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 \(\varPhi \). 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 exists one graph satisfying \(\varPhi \)? We approach this question by using formal languages for specifying families of graphs. In particular, we show that certain graph properties can be decided by studying the syntactic monoid of the specification language.