Overlap–Density Vertex Rough Graph

Rough set theory models imprecision by approximating a concept with lower and upper bounds induced by an indiscernibility relation. We extend vertex rough graphs with quantitative edge screening and formalize the Overlap–Density Vertex Rough Graph (ODVRG). This paper makes three main contributions: (i) we introduce an overlap coefficient for each candidate edge together with a local-density threshold that curbs overly loose upper approximations; (ii) we prove set-theoretic properties for OD lower and upper graphs and establish conditions under which intersection and union distribute over these approximations; and (iii) we derive clique-size bounds that link the boundary region to the maximum clique. A worked public-health example with a fully specified dataset illustrates how ODVRG separates core from boundary membership and how density control tightens edge acceptance. The framework preserves rough-set semantics on vertices while adding interpretable, data-driven control on edges, yielding crisper upper approximations and a more informative boundary structure in uncertain networks.