Fault-Tolerant Routings in Chordal Ring Networks Lali Barrière Departament de Matemàtica Aplicada i Telemàtica, Universitat Politècnica de Catalunya, Barcelona, Spain Josep Fàbrega Departament de Matemàtica Aplicada i Telemàtica, Universitat Politècnica de Catalunya, Barcelona, Spain Ester Simó Departament de Matemàtica Aplicada i Telemàtica, Universitat Politècnica de Catalunya, Barcelona, Spain Marisa Zaragozá Departament de Matemàtica Aplicada i Telemàtica, Universitat Politècnica de Catalunya, Barcelona, Spain This paper studies routing vulnerability in networks modeled by chordal ring graphs. In a chordal ring graph, the vertices are labeled in Z 2n 2n 2n and each even vertex i is adjacent to the vertices i + a, i + b, i + c i + c i + c, where a, b, and c are different odd integers. Our study is based on a geometrical representation that associates to the graph a tile which periodically tessellates the plane. Us- ing this approach, we present some previous results on triple-loop graphs, including an algorithm to calcu- late the coordinates of a given vertex in the tile. Then, an optimal consistent fault-tolerant routing of shortest paths is defined for a chordal ring graph with odd di- ameter and maximum order. This is accomplished by associating to the chordal ring graph a triple-loop one. When some faulty elements are present in the network, we give a method to obtain central vertices, which are vertices that can be used to reroute any communication affected by the faulty elements. This implies that the di- ameter of the corresponding surviving route graph is optimum. © 2000 John Wiley & Sons, Inc. Keywords: chordal ring networks; fault-tolerance; routing vul- nerability; plane tessellations 1. INTRODUCTION Loop networks have been widely considered in recent years as good models for interconnection or communi- cation computer networks, due to their regularity, simple Received March 1998; accepted May 2000 Correspondence to: J. Fàbrega; e-mail: matjfc@mat.upc.es Spanish Research Council (Comisión Interministerial de Ciencia y Tec- nología, CICYT); grant nos. TIC 94-0592 and TIC 97-0963 c  2000 John Wiley & Sons, Inc. structure, and symmetry. See Bermond et al. [5] for an exhaustive survey on this topic. Within this context, two problems that have been considered in the literature— because they are closely related to the network band- width and the transmission delay—are the minimization of the diameter for a given number of nodes and max- imum degree and the maximization of the number of nodes given the diameter and the maximum degree. Au- thors that have contributed to the study of these problems are, among others, Aguiló and Fiol [1], Aguiló et al. [2], Bermond and Tzvieli [7], Chen and Jia [8], Du et al. [11], Erdös and Hsu [12], Esqué et al. [14], Fiol et al. [17], Hsu and Shapiro [19, 20], Tzivieli [29], and Yebra et al. [31]. See also the survey of Hwang [21]. Chordal ring graphs of degree 3 constitute a family of generalized loop networks that, although they were first introduced by Coxeter [9] more than 50 years ago, were first considered from the point of view of intercon- nection networks by Arden and Lee in [3]. These graphs are simply obtained by adding chords in a regular man- ner to an undirected cycle. Yebra et al. [31] improved the number of vertices (in relation to its diameter) in the construction of Arden and Lee. Moreover, several op- timal constructions of generalized chordal ring graphs were presented in this reference. Following this work, and also Morillo [25], our study is based on a geometri- cal representation that associates to a generalized chordal ring graph a tile which periodically tessellates the plane. This geometrical approach leads us to the study of lat- tices and congruences in Z 2 . Within this framework, see the contributions of Fiol [16] and Morillo [26]. It is worth NETWORKS, Vol. 36(3), 180–190 2000