Conditional (t,k)-Diagnosis in Graphs by Using the Comparison Diagnosis Model

Abstract

(t, k)-Diagnosis, which is a generalization of sequential diagnosis, requires that at least k faulty processors be identified and repaired in each iteration when there are at most t faulty processors, where t ≥ k. Based on the assumption that each vertex is adjacent to at least one fault-free vertex, the conditional (t, k)-diagnosis of graphs was investigated by using the comparison diagnosis model. Lower bounds on the conditional (t, k)-diagnosability of graphs were derived, and applied to obtain the following results. 1) Symmetric d-dimensional grids are conditionally (N/2d+1 -1, 2d -1)-diagnosable when d ≥ 2 and N (the number of vertices) ≥ 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . 2) Symmetric d-dimensional tori are conditionally (1/5 (N + min{8/5 N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> , 2N-20/15} - 2), 6)-diagnosable when d = 2 and N ≥ 49 and ( N/2d+1 -1, 4d-2)-diagnosable when d ≥ 3 and N ≤ 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . 3) Cube-connected cycles are conditionally (N/4 - 1, 4)-diagnosable. 4) k-ary trees are conditionally (N/k+1 - 1)-diagnosable.