Weber function: Difference between revisions

Latest revision as of 03:42, 19 March 2013

In mathematics, a permutation graph is a graph whose vertices represent the elements of a permutation, and whose edges represent pairs of elements that are reversed by the permutation. Permutation graphs may also be defined geometrically, as the intersection graphs of line segments whose endpoints lie on two parallel lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the modular decomposition.^[1]

Definition and characterization

If σ = (σ₁,σ₂, ..., σ_n) is any permutation of the numbers from 1 to n, then one may define a permutation graph from σ in which there are n vertices v₁, v₂, ..., v_n, and in which there is an edge v_iv_j for any two indices i and j for which i < j and σ_i > σ_j. That is, two indices i and j determine an edge in the permutation graph exactly when they determine an inversion in the permutation.

Given a permutation σ, one may also determine a set of line segments s_i with endpoints (i,0) and (σ_i,1). The endpoints of these segments lie on the two parallel lines y = 0 and y = 1, and two segments have a non-empty intersection if and only if they correspond to an inversion in the permutation. Thus, the permutation graph of σ coincides with the intersection graph of the segments. For every two parallel lines, and every finite set of line segments with endpoints on both lines, the intersection graph of the segments is a permutation graph; in the case that the segment endpoints are all distinct, a permutation for which it is the permutation graph may be given by numbering the segments on one of the two lines in consecutive order, and reading off these numbers in the order that the segment endpoints appear on the other line.

Permutation graphs have several other equivalent characterizations:

A graph G is a permutation graph if and only if G is a circle graph that admits an equator, i.e., an additional chord that intersects every other chord.^[2]
A graph G is a permutation graph if and only if both G and its complement $\overline{G}$ are comparability graphs.^[3]
A graph G is a permutation graph if and only if it is the comparability graph of a partially ordered set that has order dimension at most two.^[4]
If a graph G is a permutation graph, so is its complement. A permutation that represents the complement of G may be obtained by reversing the permutation representing G.

Efficient algorithms

It is possible to test whether a given graph is a permutation graph, and if so construct a permutation representing it, in linear time.^[5]

As a subclass of the perfect graphs, many problems that are NP-complete for arbitrary graphs may be solved efficiently for permutation graphs. For instance:

the largest clique in a permutation graph corresponds to the longest decreasing subsequence in the permutation defining the graph, so the clique problem may be solved in polynomial time for permutation graphs by using a longest decreasing subsequence algorithm.^[6]

likewise, an increasing subsequence in a permutation corresponds to an independent set of the same size in the corresponding permutation graph.

the treewidth and pathwidth of permutation graphs can be computed in polynomial time; these algorithms exploit the fact that the number of inclusion minimal vertex separators in a permutation graph is polynomial in the size of the graph.^[7]

Relation to other graph classes

Permutation graphs are a special case of circle graphs, comparability graphs, the complements of comparability graphs, and trapezoid graphs.

The subclasses of the permutation graphs include the bipartite permutation graphs and the cographs.

Notes

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References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

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22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

