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第86行： − − ==Properties== − − − − ===Characterization=== − − Bipartite graphs may be characterized in several different ways: − − Bipartite graphs may be characterized in several different ways: − − 二部图可以用几种不同的方式来表示: − − * A graph is bipartite [[if and only if]] it does not contain an [[Cycle (graph theory)|odd cycle]].<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, Theorem 2.1.3, p. 8. Asratian et al. attribute this characterization to a 1916 paper by [[Dénes Kőnig]]. For infinite graphs, this result requires the [[axiom of choice]].</ref> − − * A graph is bipartite if and only if it is 2-colorable, (i.e. its [[chromatic number]] is less than or equal to 2).<ref name="adh98-7"/> − − * The [[Spectral graph theory|spectrum]] of a graph is symmetric if and only if it's a bipartite graph.<ref>{{citation − − | last = Biggs | first = Norman − − | last = Biggs | first = Norman − − | last = Biggs | first = Norman − − | edition = 2nd − − | edition = 2nd − − 2nd − − | isbn = 9780521458979 − − | isbn = 9780521458979 − − 9780521458979 − − | page = 53 − − | page = 53 − − 53 − − | publisher = Cambridge University Press − − | publisher = Cambridge University Press − − 剑桥大学出版社 − − | series = Cambridge Mathematical Library − − | series = Cambridge Mathematical Library − − | series = Cambridge Mathematical Library − − | title = Algebraic Graph Theory − − | title = Algebraic Graph Theory − − | title = 代数图论 − − | url = https://books.google.com/books?id=6TasRmIFOxQC&pg=PA53 − − | url = https://books.google.com/books?id=6TasRmIFOxQC&pg=PA53 − − Https://books.google.com/books?id=6tasrmifoxqc&pg=pa53 − − | year = 1994}}.</ref> − − | year = 1994}}.</ref> − − | year = 1994} . </ref > − − − − ===Kőnig's theorem and perfect graphs=== − − In bipartite graphs, the size of [[minimum vertex cover]] is equal to the size of the [[maximum matching]]; this is [[Kőnig's theorem (graph theory)|Kőnig's theorem]].<ref>{{cite journal − − In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching; this is Kőnig's theorem.<ref>{{cite journal − − 在二部图中，最小顶点覆盖的大小等于最大匹配的大小，这就是 k-nig 定理。{ cite journal − − | author = Kőnig, Dénes − − | author = Kőnig, Dénes − − 作者: k nig，d é nes − − | authorlink = Dénes Kőnig − − | authorlink = Dénes Kőnig − − | authorlink = d é nes k nig − − | title = Gráfok és mátrixok − − | title = Gráfok és mátrixok − − | title = Gráfok és mátrixok − − | journal = Matematikai és Fizikai Lapok − − | journal = Matematikai és Fizikai Lapok − − | journal = Matematikai és Fizikai Lapok − − | volume = 38 − − | volume = 38 − − 38 − − | year = 1931 − − | year = 1931 − − 1931年 − − | pages = 116–119}}.</ref><ref>{{citation − − | pages = 116–119}}.</ref><ref>{{citation − − 116-119} . </ref > < ref > { citation − − | last1 = Gross | first1 = Jonathan L. − − | last1 = Gross | first1 = Jonathan L. − − 1 = Gross | first1 = Jonathan l. − − | last2 = Yellen | first2 = Jay − − | last2 = Yellen | first2 = Jay − − 2 = Yellen | first2 = Jay − − | edition = 2nd − − | edition = 2nd − − 2nd − − | isbn = 9781584885054 − − | isbn = 9781584885054 − − 9781584885054 − − | page = 568 − − | page = 568 − − 568 − − | publisher = CRC Press − − | publisher = CRC Press − − | publisher = CRC Press − − | series = Discrete Mathematics And Its Applications − − | series = Discrete Mathematics And Its Applications − − 系列 = 离散数学及其应用 − − | title = Graph Theory and Its Applications − − | title = Graph Theory and Its Applications − − | title = 图论及其应用 − − | url = https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA568 − − | url = https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA568 − − Https://books.google.com/books?id=-7q_pogh-2cc&pg=pa568 − − | year = 2005}}.