Uphill graph topologies and degree-constrained realizations of posets

We define a family of topologies on graphs using degree-constrained reachability. Open sets are closed under paths of nondecreasing or strictly increasing vertex degree; for multigraphs, counting incident edges or distinct neighbours gives further variants. These topologies encode the reachability determined by adjacency and degree comparisons, linking combinatorial constraints to topological structure. For the non-strict uphill topology on simple undirected graphs, we prove that its T0 plateau quotients are exactly the well-founded set-sized posets; the minimum degree-spectrum order type equals ordinal height. Every finite poset of height h has a polynomial-size bipartite realization using the optimal h degree values 3, …, h+2. The bound h+2 on maximum degree is uniformly sharp over height-h posets for h ≥ 2, independently of cover-graph branching. A countable well-founded poset has a countable locally finite realization exactly when each element has finite rank. Consequences include finite simplicial weak homotopy types with degree at most dimension plus three, and #P-complete open-set counting for bipartite graphs with degrees exactly three and four. Compactness, countable compactness and sequential compactness coincide in every subspace and are characterized by minimal plateaus. The four-point crown has a unique smallest bipartite realization up to isomorphism: seven vertices and eleven edges. Under the standard weak patch refinement, degree at most three forces a discrete plateau quotient; a degree-four tree contains a convergent-sequence quotient subspace. Another countable locally finite realization has a metrizable refined quotient without isolated points. Supplementary interactive explorer: Open Uphill Topology Explorer, version 1.12.1. Download the frozen software and reproducible source. The simulator is an optional finite illustration and does not replace the proofs. Correspondence: [email protected].