Interlocking directorships-where individuals simultaneously serve on the boards of multiple corporations-can facilitate the exchange of expertise and strategic alignment but also present risks, including conflicts of interest, economic 'oligarchy', and regulatory non-compliance. In contexts such as large, family-controlled corporate conglomerates in India, the manual detection of interlocks is hindered by the high volume of corporate entities and the complex involvement of extended familial networks. This study introduces a scalable, graph-theoretic framework for the systematic identification and analysis of interlocking directorships. Using Breadth-First Search (BFS) traversal, we examined a curated dataset comprising over 50,000 directors, 85,000 companies, and 300,000 director-company affiliations, yielding a comprehensive representation of corporate network structures. Large Language Models (LLMs) were integrated into the analytical pipeline to characterize both personal and professional linkages among directors. Empirical results indicate that 17% of directors hold positions in exactly two companies, while 58.6% maintain directorships in two or more companies. The combined BFS-LLM methodology enables the detection of recurrent director-company clusters, indicative of strong network cohesion, and provides qualitative insights into potential underlying drivers of these interlocks. The proposed approach enhances the capacity for automated, data-driven detection of complex intercorporate relationships, offering actionable implications for corporate governance, regulatory monitoring, and systemic risk assessment.