Properties of graphs specified by a regular language

Abstract Traditionally, graph algorithms get a single as input, and then they should decide if this satisfies certain property $$\varPhi $$ Φ . What happens question is modified in way that we possibly infinite family of graphs an the there satisfying family? We approach by using formal languages for specifying families graphs, particular regular sets words. show properties can be decided studying syntactic monoid specification language L torsion condition satisfied. This holds trivially regular. More specifically, use natural binary encoding finite over alphabet $$\varSigma Σ , define set $$\mathbb {G}\subseteq \varSigma ^*$$ G ⊆ ∗ such every nonempty word $$w\in \mathbb {G}$$ w ∈ defines graph. Also, syntactically defined Then, ask whether automaton $$\mathcal {A}$$ A specifies some Our structural results answer all “typical” properties. In order to our results, split into union subsets subset F where edges vertices are marked. The marked turn $$F^\infty F ∞ therefore subgraphs appears induced subgraph. yields geometric description specified based on splitting finitely many pieces; notion retraction, obtain easily understandable each piece.