Chen Jingrun: Difference between revisions

Revision as of 01:53, 12 September 2013

Template:Distinguish In graph theory, two graphs $G$ and $G'$ are homeomorphic if there is an graph isomorphism from some subdivision of $G$ to some subdivision of $G'$ . If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the sense in which the term is used in topology.

Subdivisions

In general, a subdivision of a graph G (sometimes known as an expansion^[1]) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v} yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w} and {w,v}.

For example, the edge e, with endpoints {u,v}:

can be subdivided into two edges, e₁ and e₂, connecting to a new vertex w:

The reverse operation, smoothing out or smoothing a vertex w with regards to the pair of edges (e,f) incident on w, removes both edges containing w and replaces (e,f) with a new edge that connects the other endpoints of the pair. Here it is emphasized that only 2-valent vertices can be smoothed.

For example, the simple connected graph with two edges, e₁ {u,w} and e₂ {w,v}:

has a vertex (namely w) that can be smoothed away, resulting in::

Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.^[2]

Barycentric Subdivisions

The barycentric subdivision subdivides each edge of the graph. This is a special subdivision, as it always results in a bipartite graph. This procedure can be repeated, so that the n^th barycentric subdivision is the barycentric subdivision of the n-1^th barycentric subdivision of the graph. The second such subdivision is always a simple graph.

Embedding on a surface

It is evident that subdividing a graph preserves planarity. Kuratowski's theorem states that

a finite graph is planar if and only if it contains no subgraph homeomorphic to K₅ (complete graph on five vertices) or K_3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three).

In fact, a graph homeomorphic to K₅ or K_3,3 is called a Kuratowski subgraph.

A generalization, flowing from the Robertson–Seymour theorem, asserts that for each integer g, there is a finite obstruction set of graphs $L(g)=\{G_{i}^{(g)}\}$ such that a graph H is embeddable on a surface of genus g if and only if H contains no homeomorphic copy of any of the $G_{i}^{(g)}$ . For example, $L(0)=\{K_{5},K_{3,3}\}$ contains the Kuratowski subgraphs.

Example

In the following example, graph G and graph H are homeomorphic.

G

H

If G' is the graph created by subdivision of the outer edges of G and H' is the graph created by subdivision of the inner edge of H, then G' and H' have a similar graph drawing:

G', H'

Therefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.

References

↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
↑ The more commonly studied problem in the literature, under the name of the subgraph homeomorphism problem, is whether a subdivision of H is isomorphic to a subgraph of G. The case when H is an n-vertex cycle is equivalent to the Hamiltonian cycle problem, and is therefore NP-complete. However, this formulation is only equivalent to the question of whether H is homeomorphic to a subgraph of G when H has no degree-two vertices, because it does not allow smoothing in H. The stated problem can be shown to be NP-complete by a small modification of the Hamiltonian cycle reduction: add one vertex to each of H and G, adjacent to all the other vertices. Thus, the one-vertex augmentation of a graph G contains a subgraph homeomorphic to an (n + 1)-vertex wheel graph, if and only if G is Hamiltonian. For the hardness of the subgraph homeomorphism problem, see e.g. 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.

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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

[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[2] The more commonly studied problem in the literature, under the name of the subgraph homeomorphism problem, is whether a subdivision of H is isomorphic to a subgraph of G. The case when H is an n-vertex cycle is equivalent to the Hamiltonian cycle problem, and is therefore NP-complete. However, this formulation is only equivalent to the question of whether H is homeomorphic to a subgraph of G when H has no degree-two vertices, because it does not allow smoothing in H. The stated problem can be shown to be NP-complete by a small modification of the Hamiltonian cycle reduction: add one vertex to each of H and G, adjacent to all the other vertices. Thus, the one-vertex augmentation of a graph G contains a subgraph homeomorphic to an (n + 1)-vertex wheel graph, if and only if G is Hamiltonian. For the hardness of the subgraph homeomorphism problem, see e.g. 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.

[1]

[2]

