Structural and Computational Analysis of Weak Hausdorff Type Separation in Alpha Topologized Graphs

Abstract

Weak Hausdorff-type conditions on graph-associated topologies are usually expressed as qualitative yes/no properties, even though finite graph models allow the unavoidable overlap of neighborhoods to be measured exactly. This paper develops a structural and computational framework for alpha-topologized graphs. For the incidence topology generated by edge-endpoint sets, three quantitative invariants are studied: the alpha-overlap order, the mean alpha-overlap, and the zero-overlap separation ratio. A bounded alpha-weak Hausdorff hierarchy alpha-WH(k) is introduced to avoid the collapse of the ordinary finite-intersection condition on finite spaces. The incidence model is shown to be alpha-stable, and its complete separation structure is characterized by pendant vertices. The total overlap is expressed by a pendant separation load, from which sharp extremal results and closed formulas for standard graph families are obtained. To ensure that the alpha operator plays a genuinely nontrivial role, a second model based on the II-induced topology is analyzed. In that model the alpha-refinement changes the topology for 463 of the 994 connected non-isomorphic graphs of orders three through seven, and overlap orders as large as seven occur. Exhaustive Graph Atlas calculations compare the two models, while a reproducible benchmark on random labeled trees demonstrates the computational advantage of the linear-time pendant-load algorithm over a quadratic pairwise scan. The results provide a quantitative bridge between weak topological separation and measurable graph structure, while clearly separating alpha-stable and non-alpha-stable graph topologies.