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The Int_reg-problem of a combinatorial problem P asks, given anondeterministic automaton M as input, whether the language L(M) accepted by Mcontains any positive instance of the problem P. We consider theInt_reg-problem for a number of different graph problems and give generalcriteria that give decision procedures for these Int_reg-problems. To achievethis goal, we consider a natural graph encoding so that the language of allgraph encodings is regular. Then, we draw the connection between classicalpumping- and interchange-arguments from the field of formal language theorywith the graph operations induced on the encoded graph. Our techniques applyamong others to the Int_reg-problem of well-known graph problems like VertexCover and Independent Set, as well as to subgraph problems, graph-edit problemsand graph-partitioning problems, including coloring problems.

Regular Intersection Emptiness of Graph Problems: Finding a Needle in a Haystack of Graphs with the Help of Automata