Properties of Graphs Specified by a Regular Language

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