Properties of Graphs Specified by a Regular Language

Abstract

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property Φ. 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 if is there exists one graph satisfying Φ? 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 L if a certain torsion condition is satisfied. This condition holds trivially if L is regular.In order to show our results, we split L into a finite union of subsets Lα. Every Lα defines in a natural way a single finite graph Fα where some edges and vertices are marked. The marked graph in turn defines a set of graphs with a geometric description using the notion of graph retraction and where Fα appears as an induced subgraph. ACM Subject ClassificationTheory of computation → Formal languages and automata theory Keywords and phrases Graph properties, Regular languages, Periodic semigroups Funding Petra Wolf : DFG project FE 560/9-1 1 The notation int Reg refers to intersection non-emptiness with regular languages.