Maximum cardinality search

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

Usage

max_cardinality(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.

Author

Gabor Csardi csardi.gabor@gmail.com

igraph_maximum_cardinality_search().

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 0b198ba:
#> [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 6f9cd39:
#>  [1] J H I G C F D B E A
#> 
is_chordal(g2, fillin = TRUE)
#> $chordal
#> [1] TRUE
#> 
#> $fillin
#> numeric(0)
#> 
#> $newgraph
#> NULL
#>