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The closed curve represented by the dotted line crossing the graph corresponds to a cut of the graph.

Formally, a cut of a graph G ( V , E ) is a partition of the graph vertices into subsets V 1 , V 2 V such that V 1 V 2 = V and V 1 V 2 = , as demonstrated by [link] . The corresponding cut set is the set of edges C such that C = ( u , v ) E | u V 1 , v V 2 . Hence, the cut set induces a bipartite subgraph.

The partitions of graph vertices that correspond to this cut are highlighted in red and blue.

The size of a cut equals the sum of the weights of edges in the cut set, which in the case of unweighted graphs is simply the number of edges in the cut set. With this definition of size, the maximum cut of a graph, like those shown in [link] , is a cut not smaller than any other cut in the graph, and it corresponds to the largest bipartite subgraph of the graph. The maximum cut of a graph is not necessarily unique and is not unique in either of the examples.

Maximum cuts for the two example graphs are shown above.

Alternatively, the problem can be formulated in terms of the edges in the complement of the cut set. The complement of a set of edges that intersects every odd cycle in a graph induces a graph with no subgraphs that are odd cycles, which is therefore a bipartite graph. Thus, the complement of the minimum set of edges intersecting every odd cycle induces the largest bipartite subgraph of the graph and hence is the maximum cut set, as illustrated in [link] .

The complement of the minimum set (red) of edges intersecting all odd cycles of a graph is the maximum cut set.

Finding the maximum cut of a graph was one of the earliest problems proven to be np-complete, which, ignoring the formal details of what that means, indicates that no currently known algorithms terminate in a polynomial bounded number of operations in all cases [link] . There are, however, several types of graphs for which polynomial bounded solutions are known, such as graphs embeddable on the plane [link] . Since computing the maxcut of large graphs often requires extremely long lengths of time, randomized ρ -approximation algorithms, such as that of Goemans and Williamson, may be employed for situations in which optimality is not required and a good estimate will suffice [link] .

Applications of the maxcut problem include minimization of number of holes on circuit boards or number of chip contacts in VLSI circuit layout design, energy minimization problems in computer vision programs, and modeling of the interactions of spin glasses with magnetic fields in statistical physics [link] .

Several algorithms

The most direct and straightforward way to find maximum cuts of a graph would be to perform an exhaustive search of all bipartitions of the graph vertices. The maximum cut may be found by iterating over all distinct bipartitions of the graph vertices, summing the weights of edges connecting vertices in opposite partitions to calculate the size of the corresponding cut, comparing this value to the largest cut size previously found, and updating the maximum accordingly.

The exhaustive algorithm, which has computational complexity O ( | E | 2 | V | ) , examines the same number of bipartitions for a tree, for which the maximum cut always equals the number of edges, as it does for a complete graph on the same number of vertices. Thus, it is clear that the exhaustive algorithm is not completely satisfactory, especially for graphs with few edges relative to other graphs with a given number of vertices.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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