This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

Roughly, a graph has small “tree-width ” if it can be constructed by piecing small graphs together in a tree structure. Here we study the obstructions to the existence of such a tree structure. We find, for instance: (i) a minimax formula relating tree-width with the largest such obstructions (ii) an association between such obstructions and large grid minors of the graph (iii) a “tree-decomposition ” of the graph into pieces corresponding with the obstructions. These results will be of use in later papers. 0 1991 Academic Press, Inc. 1. TANGLES Graphs in this paper are finite and undirected and may have loops or multiple edges. The vertex- and edge-sets of a graph G are denoted by V(G) and E(G). If G, = ( V1, E,), G2 = ( V2, E2) are subgraphs of a graph G, we denote the graphs (V1n V2,E1nE,) and (V,u V2, EluEZ) by G,nG, and G, u GZ, respectively. A separation of a graph G is a pair (G,, G2) of subgraphs with G1 u G2 = G and E(G1 n G2) = 0, and the order of this separation is f V(G, n G2)(. It sometimes happens with a graph G that for each separation (G, , G2) of G of low order, we may view one of G1, G, as the “main part ” of G, in * This work ’ was performed under a consulting agreement with Bellcore.

The main question studied in this article may be viewed as a non-linear analog of Dvoretzky's Theorem in Banach space theory or as part of Ramsey Theory in combinatorics.

For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.

Graph expansion has proved to be a powerful general tool for analyzing the behavior of routing algorithms and the inter--connection networks on which they run. We develop new routing algorithms and structural results for bounded--degree expander graphs. Our results are unified by the fact that they are all based upon, and extend, a body of work asserting that expanders are rich in short, disjoint paths. In particular, our work has consequences for the disjoint paths problem, multicommodity flow, and graph minor containment. We show: (i) A greedy algorithm for approximating the maximum disjoint paths problem achieves a polylogarithmic approximation ratio in bounded--degree expanders. Although our algorithm is both deterministic and on-line, its performance guarantee is an improvement over previous bounds in expanders. (ii) For a multicommodity flow problem with arbitrary demands on a bounded--degree expander, there is a (1+ ")--optimal solution using only flow paths of polylogarithmi...

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In this work we relate the theory of quasi-orders to the theory of algorithms over some combinatorial objects. First we develope the theory of well-quasi-orderings, wqo's, and relate it to the theory of worst-case complexity. Then we give a general 0-1-law for hereditary properties that has implications for average case complexity. This result on average-case complexity is applied to the class of finite graphs equipped with the induced subgraph relation. We obtain that a wide class of problems, including e.g. perfectness, has average constant time algorithms. Then we show, by extending a result of Damaschke on cographs, that the classes of finite orders resp. graphs with bounded decomposition diameter form wqo's with respect to the induced suborder resp. induced subgraph relation. This leads to linear time algorithms for the recognition of any hereditary property on these objects. Our main result is then that the set of finite posets is a wqo with respect to a certain relation , calle...

A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minor-closed class of graphs does not contain all graphs, then every graph in it is glued together in a tree-like fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.

Graph minors and the theory of graphs embedded in surfaces are

Let K be an induced non-separating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2-cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded branch size up to a bounded number of “almost disjoint ” millipedes. Moreover, B contains a nice subgraph ˜ B with the following properties. If K is 2-cell embedded in some surface and F is a face of K, then ˜ B admits exactly the same types of embeddings in F as B. A linear time algorithm to construct such a universal obstruction ˜ B is presented. At the same time, for every type of embeddings of ˜ B, an embedding of B ofthesametypeis determined. 1