# is.chordal

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##### 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.

Keywords
graphs
##### 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 maximum.cardinality.search, or from
alpham1
Numeric vector, the inverse of alpha. If it is NULL, then it is automatically calculated by calling maximum.cardinality.search, or from alpha
fillin
Logical scalar, whether to calculate the fill-in edges.
newgraph
Logical scalar, whether to calculate the triangulated graph.
##### 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.

##### Value

• A list with three members:
• chordalLogical scalar, it is TRUE iff the input graph is chordal.
• fillinIf requested, then a numeric vector giving the fill-in edges. NULL otherwise.
• newgraphIf requested, then the triangulated graph, an igraph object. NULL otherwise.

##### 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.

maximum.cardinality.search

• is.chordal
##### 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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