A graph is chordal (or triangulated) if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle. An equivalent definition is that any chordless cycles have at most three nodes.

## Usage

``````is_chordal(
graph,
alpha = NULL,
alpham1 = NULL,
fillin = FALSE,
newgraph = FALSE
)``````

## Arguments

graph

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

alpha

Numeric vector, the maximal chardinality ordering of the vertices. If it is `NULL`, then it is automatically calculated by calling `max_cardinality()`, or from `alpham1` if that is given..

alpham1

Numeric vector, the inverse of `alpha`. If it is `NULL`, then it is automatically calculated by calling `max_cardinality()`, or from `alpha`.

fillin

Logical scalar, whether to calculate the fill-in edges.

newgraph

Logical scalar, whether to calculate the triangulated graph.

## Value

A list with three members:

chordal

Logical scalar, it is `TRUE` iff the input graph is chordal.

fillin

If requested, then a numeric vector giving the fill-in edges. `NULL` otherwise.

newgraph

If requested, then the triangulated graph, an `igraph` object. `NULL` otherwise.

## Details

The chordality of the graph is decided by first performing maximum cardinality search on it (if the `alpha` and `alpham1` arguments are `NULL`), and then calculating the set of fill-in edges.

The set of fill-in edges is empty if and only if the graph is chordal.

It is also true that adding the fill-in edges to the graph makes it 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.

`max_cardinality()`

Other chordal: `max_cardinality()`

## 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 6a5bad9:
#> [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 22d96f8:
#>  [1] J H I G C F D B E A
#>
is_chordal(g2, fillin = TRUE)
#> \$chordal
#> [1] TRUE
#>
#> \$fillin
#> numeric(0)
#>
#> \$newgraph
#> NULL
#>

``````