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+[[Image:Permutation graph.svg|thumb|300px|The permutation (4,3,5,1,2) and the corresponding permutation graph]]
+In [[mathematics]], a '''permutation graph''' is a [[graph theory|graph]] whose vertices represent the elements of a [[permutation]], and whose edges represent [[Inversion (discrete mathematics)|pairs of elements that are reversed by the permutation]]. Permutation graphs may also be defined geometrically, as the [[intersection graph]]s of line segments whose endpoints lie on two [[parallel (geometry)|parallel]] lines. Different permutations may give rise to the same permutation graph; a given graph has a unique representation (up to permutation symmetry) if it is prime with respect to the [[modular decomposition]].<ref>{{harvtxt|Brandstädt|Le|Spinrad|1999}}, p.191.</ref>
+==Definition and characterization==
+If σ = (σ<sub>1</sub>,σ<sub>2</sub>, ..., σ<sub>''n''</sub>) is any [[permutation]] of the numbers from 1 to ''n'', then one may define a permutation graph from σ in which there are ''n'' [[vertex (graph theory)|vertices]] ''v''<sub>1</sub>, ''v''<sub>2</sub>, ..., ''v''<sub>''n''</sub>, and in which there is an edge ''v''<sub>''i''</sub>''v''<sub>''j''</sub> for any two indices ''i'' and ''j'' for which ''i''&nbsp;<&nbsp;''j'' and σ<sub>''i''</sub>&nbsp;>&nbsp;σ<sub>''j''</sub>. That is, two indices ''i'' and ''j'' determine an edge in the permutation graph exactly when they determine an [[Inversion (discrete mathematics)|inversion]] in the permutation.
+Given a permutation σ, one may also determine a set of [[line segment]]s ''s''<sub>''i''</sub> with endpoints (''i'',0) and (σ<sub>''i''</sub>,1). The endpoints of these segments lie on the two parallel lines ''y''&nbsp;=&nbsp;0 and ''y''&nbsp;=&nbsp;1, and two segments have a non-empty intersection if and only if they correspond to an inversion in the permutation. Thus, the permutation graph of σ coincides with the [[intersection graph]] of the segments. For every two parallel lines, and every finite set of line segments with endpoints on both lines, the intersection graph of the segments is a permutation graph; in the case that the segment endpoints are all distinct, a permutation for which it is the permutation graph may be given by numbering the segments on one of the two lines in consecutive order, and reading off these numbers in the order that the segment endpoints appear on the other line.
+Permutation graphs have several other equivalent characterizations:
+* A graph ''G'' is a permutation graph if and only if ''G'' is a [[circle graph]] that admits an ''equator,'' i.e., an additional chord that intersects every other chord.<ref>{{harvtxt|Brandstädt|Le|Spinrad|1999}}, Proposition&nbsp;4.7.1, p.57.</ref>
+*A graph ''G'' is a permutation graph if and only if both ''G'' and its [[complement graph|complement]] <math>\overline{G}</math> are [[comparability graph]]s.<ref>{{harvtxt|Dushnik||Miller|1941}}.</ref>
+*A graph ''G'' is a permutation graph if and only if it is the [[comparability graph]] of a [[partially ordered set]] that has [[order dimension]] at most two.<ref>{{harvtxt|Baker|Fishburn|Roberts|1971}}.</ref>
+*If a graph ''G'' is a permutation graph, so is its complement. A permutation that represents the complement of ''G'' may be obtained by reversing the permutation representing ''G''.
+==Efficient algorithms==
+It is possible to test whether a given graph is a permutation graph, and if so construct a permutation representing it, in [[linear time]].<ref>{{harvtxt|McConnell|Spinrad|1999}}.</ref>
+As a subclass of the [[perfect graph]]s, many problems that are [[NP-complete]] for arbitrary graphs may be solved efficiently for permutation graphs. For instance:
+* the largest [[clique (graph theory)|clique]] in a permutation graph corresponds to the [[longest increasing subsequence|longest decreasing subsequence]] in the permutation defining the graph, so the [[clique problem]] may be solved in [[polynomial time]] for permutation graphs by using a longest decreasing subsequence algorithm.<ref>{{harvtxt|Golumbic|1980}}.</ref>
+* likewise, an increasing subsequence in a permutation corresponds to an [[Independent set (graph theory)|independent set]] of the same size in the corresponding permutation graph.
+* the [[treewidth]] and [[pathwidth]] of permutation graphs can be computed in polynomial time; these algorithms exploit the fact that the number of [[Vertex_separator#Minimal_separators|inclusion minimal vertex separators]] in a permutation graph is polynomial in the size of the graph.<ref>{{harvtxt|Bodlaender|Kloks|Kratsch|1995}}</ref>
+==Relation to other graph classes==
+Permutation graphs are a special case of [[circle graph]]s, [[comparability graph]]s, the complements of comparability graphs, and  [[Trapezoid_graph|trapezoid graphs]].
+The subclasses of the permutation graphs include the [[bipartite graph|bipartite]] permutation graphs and the [[cograph]]s.
+==Notes==
+{{reflist|2}}
+==References==
+*{{citation
+ | last1 = Baker | first1 = K. A.
+ | last2 = Fishburn | first2 = P. | author2-link = Peter C. Fishburn
+ | last3 = Roberts | first3 = F. S. | author3-link = Fred S. Roberts
+ | doi = 10.1002/net.3230020103
+ | issue = 1
+ | journal = Networks
+ | pages = 11–28
+ | title = Partial orders of dimension 2
+ | volume = 2
+ | year = 1971}}.
+*{{citation
+ | last1 = Bodlaender | first1 = Hans L. | author1-link = Hans L. Bodlaender
+ | last2 = Kloks | first2 = Ton
+ | last3 = Kratsch | first3 = Dieter
+ | doi = 10.1137/S089548019223992X
+ | issue = 4
+ | journal = [[SIAM Journal on Discrete Mathematics]]
+ | pages = 606–616
+ | title = Treewidth and pathwidth of permutation graphs
+ | volume = 8
+ | year = 1995}}.
+*{{citation
+ | last1 = Brandstädt | first1 = Andreas
+ | last2 = Le | first2 = Van Bang
+ | last3 = Spinrad | first3 = Jeremy
+ | title = Graph Classes: A Survey
+ | publisher = SIAM Monographs on Discrete Mathematics and Applications
+ | year = 1999
+ | id = ISBN 0-89871-432-X}}.
+*{{citation
+ | doi = 10.2307/2371374
+ | last1 = Dushnik | first1 = Ben
+ | last2 = Miller | first2 = E. W.
+ | title = Partially ordered sets
+ | journal = [[American Journal of Mathematics]]
+ | volume = 63 | issue = 3 | year = 1941 | pages = 600–610
+ | jstor = 2371374}}.
+*{{citation|first=M. C.|last=Golumbic|authorlink=Martin Charles Golumbic|title=Algorithmic Graph Theory and Perfect Graphs|series=Computer Science and Applied Mathematics|publisher=Academic Press|year=1980|page=159}}.
+*{{citation
+ | last1 = McConnell | first1 = Ross M.
+ | last2 = Spinrad | first2 = Jeremy P.
+ | doi = 10.1016/S0012-365X(98)00319-7
+ | issue = 1-3
+ | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
+ | mr = 1687819
+ | pages = 189–241
+ | title = Modular decomposition and transitive orientation
+ | volume = 201
+ | year = 1999}}.
+== External links ==
+* {{cite web
+ | url = http://www.graphclasses.org/classes/gc_23.html
+ | title = Permutation graph
+ | work = [http://www.graphclasses.org/index.html Information System on Graph Classes and their Inclusions]}}
+* {{mathworld|id=PermutationGraph|title=Permutation Graph}}
+[[Category:Intersection classes of graphs]]
+[[Category:Perfect graphs]]
+[[Category:Geometric graphs]]

Weber function: Difference between revisions

Latest revision as of 03:42, 19 March 2013

Contents

Definition and characterization

Efficient algorithms

Relation to other graph classes

Notes

References

External links

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Weber function: Difference between revisions

Latest revision as of 03:42, 19 March 2013

Definition and characterization

Efficient algorithms

Relation to other graph classes

Notes

References

External links

Navigation menu

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