The Regional Incidence Matrix of Good Graph Drawings

This paper introduces the regional incidence matrix for good drawings of graphs, defined via planarization. The construction extends the face incidence matrix of planar graphs to drawings with crossings. Two local identities (edge and vertex) are established and combined into a global formula expressing the crossing number in terms of the regional matrix and a repetition defect parameter. This yields a functional representation of the crossing number as a minimization problem over realizable matrix–defect pairs. This is a preprint. The paper is intended for submission to a peer-reviewed journal.