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| In [[graph theory]], a '''partial cube''' is a [[graph (mathematics)|graph]] that is an [[isometry|isometric]] [[Glossary of graph theory#Subgraphs|subgraph]] of a [[hypercube graph|hypercube]].<ref>{{harvtxt|Ovchinnikov|2011}}, Definition 5.1, p. 127.</ref> In other words, a partial cube is a subgraph of a hypercube that preserves [[distance (graph theory)|distance]]s—the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with [[bit string]]s of equal length in such a way that the distance between two vertices in the graph is equal to the [[Hamming distance]] between their labels. Such a labeling is called a ''Hamming labeling''; it represents an isometric [[embedding]] of the partial cube into a hypercube. | | In today's tough economic situation, many successful professionals and every day people are quitting their jobs to pursue new business goals.<br><br>The Internet has also made it easier for many people to explore a wide range of business opportunities with minimal investment. To achieve success, it is important for you to find a lucrative business idea that can deliver desired and consistent results. You must, therefore, unleash your creativity and explore unique ideas that can make your customers happy and give you competitive edge.<br><br>Here are some unique home business ideas you probably never considered.<br>Customized Cupcakes<br>Instead of selling cupcakes online, (which is a booming business by the way) how about letting the customers get creative and order their own sweet delights? You just need to create an interactive website that will allow users to not just select the flavor but also toppings and design. To make it more appealing, you can give them more creative options.<br><br>For example, they can order their own savory cupcakes that may have bacon, guacamole and cheddar cheese. It is a smart business idea to offer exactly what the customers want. <br>Relationship Manager<br>If you are good at solving your friends' relationship problems and a good listener, you can consider making a career out of it. As a dedicated relationship manager, you will focus on understanding your client's relationship problems and finding ways to solve them.<br><br>Unlike a psychologist or counselor, your role will be more holistic. Not only will you counsel your client, but also provide a host of other services such as delivering flowers to their [http://goo.gl/W5AGuo play doh food] spouses/partners, organizing dates and sending gifts on anniversaries. <br>Branded Clothes on Rent<br>There is a large section of buyers who want to wear branded clothes but do not have enough money to buy them. By offering them the option of hiring branded clothes for a day, you can reach out to them and start a very lucrative business venture. Customers need to simply order for branded clothes online that can be home delivered within a day or so and should be returned in 2 days.<br><br>Fashion conscious women and men looking for the best prom dress or a reunion party suit will be interested in this cost-effective service. <br>A unique business idea can help you make more money in very little time. With some creativity and passion, you will be able to stay on top of your game. |
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| ==History==
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| {{harvtxt|Firsov|1965}} was the first to study isometric embeddings of graphs into hypercubes. The graphs that admit such embeddings were characterized by {{harvtxt|Djoković|1973}} and {{harvtxt|Winkler|1984}}, and were later named partial cubes. A separate line of research on the same structures, in the terminology of [[Family of sets|families of sets]] rather than of hypercube labelings of graphs, was followed by {{harvtxt|Kuzmin|Ovchinnikov|1975}} and {{harvtxt|Falmagne|Doignon|1997}}, among others.<ref>{{harvtxt|Ovchinnikov|2011}}, p. 174.</ref>
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| ==Examples==
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| [[File:2SAT median graph.svg|thumb|300px|An example of a partial cube with a Hamming labeling on its vertices. This graph is also a [[median graph]].]]
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| The [[Desargues graph]] is a partial cube, as is more generally any [[Kneser graph|bipartite Kneser graph]] ''H''<sub>{{nowrap|2''n''+1,''n''}}</sub>. In this bipartite Kneser graph, the labels consist of all possible {{nowrap|(2''n'' + 1)}}-digit bitstrings that have either ''n'' or {{nowrap|''n'' + 1}} nonzero bits; the Desargues graph is constructed in this way with {{nowrap|1=''n'' = 2}}.
