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23
Generating Linear Extensions Fast
"... One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of t ..."
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Cited by 37 (6 self)
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One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) =  E(P) . The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #Pcomplete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).
Algorithms for Square Roots of Graphs
 SIAM Journal on Discrete Mathematics
, 1991
"... The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, ..."
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Cited by 34 (0 self)
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The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2 . Here we give a linear time algorithm for finding the tree square roots of a given graph and a linear time algorithm for finding the square roots of planar graphs. We also give a polynomial time algorithm for finding the square roots of subdivision graphs, which is equivalent to the problem of the inversion of total graphs. Further, we give a linear time algorithm for finding a Hamiltonian cycle in a cubic graph, and we prove the NPcompleteness of finding the maximum cliques in powers of graphs and the chordality of powers of trees. Keywords: Square graphs, power graphs, tree square, planar square graphs. 1 Introduct...
Computing Roots of Graphs is Hard
 DISCRETE APPLIED MATHEMATICS
, 1994
"... The square of an undirected graph G is the graph G² on the same vertex set such that there is an edge between two vertices in G² if and only if they are at distance at most 2 in G. The k'th power of a graph is defined analogously. It has been conjectured that the problem of computing any square r ..."
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Cited by 25 (1 self)
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The square of an undirected graph G is the graph G² on the same vertex set such that there is an edge between two vertices in G² if and only if they are at distance at most 2 in G. The k'th power of a graph is defined analogously. It has been conjectured that the problem of computing any square root of a square graph, or even that of deciding whether a graph is a square, is NPhard. We settle this conjecture in the affirmative.
Approximating DisjointPath Problems Using Packing Integer Programs
, 1998
"... In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax ≤ b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPhard p ..."
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Cited by 15 (2 self)
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In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax ≤ b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPhard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjointpath problems using polynomialsize packing integer programs. Motivated by the...
Performance Guarantees for the TSP with a Parameterized Triangle Inequality
 IN PROC. 6TH INT. WORKSHOP ON ALGORITHMS AND DATA STRUCTURES (WADS), VOLUME 1663 OF LECTURE NOTES IN COMPUT. SCI
, 2000
"... We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter 1, the distances satisfy the inequality dist(x; y) \Delta (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We o ..."
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Cited by 8 (0 self)
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We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter 1, the distances satisfy the inequality dist(x; y) \Delta (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We obtain a 4 approximation and also show that for some ffl ? 0 it is nphard to obtain a (1 + ffl ) approximation. Our upper bound improves upon the earlier known ratio of (3 =2 + =2) [1] for all values of ? 7/3.
Toughness in graphs a survey
 Graphs and Combinatorics
, 2006
"... Abstract. In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. The ..."
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Cited by 6 (0 self)
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Abstract. In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!
Cycles in Networks
, 1993
"... We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path P ..."
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Cited by 6 (0 self)
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We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path PN and the cycle CN , in graphs that have been proposed as interconnection networks for parallel architectures. In addition to their intrinsic interest, in terms of the development of algorithms on parallel architectures, these two guest graphs are important because of the fact that many structurally richer graphs can be constructed from paths and cycles by various product constructions. A few of the results we present are original; several appear in the literature and are duly cited; many belong to the folklore of the field. Indeed this paper is motivated by a desire to find a single repository for this important, yet scattered material. Before proceeding further, we define formally the ...
Exact and Approximation Algorithms for Network Flow and DisjointPath Problems
, 1998
"... Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP ..."
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Cited by 5 (3 self)
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Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP hard variants such as disjoint paths and unsplittable flow. Given an nvertex
Hamiltonian paths in Cayley graphs
, 2008
"... The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that th ..."
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Cited by 4 (0 self)
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The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3regular Hamiltonian expanders.
An upper bound for the Hamiltonicity exponent of finite digraphs, submitted to Discrete Math. Address of the author: TU Bergakademie Freiberg Fakultat f. Mathematik und Physik BernhardvonCottaStrae 2 D09596
"... Let D be a strongly k−connected digraph of order n ≥ 2. We prove that for every l ≥ n 2k the power Dl of D is Hamiltonian. Moreover, for any n> 2k ≥ 2 we exhibit strongly kconnected digraphs D of order n such that D l−1 is nonHamiltonian for l = ⌈ n 2k ⌉. We use standard terminology, unless other ..."
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Cited by 1 (1 self)
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Let D be a strongly k−connected digraph of order n ≥ 2. We prove that for every l ≥ n 2k the power Dl of D is Hamiltonian. Moreover, for any n> 2k ≥ 2 we exhibit strongly kconnected digraphs D of order n such that D l−1 is nonHamiltonian for l = ⌈ n 2k ⌉. We use standard terminology, unless otherwise stated. A digraph D = (V, A) of order n ≥ 2 is said to be strongly qarc Hamiltonian if for any system S of mutually vertexdisjoint paths of the complete symmetric digraph with vertex set V of total length not greater than q, the digraph D ′ = (V; A ∪ S) has a Hamiltonian cycle containing S. If D is such a digraph that for every set W of vertices such that  W  ≤ p the digraph D − W is strongly qarc Hamiltonian, then we say that D is strongly (p,q)Hamiltonian. It is clear that a strongly (0, 1)Hamiltonian digraph is strongly Hamiltonian connected, that is, for every pair of vertices u, v ∈ V there is a Hamiltonian path from u to v. In [1] Bermond proved the following 1 Theorem 1 (cf [1] , [2] ) If a digraph D of order n has the property that for any two vertices x and y, either x dominates y or d + D (x) + d− D (y) ≥ n + p + q then D is strongly (p, q)Hamiltonian. It is a very well known fact that the square of a 2connected graph G of order n ≥ 3 is Hamiltonian [3]. Inspired by these results Schaar [4] defined the Hamiltonicity exponent of a strongly connected digraph D to be the minimum number eH(D) such that D eH(D) is Hamiltonian. Similarly, eHC(D) = min{l: D l is Hamiltonianconnected} and ep−H = min{l: D l is pHamiltonian}. The lth power D l of a digraph D is defined analogously as in the case of undirected graphs (cf.[4]). We shall prove the following. Theorem 2 Let D be a strongly kconnected digraph of order n ≥ 2, and let l, p and q be nonnegative integers such that l ≥ n+(p+q), p + q ≤ n − 2. Then D 2k l is strongly (p, q)Hamiltonian. It is clear that Theorem 2 implies the following two corollaries. Corollary 3 Let D be a strongly kconnected digraph of order n. Then (i) eH(D) ≤ ⌈ n ⌉ if n ≥ 2