Results 1  10
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17
Dynamic Programming On Graphs With Bounded Treewidth
, 1987
"... In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth _< k, (or equivalently, the class of partial ktrees), for fixed k. We introduce two classes of graph decision problems, LCC and ECC, and subclasses CLCC, and CECC. We show that ea ..."
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Cited by 70 (1 self)
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In this paper we study the complexity of graph decision problems, restricted to the class of graphs with treewidth _< k, (or equivalently, the class of partial ktrees), for fixed k. We introduce two classes of graph decision problems, LCC and ECC, and subclasses CLCC, and CECC. We show that each problem in LCC (or CLCC) is solvable in polynomial (O(nc)) time, when restricted to graphs with fixed up perbounds on the treewidth and degree; and that each problem in ECC (or CECC) is solvable in polynomial (O(nc)) time, when re stricted to graphs with a fixed upperbound on the treewidth (with given corresponding treedecomposition). Also, problems in CLCC and CECC are solvable in polynomial time for graphs with a logarithmic treewidth, and given corresponding treedecomposition, and in the case of CLCCproblems, a fixed upperbound on the degree of the graph. Also, we show
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
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Cited by 61 (14 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Constructive Linear Time Algorithms for Branchwidth
, 1997
"... 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 noti ..."
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Cited by 33 (7 self)
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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 wellknown 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...
The Steiner tree polytope and related polyhedra
, 1994
"... We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For ..."
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Cited by 31 (1 self)
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We consider the vertexweighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is seriesparallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facetdefining valid inequalities for the Steiner tree polytope.
Algorithms Finding TreeDecompositions of Graphs
, 1991
"... A graph G has treewidth at most w if it admits a treedecomposition of width ≤ w. It is known that once we have a treedecomposition of a graph G of bounded width, many NPhard problems can be solved for G in linear time. For w ≤ 3 we give a lineartime algorithm for finding such a decomposition an ..."
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Cited by 25 (0 self)
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A graph G has treewidth at most w if it admits a treedecomposition of width ≤ w. It is known that once we have a treedecomposition of a graph G of bounded width, many NPhard problems can be solved for G in linear time. For w ≤ 3 we give a lineartime algorithm for finding such a decomposition and for a general fixed w we obtain a probabilistic algorithm with execution time O(n log 2 n + n log n  log p), which for a graph G on n vertices and a real number p> 0 either finds a treedecomposition of width ≤ 6w or answers that the treewidth of G is ≥ w; this second answer may be wrong but with probability at most p. The second result is based on a separator technique which may be of independent interest.
Graphs with Branchwidth at most Three
 J. Algorithms
, 1997
"... In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as ..."
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Cited by 23 (2 self)
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In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition. Keywords: graph algorithms, branchwidth, obstruction set, graph minor, reduction rule. 1 Introduction This paper considers the study of the graphs with branchwidth at most three. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many ...
Polynomial Instances Of The Positive Semidefinite And Euclidean Distance Matrix Completion Problems
 SIAM J. Matrix Anal. Appl
, 1998
"... Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a ..."
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Cited by 17 (6 self)
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Given an undirected graph G = (V; E) with node set V = [1; n], a subset S ` V and a rational vector a 2 Q S[E , the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n \Theta n positive semidefinite matrix X = (x ij ) satisfying: x ii = a i (i 2 S) and x ij = a ij (ij 2 E). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fillin; the minimum fillin of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph of K 4 .
Design of Survivable Networks with Bounded Rings
, 2000
"... This dissertation is the result of a project funded by Belgacom, the Belgian telecommunication operator, dealing with the development of new models and optimization techniques for the longterm planning of the backbone network. The minimumcost twoconnected spanning network problem consists in find ..."
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Cited by 14 (2 self)
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This dissertation is the result of a project funded by Belgacom, the Belgian telecommunication operator, dealing with the development of new models and optimization techniques for the longterm planning of the backbone network. The minimumcost twoconnected spanning network problem consists in finding a network with minimal total cost for which there exist two nodedisjoint paths between every pair of nodes. This problem, arising from the need to obtain survivable communication and transportation networks, has been widely studied. In our model, the following constraint is added in order to increase the reliability of the network : each edge must belong to a cycle of length less than or equal to a given threshold value K. This condition ensures that when traffic between two nodes has to be redirected (e.g. in case of failure of an edge), we can limit the increase of the distance between these nodes. We investigate valid inequalities for this problem and provide numerical results obtai...
Weighted connected domination and Steiner trees in distancehereditary graphs (Extended abstract)
 Discrete Appl. Math
, 1996
"... this paper is to present linear time algorithms for the weighted connected domination problem with general weights and the Steiner tree problem with nonnegative weights in distancehereditary graphs. D'Atri and Moscarini [8] gave ..."
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Cited by 10 (2 self)
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this paper is to present linear time algorithms for the weighted connected domination problem with general weights and the Steiner tree problem with nonnegative weights in distancehereditary graphs. D'Atri and Moscarini [8] gave