二分图

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文件:Simple-bipartite-graph.svg

Example of a bipartite graph without cycles

Example of a bipartite graph without cycles

没有圈的二部图的例子

文件:Biclique K 3 5.svg

A complete bipartite graph with m = 5 and n = 3

A complete bipartite graph with m = 5 and n = 3

顶点数分别为m=5和n=3的两个集合所组成的一个完整二分图

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] such that every edge connects a vertex in [math]\displaystyle{ U }[/math] to one in [math]\displaystyle{ V }[/math]. Vertex sets [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length。

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets {\displaystyle U}U and {\displaystyle V}V such that every edge connects a vertex in {\displaystyle U}U to one in {\displaystyle V}V. Vertex sets {\displaystyle U}U and {\displaystyle V}V are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.[1][2]

在图论的数学领域中，二分图（或二部图）内的所有顶点可以分为两个不相交且独立的集合U和集合V，并且每个连边（无向或有向）的两个顶点分别在集合U和集合V当中。通常集合U和集合V被称为该二分图的子集。同时，二分图中不包含任何形式的奇数环，即：集合U和集合V构造的点集所形成的循环圈边数不为奇数。

The two sets [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] may be thought of as a coloring of the graph with two colors: if one colors all nodes in [math]\displaystyle{ U }[/math] blue, and all nodes in [math]\displaystyle{ V }[/math] green, each edge has endpoints of differing colors, as is required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another green, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color.

The two sets [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math] may be thought of as a coloring of the graph with two colors: if one colors all nodes in [math]\displaystyle{ U }[/math] blue, and all nodes in [math]\displaystyle{ V }[/math] green, each edge has endpoints of differing colors, as is required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another green, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color.

通常可以将集合U和集合V视为被着色的两类图集：如果其中一类所有顶点为蓝色，另外一类为绿色，那么每条连边的两端必须具有不同的颜色。这是二分图中必须满足的条件。相反，在非二分图（例如三角形）的情况下，这种着色要求是不可能达到的：因为一个顶点为蓝色，另一个为绿色，其三角形剩下的第三个顶点将需要同时连接这两个颜色，很明显这是不可能达到的，因为它违背了要求：二分图中所有顶点必须分为两个不相交且独立的集合U和V。

One often writes [math]\displaystyle{ G=(U,V,E) }[/math] to denote a bipartite graph whose partition has the parts [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math], with [math]\displaystyle{ E }[/math] denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition; in this case, the [math]\displaystyle{ (U,V,E) }[/math] notation is helpful in specifying one particular bipartition that may be of importance in an application. If [math]\displaystyle{ |U|=|V| }[/math], that is, if the two subsets have equal cardinality, then [math]\displaystyle{ G }[/math] is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then [math]\displaystyle{ G }[/math] is called biregular.

One often writes [math]\displaystyle{ G=(U,V,E) }[/math] to denote a bipartite graph whose partition has the parts [math]\displaystyle{ U }[/math] and [math]\displaystyle{ V }[/math], with [math]\displaystyle{ E }[/math] denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition; in this case, the [math]\displaystyle{ (U,V,E) }[/math] notation is helpful in specifying one particular bipartition that may be of importance in an application. If [math]\displaystyle{ |U|=|V| }[/math], that is, if the two subsets have equal cardinality, then [math]\displaystyle{ G }[/math] is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then [math]\displaystyle{ G }[/math] is called biregular.

对于二分图常用的表达方式是G=(U,V,E)，其中U和V分别代表子集，E代表二分图中的连边。如果有一个二分图内部不是连通的，则它可能具有多个二分图；在这种情况下，(U,V,E)标注有助于指定一个特殊的二分图。在实际的应用当中，这无比重要。如果|U|=|V|，即这两个子集具有相同的基数，此时G被称为均衡二分图。如果在二分图中同一侧（同一个子集）所有的顶点都具有相同的度数，则G被称为双正则。

Examples 案例

When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.

When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.

在建立两类不同对象之间的关系时，二分图往往自然而然地就出现了。比如说足球运动员和俱乐部的关系图，如果该球员曾为该俱乐部效力，则在运动员和俱乐部之间就形成了一条连边，这是隶属关系网络的示例。在社交网络分析中，隶属关系便形成一种二分图。

Another example where bipartite graphs appear naturally is in the (NP-complete) railway optimization problem, in which the input is a schedule of trains and their stops, and the goal is to find a set of train stations as small as possible such that every train visits at least one of the chosen stations. This problem can be modeled as a dominating set problem in a bipartite graph that has a vertex for each train and each station and an edge for each pair of a station and a train that stops at that station.[7]

Another example where bipartite graphs appear naturally is in the (NP-complete) railway optimization problem, in which the input is a schedule of trains and their stops, and the goal is to find a set of train stations as small as possible such that every train visits at least one of the chosen stations. This problem can be modeled as a dominating set problem in a bipartite graph that has a vertex for each train and each station and an edge for each pair of a station and a train that stops at that station.[7]

二分图自然形成的另一个例子是（NP-完全问题）铁路优化问题，其中输入的是火车的时间表及其停靠点，目标是找到尽可能小的火车站集合，以便每个火车都可以停靠至少一个选定的火车站。这个问题可以被建模为一个二分图中的主导集合问题，该图中每辆列车和每个车站都被看作是一个顶点，而当某列车停靠在某车站的时候，其形成的关系被视为一条连边。

A third example is in the academic field of numismatics. Ancient coins are made using two positive impressions of the design (the obverse and reverse). The charts numismatists produce to represent the production of coins are bipartite graphs.

