# Finding Coordinates On A Graph

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How To Find A Perfect Match Introduction How To Find A Perfect Match Introduction This article presents a script that finds a perfect match in any bipartite graph or shows why one cannot exist. Overview This article presents a script that finds a perfect match in any bipartite graph or shows why one cannot exist. Overview Recall that a bipartite graph is any undirected graph [1] where its nodes (or vertices) can be partitioned into two sets, and where edges only exist between those two sets. For this article, those sets are equal in size. Recall that a bipartite graph is any undirected graph [1] where its nodes (or vertices) can be partitioned into two sets, and where edges only exist between those two sets. For this article, those sets are equal in size. For example, consider the following bipartite graph: For example, consider the following bipartite graph: The goal is to find a perfect match, which is a collection of edges sharing no nodes and for which every node is part of some edge in that collection. The goal is to find a perfect match, which is a collection of edges sharing no nodes and for which every node is part of some edge in that collection. This script interprets a popular algorithm for finding perfect matchings [2] by first randomly selecting as many disjoint edges as possible until no more can be found: This script interprets a popular algorithm for finding perfect matchings [2] by first randomly selecting as many disjoint edges as possible until no more can be found: Here, four disjoint edges (in boldface) have been selected: (1,1), (2,2), (3,3), (4,4). Denoting this collection by M, the script will modify M repeatedly until it covers all nodes or show why it can’t be done. Note that if this algorithm fails to match at least half of all nodes, then no perfect matching exists. This follows from a simple counting argument on how many nodes would be available for building a perfect match. Here, four disjoint edges (in boldface) have been selected: (1,1), (2,2), (3,3), (4,4). Denoting this collection by M, the script will modify M repeatedly until it covers all nodes or show why it can’t be done. Note that if this algorithm fails to match at least half of all nodes, then no perfect matching exists. This follows from a simple counting argument on how many nodes would be available for building a perfect match. Denote by U and V the collection of left and right nodes that are not in M. These are the nodes that are missing from the current match. Clearly |U| = |V| since the graph contains an equal number of left and right nodes. Denote by U and V the collection of left and right nodes that are not in M. These are the nodes that are missing from the current match. Clearly |U| = |V| since the graph contains an equal number of left and right nodes.