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| All [[median graph]]s are partial cubes.<ref>{{harvtxt|Ovchinnikov|2011}}, Section 5.11, "Median Graphs", pp. 163–165.</ref> Since the median graphs include the [[squaregraph]]s, [[simplex graph]]s, and [[Fibonacci cube]]s, as well as the covering graphs of finite [[distributive lattice]]s, these are all partial cubes. The [[planar dual]] graph of an [[arrangement of lines]] in the [[Euclidean plane]], is a partial cube; more generally, for any [[hyperplane arrangement]] in [[Euclidean space]] of any number of dimensions, the graph that has a vertex for each cell of the arrangement and an edge for each two adjacent cells is a partial cube.<ref>{{harvtxt|Ovchinnikov|2011}}, Chapter 7, "Hyperplane Arrangements", pp. 207–235.</ref>
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| A partial cube in which every vertex has exactly three neighbors is known as a [[cubic graph|cubic]] partial cube. Although several infinite families of cubic partial cubes are known, together with many other sporadic examples, the only known cubic partial cube that is not a [[planar graph]] is the Desargues graph.<ref>{{harvtxt|Eppstein|2006}}.</ref>
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| The underlying graph of any [[antimatroid]], having a vertex for each set in the antimatroid and an edge for every two sets that differ by a single element, is always a partial cube.
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| The [[Cartesian product of graphs|Cartesian product]] of any finite set of partial cubes is another partial cube.<ref>{{harvtxt|Ovchinnikov|2011}}, Section 5.7, "Cartesian Products of Partial Cubes", pp. 144–145.</ref>
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| ==The Djoković–Winkler relation==
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| Many of the theorems about partial cubes are based directly or indirectly upon a certain [[binary relation]] defined on the edges of the graph. This relation, first described by {{harvtxt|Djoković|1973}} and given an equivalent definition in terms of distances by {{harvtxt|Winkler|1984}}, is denoted by <math>\Theta</math>. Two edges <math>e=\{x,y\}</math> and <math>f=\{u,v\}</math> are defined to be in the relation <math>\Theta</math>, written <math>e\mathrel{\Theta}f</math>, if
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| <math>d(x,u)+d(y,v)\not=d(x,v)+d(y,u)</math>. This relation is [[reflexive relation|reflexive]] and [[symmetric relation|symmetric]], but in general it is not [[transitive relation|transitive]].
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| Winkler showed that a [[connectivity (graph theory)|connected]] graph is a partial cube if and only if it is [[bipartite graph|bipartite]] and the relation <math>\Theta</math> is transitive.<ref>{{harvtxt|Winkler|1984}}, Theorem 4. See also {{harvtxt|Ovchinnikov|2011}}, Definition 2.13, p.29, and Theorem 5.19, p. 136.</ref> In this case, it forms an [[equivalence relation]] and each equivalence class separates two connected subgraphs of the graph from each other. A Hamming labeling may be obtained by assigning one bit of each label to each of the equivalence classes of the Djoković–Winkler relation; in one of the two connected subgraphs separated by an equivalence class of edges, all of the vertices have a 0 in that position of their labels, and in the other connected subgraph all of the vertices have a 1 in the same position.
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| ==Recognition==
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| Partial cubes can be recognized, and a Hamming labeling constructed, in <math>O(n^2)</math> time, where <math>n</math> is the number of vertices in the graph.<ref>{{harvtxt|Eppstein|2008}}.</ref> Given a partial cube, it is straightforward to construct the equivalence classes of the Djoković–Winkler relation by doing a [[breadth first search]] from each vertex, in total time <math>O(nm)</math>; the <math>O(n^2)</math>-time recognition algorithm speeds this up by using [[bit-level parallelism]] to perform multiple breadth first searches in a single pass through the graph, and then applies a separate algorithm to verify that the result of this computation is a valid partial cube labeling.
