In graph theory, a perfect matching in a graph is a matching that covers every vertex of the graph. More formally, given a graph G = (V, E), a perfect matching in G is a subset M of edge set E, such that every vertex in the vertex set V is adjacent to exactly one edge in M.
In mathematics, Hall's marriage theorem, proved by Philip Hall, is a theorem with two equivalent formulations. In each case, the theorem gives a necessary and sufficient condition for an object to exist:
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring.
In computer science, the Edmonds–Karp algorithm is an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in time. The algorithm was first published by Yefim Dinitz in 1970, and independently published by Jack Edmonds and Richard Karp in 1972. Dinitz's algorithm includes additional techniques that reduce the running time to .
In the mathematical field of graph theory, a bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.
In graph theory, a vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph.
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem.
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three.
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. Equivalently, an edge is a bridge if and only if it is not contained in any cycle. For a connected graph, a bridge can uniquely determine a cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges.
In graph theory, the degree of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex is denoted or . The maximum degree of a graph is denoted by , and is the maximum of 's vertices' degrees. The minimum degree of a graph is denoted by , and is the minimum of 's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0.
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In computer science, the minimum edge cover problem is the problem of finding an edge cover of minimum size. It is an optimization problem that belongs to the class of covering problems and can be solved in polynomial time.
In the mathematical area of graph theory, Kőnig's theorem, proved by Dénes Kőnig, describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs.
In computer science, the Hopcroft–Karp algorithm is an algorithm that takes a bipartite graph as input and produces a maximum-cardinality matching as output — a set of as many edges as possible with the property that no two edges share an endpoint. It runs in time in the worst case, where is set of edges in the graph, is set of vertices of the graph, and it is assumed that . In the case of dense graphs the time bound becomes , and for sparse random graphs it runs in time with high probability.
In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the subgraph of G induced by all vertices adjacent to v, i.e., the graph composed of the vertices adjacent to v and all edges connecting vertices adjacent to v.
In graph theory, the blossom algorithm is an algorithm for constructing maximum matchings on graphs. The algorithm was developed by Jack Edmonds in 1961, and published in 1965. Given a general graph G = (V, E), the algorithm finds a matching M such that each vertex in V is incident with at most one edge in M and |M| is maximized. The matching is constructed by iteratively improving an initial empty matching along augmenting paths in the graph. Unlike bipartite matching, the key new idea is that an odd-length cycle in the graph (blossom) is contracted to a single vertex, with the search continuing iteratively in the contracted graph.
In graph theory, a maximally matchable edge in a graph is an edge that is included in at least one maximum-cardinality matching in the graph. An alternative term is allowed edge.
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph.
In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni, Penny Haxell, Roy Meshulam, and others.
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices.
