# Rank Decompositions of graphs¶

This module wraps C code from Philipp Klaus Krause computing an optimal rank-decomposition.

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 equal 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 nodes have degree 3. In a tree, any edge naturally corresponds to a bipartition of the vertex set : indeed, the removal 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 is a program that calculates rank-width and rank-decompositions. It is based on ideas from:

• “Computing rank-width exactly” by Sang-il Oum [Oum2009]

• “Sopra una formula numerica” by Ernesto Pascal

• “Generation of a Vector from the Lexicographical Index” by B.P. Buckles and M. Lybanon [BL1977]

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 comparable 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 !

EXAMPLES:

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


AUTHORS:

• Philipp Klaus Krause : Implementation of the C algorithm

• Nathann Cohen : Interface with Sage and documentation

## Methods¶

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

Return the graph corresponding to the current rank-decomposition.

(This function is for internal use)

EXAMPLES:

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)

Compute 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 (default: False); 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).

EXAMPLES:

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)