A matching in a graph means the selection of a set of edges that are pairwise non-adjacent, i.e. they have no common incident vertices. A matching is maximal if it is not a proper subset of any other matching.

## Usage

```
is_matching(graph, matching, types = NULL)
is_max_matching(graph, matching, types = NULL)
max_bipartite_match(
graph,
types = NULL,
weights = NULL,
eps = .Machine$double.eps
)
```

## Arguments

- graph
The input graph. It might be directed, but edge directions will be ignored.

- matching
A potential matching. An integer vector that gives the pair in the matching for each vertex. For vertices without a pair, supply

`NA`

here.- types
Vertex types, if the graph is bipartite. By default they are taken from the ‘

`type`

’ vertex attribute, if present.- weights
Potential edge weights. If the graph has an edge attribute called ‘

`weight`

’, and this argument is`NULL`

, then the edge attribute is used automatically. In weighted matching, the weights of the edges must match as much as possible.- eps
A small real number used in equality tests in the weighted bipartite matching algorithm. Two real numbers are considered equal in the algorithm if their difference is smaller than

`eps`

. This is required to avoid the accumulation of numerical errors. By default it is set to the smallest \(x\), such that \(1+x \ne 1\) holds. If you are running the algorithm with no weights, this argument is ignored.

## Value

`is_matching()`

and `is_max_matching()`

return a logical
scalar.

`max_bipartite_match()`

returns a list with components:

- matching_size
The size of the matching, i.e. the number of edges connecting the matched vertices.

- matching_weight
The weights of the matching, if the graph was weighted. For unweighted graphs this is the same as the size of the matching.

- matching
The matching itself. Numeric vertex id, or vertex names if the graph was named. Non-matched vertices are denoted by

`NA`

.

## Details

`is_matching()`

checks a matching vector and verifies whether its
length matches the number of vertices in the given graph, its values are
between zero (inclusive) and the number of vertices (inclusive), and
whether there exists a corresponding edge in the graph for every matched
vertex pair. For bipartite graphs, it also verifies whether the matched
vertices are in different parts of the graph.

`is_max_matching()`

checks whether a matching is maximal. A matching
is maximal if and only if there exists no unmatched vertex in a graph
such that one of its neighbors is also unmatched.

`max_bipartite_match()`

calculates a maximum matching in a bipartite
graph. A matching in a bipartite graph is a partial assignment of
vertices of the first kind to vertices of the second kind such that each
vertex of the first kind is matched to at most one vertex of the second
kind and vice versa, and matched vertices must be connected by an edge
in the graph. The size (or cardinality) of a matching is the number of
edges. A matching is a maximum matching if there exists no other
matching with larger cardinality. For weighted graphs, a maximum
matching is a matching whose edges have the largest possible total
weight among all possible matchings.

Maximum matchings in bipartite graphs are found by the push-relabel algorithm with greedy initialization and a global relabeling after every \(n/2\) steps where \(n\) is the number of vertices in the graph.

## See also

Other structural.properties:
`bfs()`

,
`component_distribution()`

,
`connect()`

,
`constraint()`

,
`coreness()`

,
`degree()`

,
`dfs()`

,
`distance_table()`

,
`edge_density()`

,
`feedback_arc_set()`

,
`girth()`

,
`is_acyclic()`

,
`is_dag()`

,
`k_shortest_paths()`

,
`knn()`

,
`reciprocity()`

,
`subcomponent()`

,
`subgraph()`

,
`topo_sort()`

,
`transitivity()`

,
`unfold_tree()`

,
`which_multiple()`

,
`which_mutual()`

## Author

Tamas Nepusz ntamas@gmail.com

## Examples

```
g <- graph_from_literal(a - b - c - d - e - f)
m1 <- c("b", "a", "d", "c", "f", "e") # maximal matching
m2 <- c("b", "a", "d", "c", NA, NA) # non-maximal matching
m3 <- c("b", "c", "d", "c", NA, NA) # not a matching
is_matching(g, m1)
#> [1] TRUE
is_matching(g, m2)
#> [1] TRUE
is_matching(g, m3)
#> [1] FALSE
is_max_matching(g, m1)
#> [1] TRUE
is_max_matching(g, m2)
#> [1] FALSE
is_max_matching(g, m3)
#> [1] FALSE
V(g)$type <- rep(c(FALSE, TRUE), 3)
print_all(g, v = TRUE)
#> IGRAPH cf22f16 UN-B 6 5 --
#> + attr: name (v/c), type (v/l)
#> + vertex attributes:
#> | name type
#> | [1] a FALSE
#> | [2] b TRUE
#> | [3] c FALSE
#> | [4] d TRUE
#> | [5] e FALSE
#> | [6] f TRUE
#> + edges from cf22f16 (vertex names):
#> [1] a--b b--c c--d d--e e--f
max_bipartite_match(g)
#> $matching_size
#> [1] 3
#>
#> $matching_weight
#> [1] 3
#>
#> $matching
#> a b c d e f
#> "b" "a" "d" "c" "f" "e"
#>
g2 <- graph_from_literal(a - b - c - d - e - f - g)
V(g2)$type <- rep(c(FALSE, TRUE), length.out = vcount(g2))
print_all(g2, v = TRUE)
#> IGRAPH 03f59e2 UN-B 7 6 --
#> + attr: name (v/c), type (v/l)
#> + vertex attributes:
#> | name type
#> | [1] a FALSE
#> | [2] b TRUE
#> | [3] c FALSE
#> | [4] d TRUE
#> | [5] e FALSE
#> | [6] f TRUE
#> | [7] g FALSE
#> + edges from 03f59e2 (vertex names):
#> [1] a--b b--c c--d d--e e--f f--g
max_bipartite_match(g2)
#> $matching_size
#> [1] 3
#>
#> $matching_weight
#> [1] 3
#>
#> $matching
#> a b c d e f g
#> "b" "a" "d" "c" "f" "e" NA
#>
#' @keywords graphs
```