Terse walk sets in graphs and induced closure operators

Given a graph G, for every ordinal α>1, we introduce and study closure operators on G induced by sets of α-indexed walks. For such sets, we define a property called terseness and investigate how it affects the induced closure operators. We show, among others, that the induction, if regarded as a map, is one-to-one for terse walk sets. We also determine a poset of closure operators (on a given graph) that is a direct limit of a direct system of sets of terse α-indexed walks ordered by set inclusion for certain ordinals α>1. Possible applications of the closure operators studied in digital topology are indicated.