Chordality of a graph

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.

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.

See also

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 153b043:
#> [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 0c86dac:
#>  [1] J H I G C F D B E A
#> 
is_chordal(g2, fillin = TRUE)
#> $chordal
#> [1] TRUE
#> 
#> $fillin
#> numeric(0)
#> 
#> $newgraph
#> NULL
#>