  The Int_reg-problem of a combinatorial problem P asks, given a
nondeterministic automaton M as input, whether the language L(M) accepted by M
contains any positive instance of the problem P. We consider the
Int_reg-problem for a number of different graph problems and give general
criteria that give decision procedures for these Int_reg-problems. To achieve
this goal, we consider a natural graph encoding so that the language of all
graph encodings is regular. Then, we draw the connection between classical
pumping- and interchange-arguments from the field of formal language theory
with the graph operations induced on the encoded graph. Our techniques apply
among others to the Int_reg-problem of well-known graph problems like Vertex
Cover and Independent Set, as well as to subgraph problems, graph-edit problems
and graph-partitioning problems, including coloring problems.