A third example is in the academic field of numismatics. Ancient coins are made using two positive impressions of the design (the obverse and reverse). The charts numismatists produce to represent the production of coins are bipartite graphs.

第三个例子涉及到钱币学领域。古代的硬币被设计为正反两面，不同时期和地区的政府会使用不同的正反面组合，因此，将所有可能的组合画成图就是一个二分图的结构。

More abstract examples include the following:

More abstract examples include the following:

更多抽象的例子:

• Every tree is bipartite.

每棵树都是二分的。

• Cycle graphs with an even number of vertices are bipartite.

具有偶数个顶点的环图都是二分的

• Every planar graph whose faces all have even length is bipartite. Special cases of this are grid graphs and squaregraphs, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors.

平面图（图论）中，所有面都具有偶数条边的平面图是二分的。特殊情况是网格图和方图，其中每个内面都由4个边组成，每个内顶点都有4个或更多的相邻点。

• The complete bipartite graph on m and n vertices, denoted by K_n,m is the bipartite graph [math]\displaystyle{ G = (U, V, E) }[/math], where U and V are disjoint sets of size m and n, respectively, and E connects every vertex in U with all vertices in V. It follows that K_m,n has mn edges. Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs.

The biadjacency matrix of a bipartite graph [math]\displaystyle{ (U,V,E) }[/math] is a (0,1) matrix of size [math]\displaystyle{ |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]\displaystyle{ (U,V,E) }[/math] may be used to model a hypergraph in which U is the set of vertices of the hypergraph, V is the set of hyperedges, and E contains an edge from a hypergraph vertex v to a hypergraph edge e exactly when v is one of the endpoints of e. 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.

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]\displaystyle{ (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 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 n vertices can be any (0,1) matrix of size [math]\displaystyle{ n\times n }[/math], which can then be reinterpreted as the adjacency matrix of a bipartite graph with n vertices on each side of its bipartition. 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。

Algorithms

Testing bipartiteness

It is possible to test whether a graph is bipartite, and to return either a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time, using depth-first search. The main idea is to assign to each vertex the color that differs from the color of its parent in the depth-first search forest, assigning colors in a preorder traversal of the depth-first-search forest. This will necessarily provide a two-coloring of the spanning forest consisting of the edges connecting vertices to their parents, but it may not properly color some of the non-forest edges. In a depth-first search forest, one of the two endpoints of every non-forest edge is an ancestor of the other endpoint, and when the depth first search discovers an edge of this type it should check that these two vertices have different colors. If they do not, then the path in the forest from ancestor to descendant, together with the miscolored edge, form an odd cycle, which is returned from the algorithm together with the result that the graph is not bipartite. However, if the algorithm terminates without detecting an odd cycle of this type, then every edge must be properly colored, and the algorithm returns the coloring together with the result that the graph is bipartite.

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Alternatively, a similar procedure may be used with breadth-first search in place of depth-first search. Again, each node is given the opposite color to its parent in the search forest, in breadth-first order. If, when a vertex is colored, there exists an edge connecting it to a previously-colored vertex with the same color, then this edge together with the paths in the breadth-first search forest connecting its two endpoints to their lowest common ancestor forms an odd cycle. If the algorithm terminates without finding an odd cycle in this way, then it must have found a proper coloring, and can safely conclude that the graph is bipartite.

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For the intersection graphs of [math]\displaystyle{ n }[/math] line segments or other simple shapes in the Euclidean plane, it is possible to test whether the graph is bipartite and return either a two-coloring or an odd cycle in time [math]\displaystyle{ O(n\log n) }[/math], even though the graph itself may have up to [math]\displaystyle{ O(n^2) }[/math] edges.

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Odd cycle transversal

Main article: Odd cycle transversal

文件:Odd Cycle Transversal of size 2.png

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph.

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph.

一个具有大小为2的奇圈横截的图: 去掉两个蓝色底顶点后得到一个二部图。

Odd cycle transversal is an NP-complete algorithmic problem that asks, given a graph G = (V,E) and a number k, whether there exists a set of k vertices whose removal from G would cause the resulting graph to be bipartite. The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k. The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k. The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles. Hence, to delete vertices from a graph in order to obtain a bipartite graph, one needs to "hit all odd cycle", or find a so-called odd cycle transversal set. In the illustration, every odd cycle in the graph contains the blue (the bottommost) vertices, so removing those vertices kills all odd cycles and leaves a bipartite graph.

| year = 2004| citeseerx = 10.1.1.112.6357}}.</ref> The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles.  Hence, to delete vertices from a graph in order to obtain a bipartite graph, one needs to "hit all odd cycle", or find a so-called odd cycle transversal set.  In the illustration, every odd cycle in the graph contains the blue (the bottommost) vertices, so removing those vertices kills all odd cycles and leaves a bipartite graph.

