Maximum cardinality search is a simple ordering a vertices that is useful in determining the chordality of a graph.

## Arguments

- graph
The input graph. It may be directed, but edge directions are ignored, as the algorithm is defined for undirected graphs.

## Value

A list with two components:

- alpha
Numeric vector. The 1-based rank of each vertex in the graph such that the vertex with rank 1 is visited first, the vertex with rank 2 is visited second and so on.

- alpham1
Numeric vector. The inverse of

`alpha`

. In other words, the elements of this vector are the vertices in reverse maximum cardinality search order.

## Details

Maximum cardinality search visits the vertices in such an order that every time the vertex with the most already visited neighbors is visited. Ties are broken randomly.

The algorithm provides a simple basis for deciding whether a graph is
chordal, see References below, and also `is_chordal()`

.

## References

Robert E Tarjan and Mihalis Yannakakis. (1984). Simple
linear-time algorithms to test chordality of graphs, test acyclicity of
hypergraphs, and selectively reduce acyclic hypergraphs. *SIAM Journal
of Computation* 13, 566--579.

## See also

Other chordal:
`is_chordal()`

## Author

Gabor Csardi csardi.gabor@gmail.com

## Examples

```
## The examples from the Tarjan-Yannakakis paper
g1 <- graph_from_literal(
A - B:C:I, B - A:C:D, C - A:B:E:H, D - B:E:F,
E - C:D:F:H, F - D:E:G, G - F:H, H - C:E:G:I,
I - A:H
)
max_cardinality(g1)
#> $alpha
#> [1] 9 4 6 8 3 5 7 2 1
#>
#> $alpham1
#> + 9/9 vertices, named, from d5e881a:
#> [1] G F D B E C H I A
#>
is_chordal(g1, fillin = TRUE)
#> $chordal
#> [1] FALSE
#>
#> $fillin
#> [1] 2 6 8 7 5 7 2 7 6 1 7 1
#>
#> $newgraph
#> NULL
#>
g2 <- graph_from_literal(
A - B:E, B - A:E:F:D, C - E:D:G, D - B:F:E:C:G,
E - A:B:C:D:F, F - B:D:E, G - C:D:H:I, H - G:I:J,
I - G:H:J, J - H:I
)
max_cardinality(g2)
#> $alpha
#> [1] 10 8 9 6 7 5 4 2 3 1
#>
#> $alpham1
#> + 10/10 vertices, named, from d1941ac:
#> [1] J H I G C F D B E A
#>
is_chordal(g2, fillin = TRUE)
#> $chordal
#> [1] TRUE
#>
#> $fillin
#> numeric(0)
#>
#> $newgraph
#> NULL
#>
```