We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

Abstract not found

In this paper, we investigate which processor networks allow k- label Interval Routing Schemes, under the assumption that costs of edges may vary. We show that for each fixed k 1, the class of graphs allowing such routing schemes is closed under minor-taking in the domain of connected graphs, and hence has a linear time recognition algorithm. This result connects the theory of compact routing with the theory of graph minors and treewidth. We show that every graph that does not contain K 2;r as a minor has treewidth at most 2r \Gamma 2. In case the graph is planar, this bound can be lowered to r + 2. As a consequence, graphs that allow k-label Interval Routing Schemes under dynamic cost edges have treewidth at most 4k, and treewidth at most 2k + 3 if they are planar. Similar results are shown for other types of Interval Routing Schemes.

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class W_k(G), where for each graph G in W_k(G), the removal of a set of at most k vertices from G results in a graph in the base graph class G. (If G ist the class of edgeless graphs,...

The cutwidth of a graph G is defined to be the smallest integer k such that the vertices of G can be arranged in a vertex ordering...

This document is mainly based on "A Compendium of Parameterized Complexity Results", version 2.0 (May 22, 1996), by Michael T. Hallett and H. Todd Wareham, and on Downey and Fellows' book [53]. However, this document includes several new results that have been published in the last few years