</ref> An alternative and equivalent form of this theorem is that the size of the [[maximum independent set]] plus the size of the maximum matching is equal to the number of vertices. − − | year = 2005}}.</ref> An alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. − − 2005}}.</ref > 这个定理的另一种等价形式是最大独立集的大小加上最大匹配的大小等于顶点的数目。 − − In any graph without [[isolated vertex|isolated vertices]] the size of the [[minimum edge cover]] plus the size of a maximum matching equals the number of vertices.<ref>{{citation − − In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices.<ref>{{citation − − 在任何没有孤立顶点的图中，最小边覆盖的大小加上最大匹配的大小等于顶点的数目。< ref > { citation − − | last1 = Chartrand | first1 = Gary − − | last1 = Chartrand | first1 = Gary − − 1 = Chartrand | first1 = Gary − − | last2 = Zhang | first2 = Ping − − | last2 = Zhang | first2 = Ping − − 2 = Zhang | first2 = Ping − − | isbn = 9780486483689 − − | isbn = 9780486483689 − − 9780486483689 − − | pages = 189–190 − − | pages = 189–190 − − | 页数 = 189-190 − − | publisher = Courier Dover Publications − − | publisher = Courier Dover Publications − − 2012年10月21日 | 出版商 = 多佛出版社 − − | title = A First Course in Graph Theory − − | title = A First Course in Graph Theory − − 图论第一课 − − | url = https://books.google.com/books?id=ocIr0RHyI8oC&pg=PA189 − − | url = https://books.google.com/books?id=ocIr0RHyI8oC&pg=PA189 − − Https://books.google.com/books?id=ocir0rhyi8oc&pg=pa189 − − | year = 2012}}.</ref> Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices. − − | year = 2012}}.</ref> Combining this equality with Kőnig's theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices. − − 2012}}.</ref > 将这个等式与 k nig 定理结合起来得到这样一个结论: 在二部图中，最小边覆盖的大小等于最大独立集的大小，最小边覆盖的大小加上最小顶点覆盖的大小等于顶点的数目。 − − − − Another class of related results concerns [[perfect graph]]s: every bipartite graph, the [[complement (graph theory)|complement]] of every bipartite graph, the [[line graph]] of every bipartite graph, and the complement of the line graph of every bipartite graph, are all perfect. Perfection of bipartite graphs is easy to see (their [[chromatic number]] is two and their [[maximum clique]] size is also two) but perfection of the [[complement (graph theory)|complements]] of bipartite graphs is less trivial, and is another restatement of Kőnig's theorem. This was one of the results that motivated the initial definition of perfect graphs.<ref>{{citation|title=Modern Graph Theory − − Another class of related results concerns perfect graphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement of the line graph of every bipartite graph, are all perfect. Perfection of bipartite graphs is easy to see (their chromatic number is two and their maximum clique size is also two) but perfection of the complements of bipartite graphs is less trivial, and is another restatement of Kőnig's theorem. This was one of the results that motivated the initial definition of perfect graphs.<ref>{{citation|title=Modern Graph Theory − − 另一类相关结果涉及完美图: 每一个二部图、每一个二部图的补图、每一个二部图的线图以及每一个二部图的线图的补图都是完美的。二部图的完美性是显而易见的(它们的色数是2，它们的最大团大小也是2) ，但二部图的补图的完美性是不那么琐碎的，是对 k nig 定理的另一种重述。这是激发完美图最初定义的结果之一 − − |volume= 184 |series= Graduate Texts in Mathematics − − |volume= 184 |series= Graduate Texts in Mathematics − − 184 | series = 数学研究生教材 − − |author=Béla Bollobás|publisher =Springer|year= 1998 − − |author=Béla Bollobás|publisher =Springer|year= 1998 − − 1998年 − − |isbn= 9780387984889|url=https://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA165|page=165|author-link= Béla Bollobás }}.</ref> Perfection of the complements of line graphs of perfect graphs is yet another restatement of Kőnig's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of Kőnig, that every bipartite graph has an [[edge coloring]] using a number of colors equal to its maximum degree. − − |isbn= 9780387984889|url=https://books.google.com/books?id=SbZKSZ-1qrwC&pg=PA165|page=165|author-link= Béla Bollobás }}.