subdivision

Subdivision has multiple meanings in different disciplines. In mathematics, a subdivision of a graph is obtained by replacing edges with paths through the insertion of new vertices of degree two along edges. Repeated subdivisions can transform any graph, and the concept is central to topological graph theory; a graph is nonplanar if it contains a subdivision of K5 or K3,3 (Kuratowski's theorem). Subdivisions also appear in piecewise-linear topology and in the study of routes and networks.

In urban planning and real estate, a subdivision is the division of a tract of land into multiple parcels for development or sale. The process typically involves regulatory approvals, environmental review, and the creation of infrastructure such as streets, utilities, and drainage. Steps usually include a concept or sketch plan, a tentative or preliminary map, and a final plat recorded with the jurisdiction. Subdivision regulations govern lot sizes, street patterns, lot access, dedication of rights-of-way, and covenants.

In music, subdivision refers to the division of a higher-duration beat or note into smaller rhythmic units to define tempo, timing, or articulation. Players use subdivisions such as eighth notes, sixteenth notes, or triplets to perform evenly spaced notes within a beat, often indicated in musical notation or by counting strategies.