanks to constant technological progress, multiproces- sor systems with ever increasing number of interconnected computing nodes are becoming a reality. To address the reliability concern of such a system, it is ideal, and tech- nically feasible, to have a self-diagnosable system where the computing nodes are able to detect faulty ones by themselves in the form of a diagnosis. One major approach to this regard is called the comparison diagnosis model [20], [21], where each node performs a diagnosis by sending the same input to all pairs of its distinct neighbors and then comparing their responses. Based on such comparison results made by all the processors, the faulty status of the system can be decided. The number of detectable faulty nodes in such a multiprocessor system certainly depends on the topology of its associated interprocessor structure, as well as the modeling assumptions, and the maximum number of detectable faulty nodes in such a network is called its diagnosability. Such a measurement directly characterizes the fault-tolerance ability of an interconnection network and is thus of great interest [18], [19], [23], [30]. When all the neighbors of some processor in a network are faulty simultaneously, it is impossible to determine the faulty status of this processor, as well as the whole system. Hence, the unrestricted diagnosability of a network, when represented with a graph G, is limited by the minimum degree of G, often too small thus unsatisfying. On the other hand, with the often made statistical assumption of independent and identical distribution (i.i.d.) of failures among processors, it is simply unlikely that all the neighbors of a certain processor will fail at the same time, hence the notion of conditional diagnosability was introduced in [18] which assumes that no conditional faulty set contains all the neighbors of any processor. This more realistic notion leads to an improved characterization of a network’s fault- tolerance properties, and has since been identified for several networks, including hypercubes [14], folded hypercubes [24], augmented cubes [13], [24], Cayley graphs generated by transposition trees [19], alternating group networks [30], BC Networks [15], [31], pancake graphs [24], and (n, k)-star graphs [8], all under the above comparison diagnosis model. Much work has also been done under another diagnostic model, the PMC model [22], where diagnosis is made by testing adjacent nodes. Although it is pointed out in [23] that the comparison model generalizes the PMC model, diagnosability results achieved under the comparison model are often smaller than those achieved under the PMC model. Initially, ad-hoc methods were used in determining the conditional diagnosability as in [18], [19], [29]–[31]. This topic has been slowly converging to an unified approach as developed in [8], [13]. In this note, we employ such an approach and utilize existing results to derive the exact value of the conditional diagnosability for Cayley graphs generated by 2-trees and other interconnection networks. II. FUNDAMENTAL NOTIONS AND RESULTS In this paper, we follow the usual graph theory terminol- ogy which can be found in [3]. In particular, let G be a graph and v ∈ V (G), N G (v) is the set of all the vertices adjacent to v and N G (S)=  v∈S N (v) \ S where S ⊂ V (G). 2012 International Symposium on Pervasive Systems, Algorithms and Networks 1087-4089/12 $26.00 © 2012 IEEE DOI 10.1109/I-SPAN.2012.15 58