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The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. 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 NPCompleteness," W. H. Freem ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. 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 NPCompleteness," 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, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
An Improved Linear Edge Bound for Graph Linkages
 EUROP. J. COMBINATORICS
, 2004
"... A graph is said to be klinked if it has at least 2k vertices and for every sequence s1,...,s k,t 1,...,t k of distinct vertices there exist disjoint paths P1,...,P k such that the ends of P i are s i and t i . Bollobas and Thomason showed that if a simple graph G on n vertices is 2kconnected and ..."
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A graph is said to be klinked if it has at least 2k vertices and for every sequence s1,...,s k,t 1,...,t k of distinct vertices there exist disjoint paths P1,...,P k such that the ends of P i are s i and t i . Bollobas and Thomason showed that if a simple graph G on n vertices is 2kconnected and G has at least 11kn edges, then G is klinked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2kconnectivity suffice, and then with more effort improve the edge bound to 5kn.
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... 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 farreaching ..."
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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 farreaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (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 minorclosed class of graphs does not contain all graphs, then every graph in it is glued together in a treelike 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 family closed under contraction
"... Let S be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in S if and only if the contraction G/H is in S. Examples of such an S: graphs with a spanning closed trail; graphs with at least k edgedisjoint spanning trees; and kedgeconnected graphs ( ..."
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Let S be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in S if and only if the contraction G/H is in S. Examples of such an S: graphs with a spanning closed trail; graphs with at least k edgedisjoint spanning trees; and kedgeconnected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in S and to study its structure if it is not in S. This reduction method generalizes known special cases.
An (almost) Linear Time Algorithm For Odd Cycles Transversal
, 2009
"... We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V (G) with X  ≤ k such that G − X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number o ..."
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We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V (G) with X  ≤ k such that G − X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number of vertices and the number of edges, respectively, and the function α(m, n) is the inverse of the Ackermann function (see by Tarjan [38]). This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(nm) time algorithm for this problem. Our algorithm also implies the edge version of the problem, i.e, there is an edge set X ′ ∈ E(G) such that G − X ′ is bipartite. Using this algorithm and the recent result in [16], we give an O(mα(m, n) + n log n) algorithm for the following problem for any fixed k: Input: A graph G and an integer k. Output: Determine whether or not there is a halfintegral k disjoint odd cycles packing, i.e, k odd cycles C1,..., Ck in G such that each vertex is on at most two of these odd cycles. This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(n 3) time algorithm for this problem. We also give a much simpler and much shorter proof for the following result by Reed [28]. The ErdősPósa property holds for the halfintegral disjoint odd cycles packing problem. I.e. either G has a halfintegral k
The 1FixedEndpoint Path Cover Problem is Polynomial on Interval Graphs
, 2009
"... We consider a variant of the path cover problem, namely, the kfixedendpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V(G),akfixedendpoint path cover of G with respect to T is a set of vertexdisjoint paths P that covers the vertic ..."
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We consider a variant of the path cover problem, namely, the kfixedendpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V(G),akfixedendpoint path cover of G with respect to T is a set of vertexdisjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. ThekPC problem is to find a kfixedendpoint path cover of G of minimum cardinality; note that, if T is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1fixedendpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NPcomplete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49– 64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomialtime algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.