Largest bipartite subgraphs in triangle-free graphs with maximum degree three
TL;DR: In this article, an algorithm polynomial for determining a bipartite sous-graphe bipartitio d'un graphe G without triangle and a Boucle de degre maximum 3, contenant au moins 4/5 des aretes de G.
Abstract: On presente un algorithme polynomial permettant de determiner un sous-graphe biparti d'un graphe G sans triangle ni boucle de degre maximum 3, contenant au moins 4/5 des aretes de G. On caracterise le dodecaedre et le graphe de Petersen comme les seuls graphes connexes 3-reguliers sans triangle ni boucle pour lesquels il n'existe pas de sous graphe biparti ayant un nombre d'aretes superieur a cette proportion
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TL;DR: It is shown that partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously, have algorithmic counterparts in a number of domains.
Abstract: We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.
96 citations
Cites background from "Largest bipartite subgraphs in tria..."
...Unless otherwise indicated, throughout the paper G will denote a graph with n vertices, m edges and maximum degree ∆....
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TL;DR: The authors propose the first parallel improvement algorithm using the maximum neural network model for the bipartite subgraph problem, with the simulation result showing that the algorithm finds a solution within 200 iteration steps and the solution quality is superior to that of the best existing algorithm.
Abstract: The authors propose the first parallel improvement algorithm using the maximum neural network model for the bipartite subgraph problem. The goal of this NP-complete problem is to remove the minimum number of edges in a given graph such that the remaining graph is a bipartite graph. A large number of instances have been simulated to verify the proposed algorithm, with the simulation result showing that the algorithm finds a solution within 200 iteration steps and the solution quality is superior to that of the best existing algorithm. The algorithm is extended for the K-partite subgraph problem where no algorithm has been proposed. >
80 citations
01 Jul 2005
TL;DR: In this survey, recent extremal results on a variety of questions concerning judicious partitions, and related problems such as Max Cut are discussed.
Abstract: Many classical partitioning problems in combinatorics ask for a single quantity to be maximized or minimized over a set of partitions of a combinatorial object For instance, Max Cut asks for the largest bipartite subgraph of a graph G , while Min Bisection asks for the minimum size of a cut into two equal pieces In judicious partitioning problems , we seek to maximize or minimize a number of quantities simultaneously For instance, given a graph G with m edges, we can ask for the smallest f ( m ) such that G must have a bipartition in which each vertex class contains at most f ( m ) edges In this survey, we discuss recent extremal results on a variety of questions concerning judicious partitions, and related problems such as Max Cut Introduction A wide variety of combinatorial optimization problems ask for an “optimal” partition of the vertex set of a graph or hypergraph A good example is the Max Cut problem: given a graph G , what is the maximum of e ( V 1 , V 2 ) over partitions V(G) = V 1 ∪ V 2 , where e ( V 1 , V 2 ) is the number of edges between V 1 and V 2 ? Similarly, Min Bisection asks for the minimum of e ( V 1 , V 2 ) over partitions V ( G ) = V 1 ∪ V 2 with | V 1 | ≤ | V 2 | ≤ | V 1 | + 1 (there are k -partite versions Max k -Cut and Min k -Section of both problems) Both of these problems involve maximizing or minimizing a single quantity over graphs from a certain class
60 citations
TL;DR: In this article, an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3 was presented, and the approximation ratio of the new algorithm is at least 0.9326.
51 citations
01 Jan 1985
TL;DR: In this article, a survey of packing/covering parameters and representation parameters of graphs and posets is presented, starting from the poset parameters known as width and dimension, and generalizations, related ques¬tions, and analogous parameters for graphs and/or directed graphs.
Abstract: This paper surveys results concerning packing/covering parameters and representation parameters of graphs and posets We start from the poset param¬eters known as width and dimension We consider generalizations, related ques¬tions, and analogous parameters for graphs and/or directed graphs
45 citations
References
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01 Jan 1972
TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
Abstract: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.
7,714 citations
Book•
01 Jan 1976
TL;DR: In this paper, the authors present Graph Theory with Applications: Graph theory with applications, a collection of applications of graph theory in the field of Operational Research and Management. Journal of the Operational research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
Abstract: (1977). Graph Theory with Applications. Journal of the Operational Research Society: Vol. 28, Volume 28, issue 1, pp. 237-238.
7,497 citations
5,837 citations
TL;DR: This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete.
2,200 citations
01 Apr 1941
TL;DR: Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node.
Abstract: Let N be a network (or linear graph) such that at each node not more than n lines meet (where n > 2), and no line has both ends at the same node. Suppose also that no connected component of N is an n-simplex. Then it is possible to colour the nodes of N with n colours so that no two nodes of the same colour are joined.
938 citations