An axiomatization of graphs

Fuzzy graphs: algebraic structure and syntactic recognition

“…[1,2,9,11]) is a doubly ranked set M = (M m,n ) equipped with two operations denoted by • (circle) and D (box):…”

“…Since for every graph an infinite number of such expressions exist, at the same paper, the authors stated the open problem of finding a complete set of equations (rewriting rules) with the property that two expressions represent the same graph if and only if one can be transformed into the other by these equations. This problem was solved in [9] by appropriately adopting magmoids as the necessary algebraic structure for the representation of graphs and led to the construction, for the first time, of automata operating on arbitrary graphs [10,12,23].…”

“…A magmoid, introduced by Arnold and Dauchet in 1978, is a doubly ranked set endowed with two operations which are associative, unitary, and compatible to each other [1,2,8,9,11]. It generalizes the ordinary monoid structure and a natural regularity notion, analogously with the string case, derives from this.…”

Varieties of Graphoids and Birkoff's Theorem for Graphs

“…Actually it is the free such structure over Σ (cf. [4]). This important result allows us to construct free objects within a variety of graphoids and it is a cornerstone in the present theory.…”

Fuzzy Graph Language Recognizability

“…Since for every graph an infinite number of such expressions exist, at the same paper, the authors stated the open problem of finding a complete set of equations (rewriting rules) with the property that two expressions represent the same graph if and only if one can be transformed into the other by these equations. We solved this problem in [8] by appropriately adopting magmoids as the necessary algebraic structure for the representation of graphs and graph operations. This result led to the introduction of graphoids and in the construction, for the first time, of automata operating on arbitrary graphs (cf.…”