Embeddings of k-connected graphs of pathwidth k

We present O(n3) embedding algorithms (subgraph isomorphism and its generalizations) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomial-time algorithm for minor containment and the first O(nc) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees, as well as an O(n2) algorithm for subgraph isomorphism. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions. Finally, we describe nk+O(1) algorithms for solving these problems on arbitrary graphs of pathwidth at most k.