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# Rank Decompositions of graphs

Rank Decompositions of graphs

This modules wraps a C code from Philipp Klaus Krause computing a an optimal rank-decomposition [RWKlause].

Definitions :

Given a graph \(G\) and a subset \(S\subseteq V(G)\) of vertices, the rank-width of \(S\) in \(G\), denoted \(rw_G(S)\), is equal to the rank in \(GF(2)\) of the \(|S| \times (|V|-|S|)\) matrix of the adjacencies between the vertices of \(S\) and \(V\backslash S\). By definition, \(rw_G(S)\) is qual to \(rw_G(\overline S)\) where \(\overline S\) is the complement of \(S\) in \(V(G)\).

A rank-decomposition of \(G\) is a tree whose \(n\) leaves are the elements of \(V(G)\), and whose internal noes have degree 3. In a tree, ay edge naturally corresponds to a bipartition of the vertex set : indeed, the reoal of any edge splits the tree into two connected components, thus splitting the set of leaves (i.e. vertices of \(G\)) into two sets. Hence we can define for any edge \(e\in E(G)\) a width equal to the value \(rw_G(S)\) or \(rw_G(\overline S)\), where \(S,\overline S\) is the bipartition obtained from \(e\). The rank-width associated to the whole decomposition is then set to the maximum of the width of all the edges it contains.

A rank-decomposition is said to be optimal for \(G\) if it is the decomposition achieving the minimal rank-width.

RW – The original source code :

RW [RWKlause] is a program that calculates rank-width and rank-decompositions. It is based on ideas from :

> - “Computing rank-width exactly” by Sang-il Oum [Oum]
> - “Sopra una formula numerica” by Ernesto Pascal
> - “Generation of a Vector from the Lexicographical Index” by B.P. Buckles and M. Lybanon [BL]
> - “Fast additions on masked integers” by Michael D. Adams and David S. Wise [AW]

OUTPUT:

The rank decomposition is returned as a tree whose vertices are subsets of \(V(G)\). Its leaves, corresponding to the vertices of \(G\) are sets of 1 elements, i.e. singletons.

    sage: g = graphs.PetersenGraph()
    sage: rw, tree = g.rank_decomposition()
    sage: all(len(v)==1 for v in tree if tree.degree(v) == 1)
    True

The internal nodes are sets of the decomposition. This way, it is easy to deduce the bipartition associated to an edge from the tree. Indeed, two adjacent vertices of the tree are comarable sets : they yield the bipartition obtained from the smaller of the two and its complement.

    sage: g = graphs.PetersenGraph()
    sage: rw, tree = g.rank_decomposition()
    sage: u = Set([8, 9, 3, 7])
    sage: v = Set([8, 9])
    sage: tree.has_edge(u,v)
    True
    sage: m = min(u,v)
    sage: bipartition = (m, Set(g.vertices()) - m)
    sage: bipartition
    ({8, 9}, {0, 1, 2, 3, 4, 5, 6, 7})

Warning

- The current implementation cannot handle graphs of \(\geq 32\) vertices.
- A bug that has been reported upstream make the code crash immediately on instances of size \(30\). If you experience this kind of bug please report it to us, what we need is some information on the hardware you run to know where it comes from !

EXAMPLE:

    sage: g = graphs.PetersenGraph()
    sage: g.rank_decomposition()
    (3, Graph on 19 vertices)

AUTHORS:

- Philipp Klaus Krause : Implementation of the C algorithm [RWKlause].
- Nathann Cohen : Interface with Sage and documentation.

REFERENCES:

> [RWKlause] | (1, 2, 3) Philipp Klaus Krause – rw v0.2 http://pholia.tdi.informatik.uni-frankfurt.de/~philipp/software/rw.shtml

> [Oum] | Sang-il Oum Computing rank-width exactly Information Processing Letters, 2008 vol. 109, n. 13, p. 745–748 Elsevier http://mathsci.kaist.ac.kr/~sangil/pdf/2008exp.pdf

> [BL] | Buckles, B.P. and Lybanon, M. Algorithm 515: generation of a vector from the lexicographical index ACM Transactions on Mathematical Software (TOMS), 1977 vol. 3, n. 2, pages 180–182 ACM

> [AW] | Adams, M.D. and Wise, D.S. Fast additions on masked integers ACM SIGPLAN Notices, 2006 vol. 41, n.5, pages 39–45 ACM http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.86.1801&rep=rep1&type=pdf

## Methods

`sage.graphs.graph_decompositions.rankwidth.``mkgraph`()

  Returns the graph corresponding the the current rank-decomposition.

  (This function is for internal use)

  EXAMPLE:

      sage: from sage.graphs.graph_decompositions.rankwidth import rank_decomposition
      sage: g = graphs.PetersenGraph()
      sage: rank_decomposition(g)
      (3, Graph on 19 vertices)

`sage.graphs.graph_decompositions.rankwidth.``rank_decomposition`(G, verbose=False)

  Computes an optimal rank-decomposition of the given graph.

  This function is available as a method of the `Graph` class. See `rank_decomposition`.

  INPUT:

  - `verbose` (boolean) – whether to display progress information while computing the decomposition.

  OUTPUT:

  A pair `(rankwidth, decomposition_tree)`, where `rankwidth` is a numerical value and `decomposition_tree` is a ternary tree describing the decomposition (cf. the module’s documentation).

  EXAMPLE:

      sage: from sage.graphs.graph_decompositions.rankwidth import rank_decomposition
      sage: g = graphs.PetersenGraph()
      sage: rank_decomposition(g)
      (3, Graph on 19 vertices)

  On more than 32 vertices:

      sage: g = graphs.RandomGNP(40, .5)
      sage: rank_decomposition(g)
      Traceback (most recent call last):
      ...
      RuntimeError: the rank decomposition cannot be computed on graphs of >= 32 vertices

  The empty graph:

      sage: g = Graph()
      sage: rank_decomposition(g)
      (0, Graph on 0 vertices)

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