Graphs generated from minimal sets of finite point-set topologies

TL;DR

This work unifies classical graph classes derived from posets—comparability and upper-bound graphs—within a single finite-topology framework by associating to each finite topology $\tau$ on a set $X$ the graph $G_i(\tau)$, where adjacency is governed by $T_i$-separation. Building six nested graph classes $\mathcal{G}_0$ through $\mathcal{G}_4$, the authors prove a poset-characterization: $\mathcal{G}_0$ corresponds to disjoint unions of cliques, $\mathcal{G}_1$ to comparability graphs, $\mathcal{G}_2$ to upper-bound/edge-simplicial graphs, $\mathcal{G}_{3'}$ to half-closed upper-bound graphs, $\mathcal{G}_{3''}$ to fully-closed upper-bound graphs, and $\mathcal{G}_4$ to extended-closed upper-bound graphs. They further illuminate $\mathcal{G}_2$ as the class of middle graphs of hypergraphs (equivalently edge-simplicial graphs) and provide a detailed clique-cover perspective; they characterize the $\mathcal{G}_2$-critical graphs via iterative anchoring of stars to $K_1$. A novel universe-based description is developed for $\mathcal{G}_{3'}$, equating graphs with an edge-cover by universes $(s,P,M)$ to graphs realized by finite topologies of height at most $3$, and the paper notes that $\mathcal{G}_{3'}$ graphs are not bipartite. Overall, the framework yields a coherent, topologically grounded taxonomy that links classical poset graph classes to new topological constructions with implications for clique covers and hypergraph representations.

Abstract

In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky characterized upper bound graphs, those derived from partially ordered sets where two elements share an edge if they have a common upper bound, by a specific edge clique cover condition. In this paper, we introduce a unifying framework for these results using finite point set topologies. Given a finite topology, we define a graph whose vertices correspond to its elements, with edges determined by intersections of their minimal containing sets, where intersection is understood in terms of the topological separation axioms. This construction yields a natural sequence of graph classes, one for each separation axiom, that connects and extends both classical results in a structured and intuitive way.

Figures (9)

• Figure 1: Visual interpretation of $x$ and $y$ being $T_i$-separated in \ref{['def: sep-axioms']}. Open sets in $\tau$ indicated in red and blue, and closed sets are indicated in gray.
• Figure 2: The six graphs corresponding to the topology $\tau$ on $[5]$ with minimal base $M(\tau) = \{\{1\},\{1,2,3\},\{3\},\{3,4,5\},\{5\}\}$.
• Figure 3: An example of $m_{\tau}$ and $m_{\tau'}$ for \ref{['lem:m(x)=m(y)']}
• Figure 4: Height $2$ reduction in \ref{['lem: height2']} (left) and height $3$ reduction in \ref{['lem: height3']} (right)
• Figure 5: Paths considered in \ref{['thm:poset-characterization']} with transitive orientation (left) and visual interpretation of the minimal base of associated sub-topology $\tau_P$ (right)

Theorems & Definitions (53)

• Theorem 1.1: Containment
• Theorem 1.2: UPPERBOUND
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.1