# maximum.cardinality.search

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##### Maximum cardinality search

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

Keywords
graphs
##### Usage
maximum.cardinality.search(graph)
##### Arguments
graph
The input graph. It may be directed, but edge directions are ignored, as the algorithm is defined for undirected graphs.
##### 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.

##### Value

• A list with two components:
• alphaNumeric vector. The vertices ordered according to the maximum cardinality search.
• alpham1Numeric vector. The inverse of alpha.

##### concept

• maximum cardinality search
• graph decomposition
• chordal graph

##### 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.

is.chordal

##### Aliases
• maximum.cardinality.search
##### Examples
## The examples from the Tarjan-Yannakakis paper
g1 <- graph.formula(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)
maximum.cardinality.search(g1)
is.chordal(g1, fillin=TRUE)

g2 <- graph.formula(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)
maximum.cardinality.search(g2)
is.chordal(g2, fillin=TRUE)
Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)

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