2–3 heap

Template:Distinguish In the mathematical field of graph theory a graph homomorphism is a mapping between two graphs that respects their structure. More concretely it maps adjacent vertices to adjacent vertices.

Definitions

A graph homomorphism $f$ from a graph $G=(V,E)$ to a graph $G'=(V',E')$, written $f:G\rightarrow G'$, is a mapping $f:V\rightarrow V'$ from the vertex set of $G$ to the vertex set of $G'$ such that $\{u,v\}\in E$ implies $\{f(u),f(v)\}\in E'$.

The above definition is extended to directed graphs. Then, for a homomorphism $f:G\rightarrow G'$, $(f(u),f(v))$ is an arc of $G'$ if $(u,v)$ is an arc of $G$.

If there exists a homomorphism $f:G\rightarrow H$ we shall write $G\rightarrow H$, and $G\not \rightarrow H$ otherwise. If $G\rightarrow H$, $G$ is said to be homomorphic to $H$ or $H$-colourable.

If the homomorphism $f:G\rightarrow G'$ is a bijection whose inverse function is also a graph homomorphism, then $f$ is a graph isomorphism.

Two graphs $G$ and $G'$ are homomorphically equivalent if $G\rightarrow G'$ and $G'\rightarrow G$.

A retract of a graph $G$ is a subgraph $H$ of $G$ such that there exists a homomorphism $r:G\rightarrow H$, called retraction with $r(x)=x$ for any vertex $x$ of $H$. A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core.

Properties

The composition of homomorphisms are homomorphisms.

Graph homomorphism preserves connectedness.

The tensor product of graphs is the category-theoretic product for the category of graphs and graph homomorphisms.

Connection to coloring and girth

A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism $f:G\rightarrow K_{k}$ from G to a complete graph K_k: the vertices of K_k correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K_k that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K_k, and every two distinct vertices of K_k are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K_k. Conversely if f is a homomorphism from G to K_k, then one can color G by using the same color for two vertices in G whenever they are both mapped to the same vertex in K_k. Because K_k has no edges that connect a vertex to itself, it is not possible for two adjacent vertices in G to both be mapped to the same vertex in K_k, so this gives a valid coloring. That is, G has a k-coloring if and only if it has a homomorphism to K_k.

If there are two homomorphisms $H\rightarrow G\rightarrow K_{k}$, then their composition $H\rightarrow K_{k}$ is also a homomorphism. In other words, if a graph G can be colored with k colors, and there is a homomorphism $H\rightarrow G$, then H can also be k-colored. Therefore, whenever a homomorphism $H\rightarrow G$ exists, the chromatic number of H is less than or equal to the chromatic number of G.

Homomorphisms can also be used very similarly to characterize the odd girth of a graph G, the length of its shortest odd-length cycle. The odd girth is, equivalently, the smallest odd number g for which there exists a homomorphism $C_{g}\rightarrow G$. For this reason, if $G\rightarrow H$, then the odd girth of G is greater than or equal to the corresponding invariant of H.

Complexity

The associated decision problem, i.e. deciding whether there exists a homomorphism from one graph to another, is NP-complete. Determining whether there is an isomorphism between two graphs is also an important problem in computational complexity theory; see graph isomorphism problem.

See also

• Hadwiger's conjecture.
• Graph rewriting
• Median graphs, definable as the retracts of hypercubes.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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.

• R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28.