Local dissymmetry on graphs and related algebraic structures

Abstract

We use the set symmetric difference between vertex subsets of a finite undirected simple graph [Formula: see text] to define a binary operation ∘ on the vertex set of a new graph [Formula: see text], that contains [Formula: see text] as subgraph and whose vertices are non-empty vertex subsets of [Formula: see text]. We show how the binary operation ∘ determines an algebraic structure on [Formula: see text] that is strictly related to the graph structure of [Formula: see text]. In fact, we show that [Formula: see text] agrees with [Formula: see text] and, next, we provide several characterizations for the algebraic structure [Formula: see text] when the graph [Formula: see text] is connected and locally dissymmetric.