László Lipták

CONDITIONAL DIAGNOSABILITY OF CAYLEY GRAPHS GENERATED BY TRANSPOSITION TREES UNDER THE COMPARISON DIAGNOSIS MODEL

The diagnosis of faulty processors plays an important role in multiprocessor systems for reliable computing, and the diagnosability of many well-known networks has been explored. Zheng et al. showed that the diagnosability of the n-dimensional star graph Sn is n - 1. Lai et al. introduced a restricted diagnosability of multiprocessor systems called conditional diagnosability. They consider the situation when no faulty set can contain all the neighbors of any vertex in the system. In this paper, we study the conditional diagnosability of Cayley graphs generated by transposition trees (which include the star graphs) under the comparison model, and show that it is 3n - 8 for n ≥ 4, except for the n-dimensional star graph, for which it is 3n - 7. Hence the conditional diagnosability of these graphs is about three times larger than their classical diagnosability.

Linearly many faults in Cayley graphs generated by transposition trees

Abstract We prove that when linearly many vertices are deleted in a Cayley graph generated by a transposition tree, the resulting graph has a large connected component containing almost all remaining vertices.

On deriving conditional diagnosability of interconnection networks

We provide general techniques to estimate an upper bound of the conditional diagnosability of a graph G, and to prove that such a bound is also tight when a certain connectivity result is available for G. As an example, we derive the exact value of the conditional diagnosability for the (n,k)-star graph.

FAULT RESILIENCY OF CAYLEY GRAPHS GENERATED BY TRANSPOSITIONS

The star graph Sn, proposed by [1], has many advantages over the n-cube. It is shown in [2] that when a large number of vertices are deleted from Sn, the resulting graph can have at most two components, one of which is small. In this paper, we show that Cayley graphs generated by transpositions have this property.

Component connectivity of the hypercubes

The r-component connectivity κ r (G) of the non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. So, κ2 is the usual connectivity. In this paper, we determine the r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and we classify all the corresponding optimal solutions.

Matching preclusion and conditional matching preclusion for regular interconnection networks

Abstract The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and neither perfect matchings nor almost-perfect matchings. In this paper, we prove general results regarding the matching preclusion number and the conditional matching preclusion number as well as the classification of their respective optimal sets for regular graphs. We then use these general results to study the problems for Cayley graphs generated by 2-trees and the hyper Petersen networks.

Matching preclusion and conditional matching preclusion for bipartite interconnection networks II: Cayley graphs generated by transposition trees and hyper-stars

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. It is natural to look for obstruction sets beyond those induced by a single vertex. The conditional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph with no isolated vertices that has no perfect matchings. In this companion paper of Cheng et al. (Networks (NET 1554)), we find these numbers for a number of popular interconnection networks including hypercubes, star graphs, Cayley graphs generated by transposition trees and hyper-stars.

Matching preclusion and conditional matching preclusion problems for tori and related Cartesian products

The matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. The conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and no perfect matchings. In this paper we study this problem for the tori by proving matching preclusion and conditional matching preclusion results of the Cartesian products of graphs involving cycles. Our results generalize the one given for k-ary n-cube by Wang et al. (2010) [10] as well as provide classification for optimal conditional matching preclusion sets for these graphs.

Matching preclusion and conditional matching preclusion for bipartite interconnection networks I: Sufficient conditions

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. This number is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this article, we prove general results regarding the matching preclusion number and the conditional matching preclusion number as well as the classification of their respective optimal sets for bipartite graphs.

Orienting Cayley graphs generated by transposition trees

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123-129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs. Cheng and Lipman [E. Cheng, M.J. Lipman, On the Day-Tripathi orientation of the star graphs: Connectivity, Inform. Process. Lett. 73 (2000) 5-10; E. Cheng, M.J. Lipman, Connectivity properties of unidirectional star graphs, Congr. Numer. 150 (2001) 33-42] studied the connectivity properties of these unidirectional star graphs. The class of star graphs is a special case of Cayley graphs generated by transposition trees. In this paper, we give directions on these graphs and study the connectivity properties of the resulting unidirectional graphs.