A matching in a graph G=(V,E) is a subset M of the edges E such that no two edges in M share a common end node. A maximum cardinality matching is matching with a maximum number of edges. A node cover is a set of nodes NC of G such that every edge of G has at least one node in NC. Matching and node cover are in some sense opposites of each other. A matching covers the nodes of G with edges such that each node is covered by at most one edge. A node cover covers the edges of G by nodes such that each edge is covered by at least one node. Interesting Fact: The maximum cardinality of a matching is at most the minimum cardinality of a node cover, that is, the maximum number of edges in a matching of G is always less than or equal the minimum number of nodes in a node cover of G. Maximum Cardinality Matching in Bipartite Graphs A graph is bipartite if it has two kinds of nodes and the edges are only allowed between nodes of different kind. We write G=(A,B,E) where A andB are the two sets of nodes and E are the edges of G. A matching in a bipartite graph assigns nodes of A to nodes of B. Matchings in bipartite graphs can be computed more efficiently than matchings in general (=non-bipartite) graphs. Interesting Fact: In a bipartite graph the maximum cardinality of a matching and the minimum cardinality of a node cover are equal. Intuition Bipartite Matching: Suppose we have a set of workers and a set of machines. These are the two kinds of nodes in our bipartite graph. We know which worker can handle which machines. This defines the edges of our bipartite graph. Our task in the maximum cardinality matching problem is assigning workers to machines in such a way that as many machines as possible are operated by a worker that can handle it. One worker can be assigned to at most one machine and we can assign at most one worker to one machine. LEDA Functions for Maximum Cardinality Matching in Bipartite Graphs MAX_CARD_BIPARTITE_MATCHING() computes a maximum cardinality matching in a bipartite graph. Remark: Additionally, LEDA provides several specialized implementations for maximum cardinality matching in bipartite graphs. These functions are mainly intended for specialists. More details can be found on the Manual Page of Maximum Cardinality Matching in Bipartite Graphs. CHECK_MCB() checks if the result of MAX_CARD_BIPARTITE_MATCHING(), a list M and a node_array NC, is maximum cardinality matching and node cover of a bipartite graph G. Example Solution A matching in a graph G=(V,E) is a subset M of the edges E such that no two edges in M share a common end node. A maximum cardinality matching is matching with a maximum number of edges. A node cover is a set of nodes NC of G such that every edge of G has at least one node in NC. Matching and node cover are in some sense opposites of each other. A matching covers the nodes of G with edges such that each node is covered by at most one edge. A node cover covers the edges of G by nodes such that each edge is .

1. A matching in a graph G=(V,E) is a subset M of the edges E such that no two edges in M share a
common end node. A maximum cardinality matching is matching with a maximum number of
edges.
A node cover is a set of nodes NC of G such that every edge of G has at least one node in NC.
Matching and node cover are in some sense opposites of each other. A matching covers the
nodes of G with edges such that each node is covered by at most one edge. A node cover covers
the edges of G by nodes such that each edge is covered by at least one node.
Interesting Fact: The maximum cardinality of a matching is at most the minimum cardinality of a
node cover, that is, the maximum number of edges in a matching of G is always less than or
equal the minimum number of nodes in a node cover of G.
Maximum Cardinality Matching in Bipartite Graphs
A graph is bipartite if it has two kinds of nodes and the edges are only allowed between nodes of
different kind. We write G=(A,B,E) where A andB are the two sets of nodes and E are the edges
of G.
A matching in a bipartite graph assigns nodes of A to nodes of B. Matchings in bipartite graphs
can be computed more efficiently than matchings in general (=non-bipartite) graphs.
Interesting Fact: In a bipartite graph the maximum cardinality of a matching and the minimum
cardinality of a node cover are equal.
Intuition Bipartite Matching: Suppose we have a set of workers and a set of machines. These are
the two kinds of nodes in our bipartite graph. We know which worker can handle which
machines. This defines the edges of our bipartite graph. Our task in the maximum cardinality
matching problem is assigning workers to machines in such a way that as many machines as
possible are operated by a worker that can handle it. One worker can be assigned to at most one
machine and we can assign at most one worker to one machine.
LEDA Functions for Maximum Cardinality Matching in Bipartite Graphs
MAX_CARD_BIPARTITE_MATCHING() computes a maximum cardinality matching in a
bipartite graph.
Remark: Additionally, LEDA provides several specialized implementations for maximum
cardinality matching in bipartite graphs. These functions are mainly intended for specialists.
More details can be found on the Manual Page of Maximum Cardinality Matching in Bipartite
Graphs.
CHECK_MCB() checks if the result of MAX_CARD_BIPARTITE_MATCHING(), a list M and
a node_array NC, is maximum cardinality matching and node cover of a bipartite graph G.
Example

2. Solution
A matching in a graph G=(V,E) is a subset M of the edges E such that no two edges in M share a
common end node. A maximum cardinality matching is matching with a maximum number of
edges.
A node cover is a set of nodes NC of G such that every edge of G has at least one node in NC.
Matching and node cover are in some sense opposites of each other. A matching covers the
nodes of G with edges such that each node is covered by at most one edge. A node cover covers
the edges of G by nodes such that each edge is covered by at least one node.
Interesting Fact: The maximum cardinality of a matching is at most the minimum cardinality of a
node cover, that is, the maximum number of edges in a matching of G is always less than or
equal the minimum number of nodes in a node cover of G.
Maximum Cardinality Matching in Bipartite Graphs
A graph is bipartite if it has two kinds of nodes and the edges are only allowed between nodes of
different kind. We write G=(A,B,E) where A andB are the two sets of nodes and E are the edges
of G.
A matching in a bipartite graph assigns nodes of A to nodes of B. Matchings in bipartite graphs
can be computed more efficiently than matchings in general (=non-bipartite) graphs.
Interesting Fact: In a bipartite graph the maximum cardinality of a matching and the minimum
cardinality of a node cover are equal.
Intuition Bipartite Matching: Suppose we have a set of workers and a set of machines. These are
the two kinds of nodes in our bipartite graph. We know which worker can handle which
machines. This defines the edges of our bipartite graph. Our task in the maximum cardinality
matching problem is assigning workers to machines in such a way that as many machines as
possible are operated by a worker that can handle it. One worker can be assigned to at most one
machine and we can assign at most one worker to one machine.
LEDA Functions for Maximum Cardinality Matching in Bipartite Graphs
MAX_CARD_BIPARTITE_MATCHING() computes a maximum cardinality matching in a
bipartite graph.
Remark: Additionally, LEDA provides several specialized implementations for maximum
cardinality matching in bipartite graphs. These functions are mainly intended for specialists.
More details can be found on the Manual Page of Maximum Cardinality Matching in Bipartite
Graphs.
CHECK_MCB() checks if the result of MAX_CARD_BIPARTITE_MATCHING(), a list M and
a node_array NC, is maximum cardinality matching and node cover of a bipartite graph G.
Example