As the scale of multiprocessor systems expands, link failures between processors become inevitable. Therefore, analyzing the reliability of the underlying topological graph G of an interconnection network is of crucial importance for the design and maintenance of such systems. To rigorously evaluate the fault tolerance of multiprocessor systems, the h-extra edge connectivity of a graph G, denoted by \(\lambda _h(G)\) , was introduced to assess their resilience. As a generalization of traditional edge connectivity, \(\lambda _h(G)\) is defined as the minimum number of edges whose removal splits a network of N processors with the underlying topological graph G into several components, each containing at least h processors, with \(h\le \left\lfloor N/2\right\rfloor \) . This specific parameter provides a refined quantitative analysis for the reliability of multiprocessor systems under link failures. The augmented 3-ary n-cube \(AQ_{n,3}\) , an interconnection network with \(N=3^n\) processors, extends the 3-ary n-cube \(Q_{n}^{3}\) by introducing complementary edges. This paper determines the values of \(\lambda _h\left( AQ_{n,3}\right) \) for all integers \(h\in \left[ 1,\left\lfloor 3^n/2 \right\rfloor \right] \) by finding the optimal solution of the edge isoperimetric problem of \(AQ_{n,3}\) , thereby refining the fault tolerance accuracy of \(AQ_{n,3}\) -based interconnection networks. The interval \(\left[ 1,\left\lfloor 3^n/2 \right\rfloor \right] \) is partitioned into several subintervals, and key properties of \(\lambda _h\left( AQ_{n,3}\right) \) are analyzed through this partitioning. Furthermore, the derivation of a recursive relation for \(\lambda _h\left( AQ_{n,3}\right) \) enables the design of an \(O\left( \log N\right) \) algorithm to compute the exact values of \(\lambda _h\left( AQ_{n,3}\right) \) for all integers \(h\in \left[ 1,\left\lfloor 3^n/2 \right\rfloor \right] \) .
