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| In [[graph theory]], the '''bipartite double cover''' of an [[undirected graph]] ''G'' is a [[bipartite graph|bipartite]] [[covering graph]] of ''G'', with twice as many vertices as ''G''. It can be constructed as the [[tensor product of graphs]] ''G'' × ''K''<sub>2</sub>. It is also called the '''Kronecker double cover''', '''canonical double cover''' or simply the '''bipartite double''' of ''G''.
| | It involves expertise and knowledge of various tools and technologies used for creating websites. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. * A community forum for debate of the product together with some other customers in the comments spot. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion ''. <br><br> |
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| It should not be confused with a [[cycle double cover conjecture|cycle double cover]] of a graph, a family of cycles that includes each edge twice.
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| ==Construction==
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| The bipartite double cover of ''G'' has two vertices ''u<sub>i</sub>'' and ''w<sub>i</sub>'' for each vertex ''v<sub>i</sub>'' of ''G''. Two vertices ''u<sub>i</sub>'' and ''w<sub>j</sub>'' are connected by an edge in the double cover if and only if ''v<sub>i</sub>'' and ''v<sub>j</sub>'' are connected by an edge in ''G''. For instance, below is an illustration of a bipartite double cover of a non-bipartite graph ''G''. In the illustration, each vertex in the tensor product is shown using a color from the first term of the product (''G'') and a shape from the second term of the product (''K''<sub>2</sub>); therefore, the vertices ''u<sub>i</sub>'' in the double cover are shown as circles while the vertices ''w<sub>i</sub>'' are shown as squares.
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| :[[Image:Covering-graph-2.svg]]
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| The bipartite double cover may also be constructed using adjacency matrices (as described below) or as the derived graph of a [[voltage graph]] in which each edge of ''G'' is labeled by the nonzero element of the two-element [[group (mathematics)|group]].
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| ==Examples==
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| The bipartite double cover of the [[Petersen graph]] is the [[Desargues graph]]: ''K''<sub>2</sub> × ''G''(5,2) = ''G''(10,3).
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| The bipartite double cover of a [[complete graph]] ''K<sub>n</sub>'' is a [[crown graph]] (a [[complete bipartite graph]] ''K<sub>n</sub>''<sub>,''n''</sub> minus a [[perfect matching]]). In particular, the bipartite double cover of the graph of a [[tetrahedron]], ''K''<sub>4</sub>, is the graph of a [[cube]].
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| The bipartite double cover of an odd-length [[cycle graph]] is a cycle of twice the length, while the bipartite double of any bipartite graph (such as an even length cycle, shown in the following example) is formed by two disjoint copies of the original graph.
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| :[[Image:Covering-graph-1.svg]]
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| ==Matrix interpretation==
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| If an undirected graph ''G'' has a matrix ''A'' as its [[adjacency matrix]], then the adjacency matrix of the double cover of ''G'' is | |
| :<math>\left[\begin{matrix}0&A\\A^T&0\end{matrix}\right],</math> | |
| and the [[biadjacency matrix]] of the double cover of ''G'' is just ''A'' itself. That is, the conversion from a graph to its double cover can be performed simply by reinterpreting ''A'' as a biadjacency matrix instead of as an adjacency matrix. More generally, the reinterpretation the adjacency matrices of [[directed graph]]s as biadjacency matrices provides a [[bijective proof|combinatorial equivalence]] between directed graphs and balanced bipartite graphs.<ref>{{harvtxt|Dulmage|Mendelsohn|1958}}; {{harvtxt|Brualdi|Harary|Miller|1980}}.</ref>
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| ==Properties==
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| The bipartite double cover of any graph ''G'' is a [[bipartite graph]]; both parts of the bipartite graph have one vertex for each vertex of ''G''. A bipartite double cover is [[connected graph|connected]] if and only if ''G'' is connected and non-bipartite.<ref>{{harvtxt|Brualdi|Harary|Miller|1980}}, Theorem 3.4.</ref>
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| The bipartite double cover is a special case of a ''double cover'' (a 2-fold [[covering graph]]). A double cover in graph theory can be viewed as a special case of a [[Double cover (topology)|topological double cover]].
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| If ''G'' is a non-bipartite [[symmetric graph]], the double cover of ''G'' is also a symmetric graph; several known [[cubic graph|cubic]] symmetric graphs may be obtained in this way. For instance, the double cover of ''K''<sub>4</sub> is the graph of a cube; the double cover of the Petersen graph is the Desargues graph; and the double cover of the graph of the [[dodecahedron]] is a 40-vertex symmetric cubic graph.<ref>{{harvtxt|Feng|Kutnar|Malnič|Marušic|2008}}.</ref>
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| It is possible for two different graphs to have [[graph isomorphism|isomorphic]] bipartite double covers. For instance, the Desargues graph is not only the bipartite double cover of the Petersen graph, but is also the bipartite double cover of a different graph that is not isomorphic to the Petersen graph.<ref name="ip08">{{harvtxt|Imrich|Pisanski|2008}}.</ref> Not every bipartite graph is a bipartite double cover of another graph; for a bipartite graph ''G'' to be the bipartite cover of another graph, it is necessary and sufficient that the [[graph automorphism|automorphism]]s of ''G'' include an [[Involution (mathematics)|involution]] that maps each vertex to a distinct and non-adjacent vertex.<ref name="ip08"/> For instance, the graph with two vertices and one edges is bipartite but is not a bipartite double cover, because it has no non-adjacent pairs of vertices to be mapped to each other by such an involution; on the other hand, the graph of the cube is a bipartite double cover, and has an involution that maps each vertex to the diametrally opposite vertex. An alternative characterization of the bipartite graphs that may be formed by the bipartite double cover construction was obtained by {{harvtxt|Sampathkumar|1975}}.