@@ Line 1: / Line 1: @@
+{{Distinguish|graph homomorphism}}
+In [[graph theory]], two graphs <math>G</math> and <math>G'</math> are '''homeomorphic''' if there is an [[graph isomorphism]] from some '''subdivision''' of <math>G</math> to some '''subdivision''' of <math>G'</math>.  If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are [[homeomorphism|homeomorphic]] in the sense in which the term is used in [[topology]].
+==Subdivisions==
+In general, a '''subdivision''' of a graph ''G'' (sometimes known as an '''expansion'''<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory |year=1993 |publisher=Dover Pub.|location=New York |isbn=978-0-486-67870-2 |url=http://store.doverpublications.com/0486678709.html |edition=Corrected, enlarged republication. |accessdate=8 August 2012 |pages=76 |quote='''Definition 20.''' If some new vertices of degree 2 are added to some of the edges of a graph ''G'', the resulting graph ''H'' is called an ''expansion'' of ''G''.}}</ref>) is a graph resulting from the subdivision of edges in ''G''. The subdivision of some edge ''e'' with endpoints {''u'',''v''} yields a graph containing one new vertex ''w'', and with an edge set replacing ''e'' by two new edges, {''u'',''w''} and {''w'',''v''}.
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+For example, the edge ''e'', with endpoints {''u'',''v''}:
+{| class="wikitable"
+|[[Image:Graph subdivision step1.svg|150px]]
+|}
+can be subdivided into two edges, ''e''<sub>1</sub> and ''e''<sub>2</sub>, connecting to a new vertex ''w'':
+{| class="wikitable"
+|[[Image:Graph subdivision step2.svg|150px]]
+|}
+The reverse operation, '''smoothing out''' or '''smoothing'''  a vertex ''w'' with regards to the pair of edges (''e'',''f'') incident on ''w'', removes both edges containing ''w'' and replaces (''e'',''f'') with a new edge that connects the other endpoints of the pair. Here it is emphasized that only 2-valent vertices can be smoothed.
+For example, the simple [[Connectivity (graph theory)|connected]] graph with two edges, ''e''<sub>1</sub> {''u'',''w''} and ''e''<sub>2</sub> {''w'',''v''}:
+{| class="wikitable"
+|[[Image:Graph subdivision step2.svg|150px]]
+|}
+has a vertex (namely ''w'') that can be smoothed away, resulting in::
+{| class="wikitable"
+|[[Image:Graph subdivision step1.svg|150px]]
+|}
+Determining whether for graphs ''G'' and ''H'', ''H'' is homeomorphic to a subgraph of ''G'', is an [[NP-complete]] problem.<ref>The more commonly studied problem in the literature, under the name of the subgraph homeomorphism problem, is whether a subdivision of ''H'' is isomorphic to a subgraph of ''G''. The case when ''H'' is an ''n''-vertex cycle is equivalent to the [[Hamiltonian cycle]] problem, and is therefore NP-complete. However, this formulation is only equivalent to the question of whether ''H'' is homeomorphic to a subgraph of ''G'' when ''H'' has no degree-two vertices, because it does not allow smoothing in ''H''. The stated problem can be shown to be NP-complete by a small modification of the Hamiltonian cycle reduction: add one vertex to each of ''H'' and ''G'', adjacent to all the other vertices. Thus, the one-vertex augmentation of a graph ''G'' contains a subgraph homeomorphic to an (''n''&nbsp;+&nbsp;1)-vertex [[wheel graph]], if and only if ''G'' is Hamiltonian. For the hardness of the subgraph homeomorphism problem, see e.g. {{citation
+ | last1 = LaPaugh | first1 = Andrea S.
+ | last2 = Rivest | first2 = Ronald L.
+ | doi = 10.1016/0022-0000(80)90057-4
+ | issue = 2
+ | journal = Journal of Computer and System Sciences
+ | mr = 574589
+ | pages = 133–149
+ | title = The subgraph homeomorphism problem
+ | volume = 20
+ | year = 1980}}.</ref>
+===Barycentric Subdivisions===
+The [[barycentric subdivision]] subdivides each edge of the graph.  This is a special subdivision, as it always results in a [[bipartite graph]].  This procedure can be repeated, so that the ''n''<sup>th</sup> barycentric subdivision is the barycentric subdivision of the ''n-1''<sup>th</sup> barycentric subdivision of the graph.  The second such subdivision is always a [[simple graph]].
+==Embedding on a surface==
+It is evident that subdividing a graph preserves planarity.  [[Kuratowski's theorem]] states that
+: a [[finite graph]] is [[planar graph|planar]] [[if and only if]] it contains no [[Glossary of graph theory#Subgraphs|subgraph]] '''homeomorphic''' to ''K''<sub>5</sub> ([[complete graph]] on five [[vertex (graph theory)|vertices]]) or ''K''<sub>3,3</sub> ([[complete bipartite graph]] on six vertices, three of which connect to each of the other three).
+In fact, a graph homeomorphic to ''K''<sub>5</sub> or ''K''<sub>3,3</sub> is called a [[Kuratowski's theorem|Kuratowski subgraph]].
+A generalization, flowing from the [[Robertson–Seymour theorem]], asserts that for each integer ''g'', there is a finite '''obstruction set''' of graphs <math>L(g) = \{G_{i}^{(g)}\}</math> such that a graph ''H'' is embeddable on a surface of [[Genus (mathematics)|genus]] ''g'' if and only if ''H'' contains no homeomorphic copy of any of the <math>G_{i}^{(g)}</math>.  For example, <math>L(0) = \{K_{5}, K_{3,3}\}</math> contains the Kuratowski subgraphs.
+==Example==
+In the following example, graph G and graph H are homeomorphic.
+G [[Image:Graph homeomorphism example 1.svg|150px|graph H]]
+H [[Image:Graph homeomorphism example 2.svg|150px|graph G]]
+If G' is the graph created by subdivision of the outer edges of G and H' is the graph created by subdivision of the inner edge of H, then G' and H' have a similar graph drawing:
+G', H' [[Image:Graph homeomorphism example 3.svg|150px|graph G]]
+Therefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.
+==See also==
+*[[Minor (graph theory)]]
+*[[Edge contraction]]
+==References==
+<references />
+*{{Citation | last1=Yellen | first1=Jay | last2=Gross | first2=Jonathan L. | title=Graph Theory and Its Applications | location=Boca Raton| publisher=Chapman & Hall/CRC | edition=2nd | series=Discrete Mathematics and Its Applications | isbn=978-1-58488-505-4 | year=2005}}
+[[Category:Graph theory]]
+[[Category:Homeomorphisms]]

Chen Jingrun: Difference between revisions

Revision as of 01:53, 12 September 2013

Contents

Subdivisions

Barycentric Subdivisions

Embedding on a surface

Example

See also

References

Navigation menu

Chen Jingrun: Difference between revisions

Revision as of 01:53, 12 September 2013

Subdivisions

Barycentric Subdivisions

Embedding on a surface

Example

See also

References

Navigation menu

Search