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Graph Structure & Topological Foundations

Abstractions about the combinatorial and topological structures underlying discrete and continuous spaces — graph properties like planarity, duality, matching, and trees, paired with the open-set axioms that define a topological space.

6 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

  • Graph Duality — Pair a planar graph with a dual by placing a vertex inside each face and joining faces that share an edge, so vertices swap with faces and problems translate across a lossless table — paths become cuts, colorings become face colorings — letting you solve whichever side is easier.
  • Matching — Cast a pairing problem as a largest (or minimum-cost, or perfect) set of pairwise vertex-disjoint edges on an explicit graph, then split on bipartiteness to select the theorems and polynomial-time algorithm that solve it.
  • Open Set — A subset in which every point has neighborhood room entirely inside it — and, via three axioms on the whole collection τ, the primitive that IS a space's topology, letting continuity, compactness, and connectedness be defined with no distance function.
  • Planarity — The graph property of admitting a crossing-free drawing in the plane — pinned by Kuratowski/Wagner to a finite obstruction (no K₅ or K₃,₃) and by Euler's formula to a density bound, which is why a shelf of NP-hard problems turns polynomial on planar graphs.
  • Topological Space — Capture the minimum data continuity needs by pairing a set with a collection of its subsets — the open sets, closed under arbitrary unions and finite intersections — so that continuity, compactness, and connectedness can be defined with no reference to distance.
  • Tree (Graph Theory) — Characterize a connected acyclic graph — equivalently, one with a unique path between every pair of vertices, or exactly n−1 edges on n vertices — so that recognizing the structure by any one handle imports the bridge-everywhere, fundamental-cycle, and cheapest-connector consequences for free.