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| ==Other double covers==
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| In general, a graph may have multiple double covers that are different from the bipartite double cover.<ref>{{harvtxt|Waller|1976}}.</ref> In the following figure, the graph ''C'' is a double cover of the graph ''H'':
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| # The graph ''C'' is a ''covering graph'' of ''H'': there is a surjective local isomorphism ''f'' from ''C'' to ''H'', the one indicated by the colours. For example, ''f'' maps both blue nodes in ''C'' to the blue node in ''H''. Furthermore, let ''X'' be the [[Neighbourhood (graph theory)|neighbourhood]] of a blue node in ''C'' and let ''Y'' be the neighbourhood of the blue node in ''H''; then the restriction of ''f'' to ''X'' is a bijection from ''X'' to ''Y''. In particular, the degree of each blue node is the same. The same applies to each colour.
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| # The graph ''C'' is a ''double'' cover (or ''2-fold cover'' or ''2-lift'') of ''H'': the preimage of each node in ''H'' has size 2. For example, there are exactly 2 nodes in ''C'' that are mapped to the blue node in ''H''.
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| However, ''C'' is not a ''bipartite'' double cover of ''H'' or any other graph; it is not a bipartite graph.
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| If we replace one triangle by a square in ''H ''the resulting graph has four distinct double covers. Two of them are bipartite but only one of them is the Kronecker cover.
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| :[[Image:Covering-graph-4.svg]]
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| As another example, the graph of the [[icosahedron]] is a double cover of the complete graph ''K''<sub>6</sub>; to obtain a covering map from the icosahedron to ''K''<sub>6</sub>, map each pair of opposite vertices of the icosahedron to a single vertex of ''K''<sub>6</sub>. However, the icosahedron is not bipartite, so it is not the bipartite double cover of ''K''<sub>6</sub>. Instead, it can be obtained as the [[orientable double cover]] of an [[graph embedding|embedding]] of ''K''<sub>6</sub> on the [[projective plane]].
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| ==See also==
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| *[[Bipartite half]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{citation
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| | last1 = Brualdi | first1 = Richard A.
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| | last2 = Harary | first2 = Frank | author2-link = Frank Harary
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| | last3 = Miller | first3 = Zevi
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| | doi = 10.1002/jgt.3190040107
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| | mr = 558453
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| | issue = 1
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| | journal = Journal of Graph Theory
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| | pages = 51–73
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| | title = Bigraphs versus digraphs via matrices
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| | volume = 4
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| | year = 1980}}.
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| *{{citation
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| | doi = 10.4153/CJM-1958-052-0
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| | last1 = Dulmage | first1 = A. L.
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| | last2 = Mendelsohn | first2 = N. S. | author2-link = Nathan Mendelsohn
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| | mr = 0097069
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| | journal = Canadian Journal of Mathematics
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| | pages = 517–534
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| | title = Coverings of bipartite graphs
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| | volume = 10
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| | year = 1958}}. The “coverings” in the title of this paper refer to the [[vertex cover]] problem, not to bipartite double covers. However, {{harvtxt|Brualdi|Harary|Miller|1980}} cite this paper as the source of the idea of reinterpreting the adjacency matrix as a biadjacency matrix.
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| *{{citation
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| | last1 = Feng | first1 = Yan-Quan
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| | last2 = Kutnar | first2 = Klavdija
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| | last3 = Malnič | first3 = Aleksander
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| | last4 = Marušic | first4 = Dragan
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| | doi = 10.1016/j.jctb.2007.07.001
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| | arxiv = math.CO/0701722 |mr=2389602 | issue = 2
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| | journal = Journal of Combinatorial Theory, Series B
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| | pages = 324–341
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| | title = On 2-fold covers of graphs
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| | volume = 98
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| | year = 2008}}.
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| *{{citation
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| | first1 = Wilfried | last1 = Imrich
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| | last2 = Pisanski | first2 = Tomaž | author2-link = Tomaž Pisanski
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| | mr = 2419215
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| | title = Multiple Kronecker covering graphs
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| | journal = European Journal of Combinatorics
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| | volume = 29 | issue = 5 | year = 2008 | pages = 1116–1122
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| | doi = 10.1016/j.ejc.2007.07.001}}.
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| *{{citation
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| | doi = 10.1017/S1446788700020619
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| | last = Sampathkumar | first = E. | author1-link = E. Sampathkumar
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| | mr = 0389681
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| | issue = 3
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| | journal = Journal of the Australian Mathematical Society, Series A, Pure Mathematics and Statistics
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| | pages = 268–273
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| | title = On tensor product graphs
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| | volume = 20
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| | year = 1975}}.
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| *{{citation
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| | doi = 10.1017/S0004972700025053
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| | last = Waller | first = Derek A.
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| | mr = 0406876
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| | issue = 2
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| | journal = Bulletin of the Australian Mathematical Society
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| | pages = 233–248
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| | title = Double covers of graphs
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| | volume = 14
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| | year = 1976}}.
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| ==External links==
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| *{{mathworld | title = Bipartite Double Graph | urlname = BipartiteDoubleGraph}}
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| [[Category:Graph operations]]
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