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arXiv · 2510.06336

On convergence structures in graphs

Paulo Magalhães Junior·Renan Maneli Mezabarba·Rodrigo Santos Monteiro
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Abstract

A closure operator on a set $X$ is a function $\operatorname{cl}: \wp(X) \to \wp(X)$ satisfying, for all $A, B \subseteq X$, the following properties: extensivity, $A \subseteq \operatorname{cl}(A)$; monotonicity, which states that if $A \subseteq B$ then $\operatorname{cl}(A) \subseteq \operatorname{cl}(B)$; and preservation of unions, $\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)$. Every graph $G$ naturally carries such an operator on its vertex set by assigning to each subset $A \subseteq V(G)$ the set $\operatorname{cl}(A) = A \cup N(A)$, where $N(A)$ denotes the vertices adjacent to a vertex in $A$. Since closure operators and pretopological spaces are equivalent notions, this operator induces a canonical convergence structure on $V(G)$. We describe this convergence in terms of nets and relate combinatorial properties of the graph to convergence-theoretic ones.

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Paulo Magalhães Junior, Renan Maneli Mezabarba, Rodrigo Santos Monteiro. 2025-10-07. On convergence structures in graphs. https://arxiv.org/abs/2510.06336

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