10.1.1.112.6357}.名称奇圈横截来自于这样一个事实: 一个图是二部图当且仅当它没有奇圈。因此，要从图中删除顶点以得到二部图，就需要“遍历所有奇圈” ，或者找到一个所谓的奇圈横截集。在图中，图中的每个奇数圈都包含蓝色(最底部的)顶点，因此去掉这些顶点会杀死所有奇数圈并留下一个二部图。

The edge bipartization problem is the algorithmic problem of deleting as few edges as possible to make a graph bipartite and is also an important problem in graph modification algorithmics. This problem is also fixed-parameter tractable, and can be solved in time O(2^k m²), where k is the number of edges to delete and m is the number of edges in the input graph.

| year = 2006}}</ref> where k is the number of edges to delete and m is the number of edges in the input graph.

| year = 2006} </ref > 其中 k 是要删除的边数，m 是输入图中的边数。

Matching

Main article: Matching (graph theory)

A matching in a graph is a subset of its edges, no two of which share an endpoint. Polynomial time algorithms are known for many algorithmic problems on matchings, including maximum matching (finding a matching that uses as many edges as possible), maximum weight matching, and stable marriage. In many cases, matching problems are simpler to solve on bipartite graphs than on non-bipartite graphs, and many matching algorithms such as the Hopcroft–Karp algorithm for maximum cardinality matching work correctly only on bipartite inputs.

| year = 1993}}.</ref> In many cases, matching problems are simpler to solve on bipartite graphs than on non-bipartite graphs, and many matching algorithms such as the Hopcroft–Karp algorithm for maximum cardinality matching work correctly only on bipartite inputs.

1993}}.在许多情况下，二部图上的匹配问题比非二部图上的匹配问题更容易解决，而且许多匹配算法，如最大基数匹配的霍普克罗夫特-卡普算法，只能在二部图输入上正确工作。

As a simple example, suppose that a set [math]\displaystyle{ P }[/math] of people are all seeking jobs from among a set of [math]\displaystyle{ J }[/math] jobs, with not all people suitable for all jobs. This situation can be modeled as a bipartite graph [math]\displaystyle{ (P,J,E) }[/math] where an edge connects each job-seeker with each suitable job. A perfect matching describes a way of simultaneously satisfying all job-seekers and filling all jobs; Hall's marriage theorem provides a characterization of the bipartite graphs which allow perfect matchings. The National Resident Matching Program applies graph matching methods to solve this problem for U.S. medical student job-seekers and hospital residency jobs.

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The Dulmage–Mendelsohn decomposition is a structural decomposition of bipartite graphs that is useful in finding maximum matchings.

The Dulmage–Mendelsohn decomposition is a structural decomposition of bipartite graphs that is useful in finding maximum matchings.

Dulmage-Mendelsohn 分解是二部图的一种结构分解，它有助于找到最大匹配。

Additional applications

Bipartite graphs are extensively used in modern coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors. A factor graph is a closely related belief network used for probabilistic decoding of LDPC and turbo codes.

| year = 2005}}.</ref> A factor graph is a closely related belief network used for probabilistic decoding of LDPC and turbo codes.

2005}}.因子图是一种密切相关的置信网络，用于 LDPC 码和 turbo 码的概率译码。

In computer science, a Petri net is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.

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In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a configuration. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their girth must be six or more.

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See also

• Bipartite dimension, the minimum number of complete bipartite graphs whose union is the given graph

• Bipartite double cover, a way of transforming any graph into a bipartite graph by doubling its vertices

• Bipartite hypergraph, a generalization of bipartiteness to hypergraphs.

• Bipartite matroid, a class of matroids that includes the graphic matroids of bipartite graphs

• Bipartite network projection, a weighting technique for compressing information about bipartite networks

• Convex bipartite graph, a bipartite graph whose vertices can be ordered so that the vertex neighborhoods are contiguous

• Multipartite graph, a generalization of bipartite graphs to more than two subsets of vertices

• Parity graph, a generalization of bipartite graphs in which every two induced paths between the same two points have the same parity

• Quasi-bipartite graph, a type of Steiner tree problem instance in which the terminals form an independent set, allowing approximation algorithms that generalize those for bipartite graphs

• Split graph, a graph in which the vertices can be partitioned into two subsets, one of which is independent and the other of which is a clique

• Zarankiewicz problem on the maximum number of edges in a bipartite graph with forbidden subgraphs

External links

• 模板:Springer

• Information System on Graph Classes and their Inclusions: bipartite graph

• 模板:Mathworld

• Bipartite graphs in systems biology and medicine

Category:Graph families

类别: 图族

Category:Parity (mathematics)

分类: 奇偶校验(数学)

This page was moved from wikipedia:en:Bipartite graph. Its edit history can be viewed at 二分图/edithistory