On convergence structures in graphs

Abstract

A closure operator on a set $X$ is a function $\mathrm{cl}:\wp (X)\to \wp (X)$ satisfying, for all $A,B\subseteq X$, the following properties: extensivity, $A\subseteq \mathrm{cl}(A)$; monotonicity, which states that if $A\subseteq B$ then $\mathrm{cl}(A)\subseteq \mathrm{cl}(B)$; and preservation of unions, $\mathrm{cl}(A\cup B)=\mathrm{cl}(A)\cup \mathrm{cl}(B)$. Every graph $G$ naturally carries such an operator on its vertex set by assigning to each subset $A\subseteq V(G)$ the set $\mathrm{cl}(A)=A\cup N(A)$, 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.