The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and faulttolerance in a multiprocessor system. The conditional diagnosability t c (G) of G is the maximum number t for which G is conditionally t-diagnosable under the comparison model, while the 2-extra connectivity κ 2 (G) of a graph G is the minimum number k for which there is a vertex-cut F with |F | = k such that every component of G − F has at least 3 vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answer this problem by proving t c (G) = κ 2 (G) for a regular graph G with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-It is well known that a topological structure of an interconnection network N can be modeled by a graph G = (V, E), where V represents the set of components such as processors and E represents the set of communication links in N (see a text-book by Xu [42]). Faults of some processors and/or communication lines in a large-scale system are inevitable. People are concerned with how to diagnose faults and to determine fault tolerance of the system.

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