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| ==Dimension==
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| The '''isometric dimension''' of a partial cube is the minimum dimension of a hypercube onto which it may be isometrically embedded, and is equal to the number of equivalence classes of the Djoković–Winkler relation. For instance, the isometric dimension of an <math>n</math>-vertex tree is its number of edges, <math>n-1</math>. An embedding of a partial cube onto a hypercube of this dimension is unique, up to symmetries of the hypercube.<ref>{{harvtxt|Ovchinnikov|2011}}, Section 5.6, "Isometric Dimension", pp. 142–144, and Section 5.10, "Uniqueness of Isometric Embeddings", pp. 157–162.</ref>
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| Every hypercube and therefore every partial cube may be embedded isometrically into an [[integer lattice]], and the '''lattice dimension''' of the partial cube is the minimum dimension of an integer lattice for which this is possible. The lattice dimension may be significantly smaller than the isometric dimension; for instance, for a tree it is half the number of leaves in the tree (rounded up to the nearest integer). The lattice dimension of any graph, and a lattice embedding of minimum dimension, may be found in [[polynomial time]] by an algorithm based on [[maximum matching]] in an auxiliary graph.<ref>{{harvtxt|Hadlock|Hoffman|1978}}; {{harvtxt|Eppstein|2005}}; {{harvtxt|Ovchinnikov|2011}}, Chapter 6, "Lattice Embeddings", pp. 183–205.</ref>
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| Other types of dimension of partial cubes have also been defined, based on embeddings into more specialized structures.<ref>{{harvtxt|Eppstein|2009}}; {{harvtxt|Cabello|Eppstein|Klavžar|2011}}.</ref>
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| ==Application to chemical graph theory==
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| Isometric embeddings of graphs into hypercubes have an important application in [[chemical graph theory]]. A ''benzenoid graph'' is a graph consisting of all vertices and edges lying on and in the interior of a cycle in a [[hexagonal lattice]]. Such graphs are the [[molecular graph]]s of the [[benzenoid hydrocarbon]]s, a large class of organic molecules. Every such graph is a partial cube. A Hamming labeling of such a graph can be used to compute the [[Wiener index]] of the corresponding molecule, which can then be used to predict certain of its chemical properties.<ref>{{harvtxt|Klavžar|Gutman|Mohar|1995}}, Propositions 2.1 and 3.1; {{harvtxt|Imrich|Klavžar|2000}}, p. 60; {{harvtxt|Ovchinnikov|2011}}, Section 5.12, "Average Length and the Wiener Index", pp. 165–168.</ref>
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| A different molecular structure formed from carbon, the [[diamond cubic]], also forms partial cube graphs.<ref>{{harvtxt|Eppstein|2009}}.</ref>
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| ==Notes==
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| {{reflist|colwidth=30em}}
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| ==References==
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| *{{citation|title=The Fibonacci dimension of a graph|first1=S.|last1=Cabello|first2=D.|last2=Eppstein|author2-link=David Eppstein|first3=S.|last3=Klavžar|arxiv=0903.2507|journal=Electronic Journal of Combinatorics|volume=18|issue=1|at=P55|year=2011}}.
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| *{{citation
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| | last1 = Djoković | first1 = D. Ž.
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| | title = Distance-preserving subgraphs of hypercubes
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| | journal = [[Journal of Combinatorial Theory]] | series = Series B
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| | volume = 14 | issue = 3 | year = 1973 | pages = 263–267
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| | mr = 0314669
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| | doi = 10.1016/0095-8956(73)90010-5
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| }}.
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| *{{citation|first=David|last=Eppstein|authorlink=David Eppstein|arxiv=cs.DS/0402028|title=The lattice dimension of a graph|journal=European Journal of Combinatorics|volume=26|issue=6|pages=585–592|year=2005|doi=10.1016/j.ejc.2004.05.001}}.
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| *{{citation|title=Cubic partial cubes from simplicial arrangements|first=David|last=Eppstein|authorlink=David Eppstein|arxiv=math.CO/0510263 |journal=Electronic Journal of Combinatorics|volume=13|issue=1|at=R79|year=2006|url=http://www.combinatorics.org/Volume_13/Abstracts/v13i1r79.html}}.
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| *{{citation|contribution=Recognizing partial cubes in quadratic time|first=David|last=Eppstein|authorlink=David Eppstein|arxiv=0705.1025 |title=Proc. 19th ACM-SIAM Symposium on Discrete Algorithms|year=2008|pages=1258–1266}}.
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| *{{citation|contribution=Isometric diamond subgraphs|first=David|last=Eppstein|authorlink=David Eppstein|arxiv=0807.2218 |title=[[International Symposium on Graph Drawing|Proc. 16th International Symposium on Graph Drawing, Heraklion, Crete, 2008]]|series=Lecture Notes in Computer Science|volume=5417|year=2009|pages=384–389|publisher=Springer-Verlag|doi=10.1007/978-3-642-00219-9_37}}.
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| *{{citation|last1=Falmagne|first1=J.-C.|author1-link=Jean-Claude Falmagne|last2=Doignon|first2=J.-P.|title=Stochastic evolution of rationality|journal=Theory and Decision|volume=43|pages=107–138|year=1997|doi=10.1023/A:1004981430688}}.
