% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/decomposition.R
\name{is_chordal}
\alias{is_chordal}
\title{Chordality of a graph}
\usage{
is_chordal(
  graph,
  alpha = NULL,
  alpham1 = NULL,
  fillin = FALSE,
  newgraph = FALSE
)
}
\arguments{
\item{graph}{The input graph. It may be directed, but edge directions are
ignored, as the algorithm is defined for undirected graphs.}

\item{alpha}{Numeric vector, the maximal chardinality ordering of the
vertices. If it is \code{NULL}, then it is automatically calculated by
calling \code{\link[=max_cardinality]{max_cardinality()}}, or from \code{alpham1} if
that is given..}

\item{alpham1}{Numeric vector, the inverse of \code{alpha}. If it is
\code{NULL}, then it is automatically calculated by calling
\code{\link[=max_cardinality]{max_cardinality()}}, or from \code{alpha}.}

\item{fillin}{Logical scalar, whether to calculate the fill-in edges.}

\item{newgraph}{Logical scalar, whether to calculate the triangulated graph.}
}
\value{
A list with three members:
\describe{
\item{chordal}{
Logical scalar, it is \code{TRUE} iff the input graph is chordal.
}
\item{fillin}{
If requested, then a numeric vector giving the fill-in edges. \code{NULL} otherwise.
}
\item{newgraph}{
If requested, then the triangulated graph, an \code{igraph} object. \code{NULL} otherwise.
}
}
}
\description{
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.
}
\details{
The chordality of the graph is decided by first performing maximum
cardinality search on it (if the \code{alpha} and \code{alpham1} arguments
are \code{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.
}
\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)
is_chordal(g1, fillin = TRUE)

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)
is_chordal(g2, fillin = TRUE)

}
\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.  \emph{SIAM Journal
of Computation} 13, 566--579.
}
\seealso{
\code{\link[=max_cardinality]{max_cardinality()}}

Other chordal: 
\code{\link{max_cardinality}()}
}
\author{
Gabor Csardi \email{csardi.gabor@gmail.com}
}
\concept{chordal}
\keyword{graphs}
