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

We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more well-known notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...

Let k be a positive integer. There is a longest finite sequence x 1 ,...,x n in k letters in which no consecutive block x i ,...,x 2i is a subsequence of any other consecutive block x j ,...,x 2j . Let n(k) be this longest length. We prove that n(1) = 3, n(2) = 11, and n(3) is incomprehensibly large. We give a lower bound for n(3) in terms of the familiar Ackerman hierarchy. We also give asymptotic upper and lower bounds for n(k). We view n(3) as a particularly elemental description of an incomprehensibly large integer. Related problems involving binary sequences (two letters) are also addressed. We also report on some recent computer explorations of R. Dougherty which we use to raise the lower bound for n(3). TABLE OF CONTENTS 1. Finiteness, and n(1),n(2). 2. Sequences of fixed length sequences. 3. The Main Lemma. 4. Lower bound for n(3). 5. The function n(k). 6. Related problems and computer explorations. 1. FINITENESS, AND n(1),n(2) We use Z for the set of all integers, Z + f...

Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1

The linear-width of a graph G is defined to be the smallest integer k such that the edges of G can be arranged in a linear ordering (e 1 ; : : : ; e r ) in such a way that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; : : : ; e i g and to fe i+1 ; : : : ; e r g. In this paper, we give a set of 57 graphs and prove that it is the set of the minimal forbidden minors for the class of graphs with linear-width at most two. Our proof also gives a linear time algorithm that either reports that a given graph has linear-width more than two or outputs an edge ordering of minimum linear-width. We further prove a structural connection between linear-width and the mixed search number which enables us to determine, for any k 1, the set acyclic forbidden minors for the class of graphs with linear-width k. Moreover, due to this connection, our algorithm can be transfered to two linear time algorithms that check whether a graph has mixe...

Let G = (V; E) be a graph. The linear-width of G is defined as the smallest integer k such that E can be arranged in a linear ordering (e 1 ; : : : ; e r ) such that for every i = 1; : : : ; r \Gamma 1, there are at most k vertices both incident to an edge that belongs to fe 1 ; : : : ; e i g as to an edge that belongs to fe i+1 ; : : : ; e r g. For each fixed constant k, a linear time algorithm is given, that decides for any graph G = (V; E) whether the linear-width of G is at most k, and if so, finds the corresponding ordering of E. Linear-width has been proven to be related with the following graph searching parameters: mixed search number, node search number, and edge search number. A consequence of this is that we obtain for fixed k, linear time algorithms that check whether a given graph can be mixed, node, or edge searched with at most k searchers, and if so, output the corresponding search strategies. 1 Introduction In this paper, we study algorithmic aspects of a relatively ...

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,...

Abstract. We show that Nash-Williams ’ theorem asserting that the countable transfinite sequences of elements of a better-quasi-ordering ordered by embeddability form a better-quasi-ordering is provable in the subsystem of second order arithmetic Π 1 1-CA0 but is not equivalent to Π 1 1-CA0. We obtain some partial results towards the proof of this theorem in the weaker subsystem ATR0 and we show that the minimality lemmas typical of wqo and bqo theory imply Π 1 1-CA0 and hence cannot be used in such a proof. The most natural generalization of the notion of well-ordering to partial orderings is the concept of well-quasi-ordering or wqo. A binary relation � on a set Q is a quasi-ordering if it is reflexive and transitive (it is a partial ordering if it is also anti-symmetric) and a quasi-ordering is wqo if it contains no infinite descending sequences and no infinite anti-chains (sets of mutually incomparable elements). Ramsey’s theorem implies that a quasi-ordering (Q, �) is wqo if and only if for any function f: N → Q there exist n and m such that n < m and f(n) � f(m). For a history of the notion of wqo see [11].

Abstract not found

The tournament N 5 can be obtained from the transitive tournament on . . . , 5}, with the natural order, by reversing the edges between successive vertices. Tournaments that do not have N 5 as a subtournament are said to omit N 5 . We describe the structure of tournaments that omit N 5 and use this with Kruskal's Tree Theorem to prove that this class of tournaments is well-quasi-ordered. The proof involves an encoding of the indecomposable tournaments omitting N 5 by a finite alphabet.