"A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is, every edge connects a vertex in U to one in V."
Bipartite graphs have two sets of vertices, and all edges are between vertices in different sets. They are often used to model relationships between two different types of objects.
Graph Theory Basics: An understanding of basic graph theory concepts, including vertices, edges, and connectivity, is essential to learning about bipartite graphs.
Bipartite Graph Definition: A bipartite graph is a type of graph in which the set of vertices can be divided into two groups such that each edge connects a vertex from one group to a vertex from the other.
Applications of Bipartite Graphs: Bipartite graphs have applications in many areas such as job scheduling, matching problems, and social network analysis.
Bipartite Graph Properties: Some essential properties of bipartite graphs include the maximum matching theorem, which states that a maximum matching of a bipartite graph can be found using only greedy algorithm methods.
The Perfect Matching Problem: In graphs, perfect matching refers to the phenomenon of finding a matching that covers all vertices of the graph.
Network Flows in Bipartite Graphs: Network flows are methods of distributing goods or services through a network, and bipartite graphs can be used to model network flow problems.
Bipartite Graph Algorithms: Different algorithms such as the Hopcroft-Karp algorithm allow for efficient resolution of bipartite matching problems.
Graph Coloring and Bipartite Graphs: Bipartite graphs are a subset of graphs that can be two-colorable, which opens up new possibilities in the domain of graph coloring problems.
Alternating Paths and Augmenting Paths: These are important concepts in bipartite matching algorithms and refer to paths in which every other edge is part of a matching.
Maximum Cardinality Matching: This refers to finding the largest possible matching within a bipartite graph, and is an important problem in network flow applications.
Complete bipartite graph: A graph in which each vertex of one set is connected to every vertex of the other set.
Bipartite cycle: A graph that forms a cycle with vertices alternating between two sets.
Bipartite planar graph: A graph that can be drawn in the plane without any crossings, where the vertices can be divided into two sets.
Perfect bipartite matching: A bipartite graph where every vertex in one set is connected with exactly one vertex in the other set.
Bipartite graph with maximum matching: A graph that has the maximum possible number of edges that do not have a common vertex.
Coordination bipartite graph: A graph that models a situation where there are two types of objects that have to be coordinated, and each edge represents the possibility of coordination between an object of one type and an object of the other type.
Bi-hypergraph: A generalization of bipartite graphs where edges can connect more than two vertices of different sets, which is also called a hypergraph or a bipartite incidence graph.
Alternating tree: A tree that can be decomposed into a sequence of alternating levels of vertices that belong to each set, where the root is in one set, the leaves are in the other set, and the edges are between adjacent levels.
Matchstick graph: A graph that can be represented by a collection of sticks, where each stick represents an edge between two vertices of opposite sets.
Expander graph: A bipartite graph that has many edges and few short cycles, which is useful for applications in computer science and cryptography.
"The two sets U and V may be thought of as a coloring of the graph with two colors."
"If one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem."
"Such a coloring is impossible in the case of a non-bipartite graph, such as a triangle."
"No, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color."
"One often writes G = (U, V, E) to denote a bipartite graph..."
"E denotes the edges of the graph."
"If a bipartite graph is not connected, it may have more than one bipartition."
"In this case, the (U, V, E) notation is helpful in specifying one particular bipartition..."
"If |U| = |V|, that is, if the two subsets have equal cardinality, then G is called a balanced bipartite graph."
"If all vertices on the same side of the bipartition have the same degree, then G is called biregular."
"A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V."
"The two sets U and V may be thought of as a coloring of the graph with two colors."
"No, a triangle is a non-bipartite graph."
"No, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle."
"One often writes G = (U, V, E) to denote a bipartite graph..."
"E denotes the edges of the graph."
"If a bipartite graph is not connected, it may have more than one bipartition."
"The (U, V, E) notation is helpful in specifying one particular bipartition."
"If |U| = |V|, that is, if the two subsets have equal cardinality, then G is called a balanced bipartite graph."