On the complexity of finding iso- and other morphisms for partial k-trees

Abstract The problems to decide whether H ⩽ G for input graphs H , G where ⩽ is ‘isomorphic to a subgraph’, ‘isomorphic to an induced subgraphs’, ‘isomorphic to a subdivision’, ‘isomorphic to a contraction’ or their combination, are NP-complete. We discuss the complexity of these problems when G is restricted to be a partial k -tree (in other terminology: to have tree-width ⩽ k , to be k -decomposable, to have dimension ⩽ k ). Under this restriction the problems are still NP-complete in general, but there are polynomial algorithms under some natural restrictions imposed on H , for example when H has bounded degrees. We also give a polynomial time algorithm for the n disjoint connecting paths problem restricted to partial k -trees (with n part of input).