</ref> Perfection of the complements of line graphs of perfect graphs is yet another restatement of Kőnig's theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of Kőnig, that every bipartite graph has an edge coloring using a number of colors equal to its maximum degree. − − 9780387984889 | url = https://books.google.com/books?id=sbzksz-1qrwc&pg=pa165|page=165|author-link= · 贝拉 · 博洛巴斯}。完全图的线图的补图的完美性是对 k nig 定理的又一次重申，线图本身的完美性是对 k nig 早期定理的重申，即每个二部图都有一个边着色，其颜色数等于它的最大度。 − − − − According to the [[strong perfect graph theorem]], the perfect graphs have a [[forbidden graph characterization]] resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its [[complement (graph theory)|complement]] as an [[induced subgraph]]. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.<ref>{{citation − − According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.<ref>{{citation − − 根据完美图问题，完美图有一个类似于二部图的禁止图: 一个图是二部图的当且仅当它作为一个子图没有奇圈，并且一个图是完美的当且仅当它没有奇圈或它作为诱导子图的补图。二部图，二部图的线图，以及它们的补图在完美图问题证明中的5个基本类型的完美图中的4个。< ref > { citation − − | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − − | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − − 1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky − − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (mathematician) − − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (mathematician) − − | last2 = Robertson | first2 = Neil | author2-link = Neil Robertson (数学家) − − | last3 = Seymour | first3 = Paul | author3-link = Paul Seymour (mathematician) − − | last3 = Seymour | first3 = Paul | author3-link = Paul Seymour (mathematician) − − 3 = Seymour | first3 = Paul | author3-link = Paul Seymour (数学家) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (mathematician) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (mathematician) − − | last4 = Thomas | first4 = Robin | author4-link = Robin Thomas (数学家) − − | doi = 10.4007/annals.2006.164.51 − − | doi = 10.4007/annals.2006.164.51 − − 2006.164.51 − − | issue = 1 − − | issue = 1 − − 1 − − | journal = [[Annals of Mathematics]] − − | journal = Annals of Mathematics − − 2012年3月24日 | 日志 = 数学纪事 − − | pages = 51–229 − − | pages = 51–229 − − | 页数 = 51-229 − − | title = The strong perfect graph theorem − − | title = The strong perfect graph theorem − − 纽约完美图问题 − − | volume = 164 − − | volume = 164 − − 164 − − | year = 2006| citeseerx = 10.1.1.111.7265| arxiv = math/0212070}}.</ref> − − | year = 2006| citeseerx = 10.1.1.111.7265| arxiv = math/0212070}}.</ref> − − 2006 | citeseerx = 10.1.1.111.7265 | arxiv = math/0212070}} . </ref > − − − − ===Degree=== − − For a vertex, the number of adjacent vertices is called the [[degree (graph theory)|degree]] of the vertex and is denoted <math>\deg(v)</math>. − − For a vertex, the number of adjacent vertices is called the degree of the vertex and is denoted <math>\deg(v)</math>. − − 对于一个顶点，相邻顶点的数目称为顶点的度数，表示为 < math > deg (v) </math > 。 − − The ''degree sum formula'' for a bipartite graph states that − − The degree sum formula for a bipartite graph states that − − 二部图的度和公式表明: − − :<math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − − <math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − − <math>\sum_{v \in V} \deg(v) = \sum_{u \in U} \deg(u) = |E|\, .</math> − − − − The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts <math>U</math> and <math>V</math>. For example, the complete bipartite graph ''K''<sub>3,5</sub> has degree sequence <math>(5,5,5),(3,3,3,3,3)</math>. Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in some cases, non-isomorphic bipartite graphs may have the same degree sequence. − − The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts <math>U</math> and <math>V</math>. For example, the complete bipartite graph K<sub>3,5</sub> has degree sequence <math>(5,5,5),(3,3,3,3,3)</math>. Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in some cases, non-isomorphic bipartite graphs may have the same degree sequence. − − 二部图的度序列是一对列表，每一个列表包含两部分的度。例如，完全二分图 k < sub > 3,5 </sub > 具有度序列 < math > (5,5,5) ，(3,3,3,3,3,3) </math > 。同构二部图具有相同的度序列。然而，度序列通常不能唯一地识别二部图，在某些情况下，不同构的二部图可能具有相同的度序列。 − − − − The [[bipartite realization problem]] is the problem of finding a simple bipartite graph with the degree sequence being two given lists of natural numbers. (Trailing zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the digraph.) − − The bipartite realization problem is the problem of finding a simple bipartite graph with the degree sequence being two given lists of natural numbers. (Trailing zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the digraph.) − − 二部实现问题是寻找度序列为两个给定自然数列表的简单二部图的问题。(尾随零可能被忽略，因为它们通过向有向图添加适当数量的孤立顶点来实现。) − − − − ===Relation to hypergraphs and directed graphs=== − − The [[Adjacency matrix of a bipartite graph|biadjacency matrix]] of a bipartite graph <math>(U,V,E)</math> is a [[(0,1) matrix]] of size <math>|U|\times|V|</math> that has a one for each pair of adjacent vertices and a zero for nonadjacent vertices.<ref>{{harvtxt|Asratian|Denley|Häggkvist|1998}}, p. 17.</ref> Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs. − − The biadjacency matrix of a bipartite graph <math>(U,V,E)</math> is a (0,1) matrix of size <math>|U|\times|V|</math> that has a one for each pair of adjacent vertices and a zero for nonadjacent vertices. Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs. − − 二部图的二邻接矩阵 < math > (u，v，e) </math > 是一个大小为 < math > | u | times | v | </math > 的(0,1)矩阵，每对相邻顶点有一个1，对不相邻顶点有一个0。双邻接矩阵可以用来描述二部图、超图和有向图之间的等价关系。 − − − − A [[hypergraph]] is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints. A bipartite graph <math>(U,V,E)</math> may be used to model a hypergraph in which {{mvar|U}} is the set of vertices of the hypergraph, {{mvar|V}} is the set of hyperedges, and {{mvar|E}} contains an edge from a hypergraph vertex {{mvar|v}} to a hypergraph edge {{mvar|e}} exactly when {{mvar|v}} is one of the endpoints of {{mvar|e}}. Under this correspondence, the biadjacency matrices of bipartite graphs are exactly the [[incidence matrix|incidence matrices]] of the corresponding hypergraphs. As a special case of this correspondence between bipartite graphs and hypergraphs, any [[multigraph]] (a graph in which there may be two or more edges between the same two vertices) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have [[degree (graph theory)|degree]] two.<ref>{{SpringerEOM|title=Hypergraph|author=A. A. Sapozhenko}}</ref> − − A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints. A bipartite graph <math>(U,V,E)</math> may be used to model a hypergraph in which is the set of vertices of the hypergraph, is the set of hyperedges, and contains an edge from a hypergraph vertex to a hypergraph edge exactly when is one of the endpoints of . Under this correspondence, the biadjacency matrices of bipartite graphs are exactly the incidence matrices of the corresponding hypergraphs. As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph (a graph in which there may be two or more edges between the same two vertices) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two. − − 一个超图是一个组合结构，就像一个无向图，有顶点和边，但其中的边可能是任意的顶点集，而不是必须有正好两个端点。二部图 < math > (u，v，e) </math > 可以用来建模一个超图，其中超图的顶点集是超边的集合，它包含一条从超图顶点到超图边的边，而这条边恰好是超图的端点之一。在此对应关系下，二部图的二邻接矩阵正好是对应超图的关联矩阵。作为二部图和超图之间对应关系的一种特殊情形，任何多重图(在同一两个顶点之间可能有两条或更多条边的图)都可以被解释为一个超图，其中一些超边具有相等的端点集，并由一个不具有多重邻接的二部图表示，在这个超图中，二部分一边的顶点都具有二次。 − − − − A similar reinterpretation of adjacency matrices may be used to show a one-to-one correspondence between [[directed graph]]s (on a given number of labeled vertices, allowing self-loops) and balanced bipartite graphs, with the same number of vertices on both sides of the bipartition. For, the adjacency matrix of a directed graph with {{mvar|n}} vertices can be any [[(0,1) matrix]] of size <math>n\times n</math>, which can then be reinterpreted as the adjacency matrix of a bipartite graph with {{mvar|n}} vertices on each side of its bipartition.<ref>{{citation − − A similar reinterpretation of adjacency matrices may be used to show a one-to-one correspondence between directed graphs (on a given number of labeled vertices, allowing self-loops) and balanced bipartite graphs, with the same number of vertices on both sides of the bipartition. For, the adjacency matrix of a directed graph with vertices can be any (0,1) matrix of size <math>n\times n</math>, which can then be reinterpreted as the adjacency matrix of a bipartite graph with vertices on each side of its bipartition.<ref>{{citation − − 一个类似的邻接矩阵翻唱可以用来显示有向图(在给定数量的标号顶点上，允许自循环)和平衡二部图之间的双射，二部图的两边顶点数相同。因为，顶点有向图的邻接矩阵可以是任意大小的(0,1)矩阵 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n 次 n。< ref > { citation − − | last1 = Brualdi | first1 = Richard A. − − | last1 = Brualdi | first1 = Richard A. − − 1 = Brualdi | first1 = Richard a. − − | last2 = Harary | first2 = Frank | author2-link = Frank Harary − − | last2 = Harary | first2 = Frank | author2-link = Frank Harary − − 2 = Harary | first2 = Frank | author2-link = Frank Harary − − | last3 = Miller | first3 = Zevi − − | last3 = Miller | first3 = Zevi − − 3 = Miller | first3 = Zevi − − | doi = 10.1002/jgt.3190040107 − − | doi = 10.1002/jgt.3190040107 − − | doi = 10.1002/jgt. 3190040107 − − | mr = 558453 − − | mr = 558453 − − 558453先生 − − | issue = 1 − − | issue = 1 − − 1 − − | journal = Journal of Graph Theory − − | journal = Journal of Graph Theory − − | Journal = Journal of Graph Theory − − | pages = 51–73 − − | pages = 51–73 − − | 页数 = 51-73 − − | title = Bigraphs versus digraphs via matrices − − | title = Bigraphs versus digraphs via matrices − − | title = Bigraphs vs 通过矩阵的有向图 − − | volume = 4 − − | volume = 4 − − 4 − − | year = 1980}}. Brualdi et al. credit the idea for this equivalence to {{citation − − | year = 1980}}. Brualdi et al. credit the idea for this equivalence to {{citation − − 1980}}.布鲁迪等人。把这个等价的概念归功于{ citation − − | doi = 10.4153/CJM-1958-052-0 − − | doi = 10.4153/CJM-1958-052-0 − − | doi = 10.4153/CJM-1958-052-0 − − | last1 = Dulmage | first1 = A. L. − − | last1 = Dulmage | first1 = A. L. − − 1 = Dulmage | first1 = A.l. − − | last2 = Mendelsohn | first2 = N. S. | author2-link = Nathan Mendelsohn − − | last2 = Mendelsohn | first2 = N. S. | author2-link = Nathan Mendelsohn − − 2 = Mendelsohn | first2 = n. s. | author2-link = Nathan Mendelsohn − − | mr = 0097069 − − | mr = 0097069 − − 0097069先生 − − | journal = Canadian Journal of Mathematics − − | journal = Canadian Journal of Mathematics − − 加拿大数学杂志 − − | pages = 517–534 − − | pages = 517–534 − − | 页数 = 517-534 − − | title = Coverings of bipartite graphs − − | title = Coverings of bipartite graphs − − | title = 二部图的覆盖 − − | volume = 10 − − | volume = 10 − − 10 − − | year = 1958}}.</ref> In this construction, the bipartite graph is the [[bipartite double cover]] of the directed graph. − − | year = 1958}}.</ref> In this construction, the bipartite graph is the bipartite double cover of the directed graph. − − | year = 1958}。 </ref > 在这个结构中，二部图是有向图的二部双覆盖。 − −
→Examples 案例
超立方体图，局部立方体和中位数图均为二分图。判别方法是将这些图中的顶点用位向量（二进制位组成的向量）进行标记，然后对比其中两个顶点的位向量，发现当且仅当位向量中只有一个位元是不同的时候，该两个顶点相邻。另外判定该图的二分性可以通过观察每个顶点的位向量，奇数位向量和偶数位向量分别为该图的二分顶点子集。树图和方图都是中位数图，而所有中位数图都是局部立方体。
超立方体图，局部立方体和中位数图均为二分图。判别方法是将这些图中的顶点用位向量（二进制位组成的向量）进行标记，然后对比其中两个顶点的位向量，发现当且仅当位向量中只有一个位元是不同的时候，该两个顶点相邻。另外判定该图的二分性可以通过观察每个顶点的位向量，奇数位向量和偶数位向量分别为该图的二分顶点子集。树图和方图都是中位数图，而所有中位数图都是局部立方体。
==Algorithms==
==Algorithms==