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| *{{citation|first=V.V.|last=Firsov|year=1965|title=On isometric embedding of a graph into a Boolean cube|journal=Cybernetics|volume=1|pages=112–113}}. As cited by {{harvtxt|Ovchinnikov|2011}}.
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| *{{citation|first1=F.|last1=Hadlock|first2=F.|last2=Hoffman|title=Manhattan trees|journal=Utilitas Mathematica|year=1978|volume=13|pages=55–67}}. As cited by {{harvtxt|Ovchinnikov|2011}}.
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| *{{citation
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| | last1 = Imrich | first1 = Wilfried
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| | last2 = Klavžar | first2 = Sandi
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| | title = Product Graphs: Structure and Recognition
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| | series = Wiley-Interscience Series in Discrete Mathematics and Optimization
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| | publisher = John Wiley & Sons
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| | year = 2000 | location = New York
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| | isbn = 978-0-471-37039-0
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| | mr = 1788124
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| }}.
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| *{{citation
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| | last1 = Klavžar | first1 = Sandi
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| | last2 = Gutman | first2 = Ivan
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| | last3 = Mohar | first3 = Bojan | author3-link = Bojan Mohar
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| | title = Labeling of benzenoid systems which reflects the vertex-distance relations
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| | url = http://www.fmf.uni-lj.si/~klavzar/preprints/labeling-benzi.pdf
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| | journal = Journal of Chemical Information and Computer Sciences
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| | volume = 35 | issue = 3 | year = 1995 | pages = 590–593
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| }}.
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| *{{citation|first1=V.|last1=Kuzmin|first2=S.|last2=Ovchinnikov|title=Geometry of preferences spaces I|journal=Automation and Remote Control|volume=36|year=1975|pages=2059–2063}}. As cited by {{harvtxt|Ovchinnikov|2011}}.
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| *{{citation|first=Sergei|last=Ovchinnikov|title=Graphs and Cubes|series=Universitext|publisher=Springer|year=2011}}. See especially Chapter 5, "Partial Cubes", pp. 127–181.
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| *{{citation
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| | last1 = Winkler | first1 = Peter M. | authorlink = Peter Winkler
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| | title = Isometric embedding in products of complete graphs
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| | journal = Discrete Applied Mathematics
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| | volume = 7 | issue = 2 | year = 1984 | pages = 221–225
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| | mr = 0727925
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| | doi = 10.1016/0166-218X(84)90069-6
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| }}.
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| [[Category:Graph families]]
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| [[Category:Mathematical chemistry]]
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In today's tough economic situation, many successful professionals and every day people are quitting their jobs to pursue new business goals.
The Internet has also made it easier for many people to explore a wide range of business opportunities with minimal investment. To achieve success, it is important for you to find a lucrative business idea that can deliver desired and consistent results. You must, therefore, unleash your creativity and explore unique ideas that can make your customers happy and give you competitive edge.
Here are some unique home business ideas you probably never considered.
Customized Cupcakes
Instead of selling cupcakes online, (which is a booming business by the way) how about letting the customers get creative and order their own sweet delights? You just need to create an interactive website that will allow users to not just select the flavor but also toppings and design. To make it more appealing, you can give them more creative options.
For example, they can order their own savory cupcakes that may have bacon, guacamole and cheddar cheese. It is a smart business idea to offer exactly what the customers want.
Relationship Manager
If you are good at solving your friends' relationship problems and a good listener, you can consider making a career out of it. As a dedicated relationship manager, you will focus on understanding your client's relationship problems and finding ways to solve them.
Unlike a psychologist or counselor, your role will be more holistic. Not only will you counsel your client, but also provide a host of other services such as delivering flowers to their play doh food spouses/partners, organizing dates and sending gifts on anniversaries.
Branded Clothes on Rent
There is a large section of buyers who want to wear branded clothes but do not have enough money to buy them. By offering them the option of hiring branded clothes for a day, you can reach out to them and start a very lucrative business venture. Customers need to simply order for branded clothes online that can be home delivered within a day or so and should be returned in 2 days.
Fashion conscious women and men looking for the best prom dress or a reunion party suit will be interested in this cost-effective service.
A unique business idea can help you make more money in very little time. With some creativity and passion, you will be able to stay on top of your game.