Mixed-level detecting arrays on graphs

The purpose of this paper was to develop a unified combinatorial framework for fault localization in heterogeneous software systems, where parameters may have different numbers of levels. Specifically, we investigated mixed-level detecting arrays (MDAs) on graphs, extending the classical detecting array model to accommodate non-uniform factor structures. In this paper, we established an optimality criterion that minimizes the number of required test cases and analyzes the structural and combinatorial properties of optimal MDAs on graphs. Furthermore, several constructive methods were proposed to generate optimal arrays, and existence results were derived that achieve the theoretical lower bounds. The findings enhance the theoretical understanding of detecting arrays in graph-based settings and provide practical guidelines for designing cost-efficient and fault-sensitive test suites in complex, heterogeneous software systems.