Note added on Jan. 2019: An improved version of this article has been published as a journal paper in Discrete Applied Mathematics.

Dynamic survey on rank-width and related width parameters of graphs

Sang-il Oum

Aug 19, 2013

This is an incomplete on-going survey on rank-width and its related parameters. I intend to expand it slowly.
By no means, this will be complete.
Please feel free to leave comments or give me suggestions.

1  Definitions

Rank-width was introduced by Oum and Seymour [35].
Clique-width is defined and investigated first by Courcelle and Olariu [5] published in 2000, but the operations for clique-width has been introduced by Courcelle, Engelfriet, and Rozenberg [4] in 1993.
NLC-width was introduced by Wanke [40] in 1994.
Boolean-width was introduced by Bui-Xuan, Telle, and Vatshelle [2].

2  Well-quasi-ordering

Oum [31] proved that graphs of bounded rank-width are well-quasi-ordered under taking pivot-minors.
This result has been generalized to

1. skew-symmetric or symmetric matrices over a fixed finite field
   by Oum [34],
    
2. σ-symmetric matrices over a fixed finite field by Kante [20].
    

3  Forbidden vertex-minors

Oum [29] showed that if a graph has rank-width k, then so is its vertex-minor.
This, together with the well-quasi-ordering theorem [31] implies that
for each k, there exists a list of finitely many graphs such that a graph has rank-width at most k if and only if none of its vertex-minors is isomorphic to a graph in the list.

Oum [29] showed that for each k, the forbidden vertex-minor for rank-width at most k has at most (6^k+1−1)/5 vertices.
So in theory, we can enumerate all of them by an algorithm for fixed k.

However, it is not clear how to find the list of forbidden vertex-minors
for graphs of linear rank-width at most k,
as there are no analogous upper bound on the size of forbidden vertex-minors. Jeong, Kwon, and Oum [16]
showed that there are at least 3^Ω(3k) forbidden vertex-minors for linear rank-width at most k.

4  Hardness

Computing rank-width is NP-hard. It can be easily deduced by combining the following known facts. This reduction is mentioned in [30].

1. Seymour and Thomas [37] proved that computing branch-width is NP-hard.
   Kloks, Kratochvíl, and Müller [21] even showed that it is NP-hard to compute branch-width of interval graphs.
    
2. Hicks and McMurray Jr. [13] and independently Mazoit and Thomassé [26] showed that if a graph G is not a forest, then the branch-width of the cycle matroid M(G) is equal to the branch-width of G.
    
3. Oum [29] showed that the branch-width of a binary matroid is equal to the rank-width of its fundamental graph. Every graphic matroid is binary.
    

It is not known to me whether computing linear rank-width is NP-hard.

Computing clique-width is NP-hard, shown by Fellows, Rosamond, Rotics, and Szeider [8].

Computing the relative clique-width is also NP-complete, shown by Müller and Urner [27]. The relative clique-width [24] is a clique-width restricted to a fixed decomposition tree.

Gurski and Wanke [12] proved that computing NLC-width is
also NP-hard via a reduction from the tree-width.

5  Finding an approximate rank-decomposition for a fixed k

Oum and Seymour [35] provided an algorithm to
find a rank-decomposition of width at most 3k+1 or confirm that the input graph has rank-width bigger than k
in time O(8^k n⁹ logn).
(Here, n is the number of vertices of the input graph.)
The running time was later improved by Oum [30]
to O(8^k n⁴).

This allows us to find a rank-decomposition of width at most c′ times rank-width
if the rank-width is at most clogn in polynomial time.
This is used in the quantum information theory (see Van den Nest, Dür, Vidal, and Briegel [38]) for their study on measurement-based quantum computation.

If we do not mind having bigger function in k, then in the same paper [30], it is possible to find a rank-decomposition of width 3k−1 or
confirm that the rank-width is bigger than k in time O(f(k)n³).

We do not know whether there is an algorithm to find an approximate rank-decomposition of width O(k) in time O(c₁^k n^c) for c ≤ 3
when an input graph has rank-width at most k.

These algorithms can be used as a tool to construct an expression for clique-width decomposition, which are essential in many algorithmic applications.

6  Deciding rank-width at most k for fixed k

There is a general algorithm of Oum and Seymour [36] which can construct a branch-decomposition of any symmetric submodular function in time O(n^8k+c), and if we apply it to rank-width, we get an algorithm of running time O(n^8k+12logn).
Note that a simple general algorithm for path-width of any symmetric submodular function
was developed by Nagamochi [28], which is applicable to linear rank-width in time O(n^2k+4).

Courcelle and Oum [6] first constructed an algorithm to decide, for fixed k, whether rank-width is at most k in time O(f(k)n³). But their algorithm uses the monadic second-order logic formula and does not provide an explicit rank-decomposition even if it exists.

This problem was solved later.
Hlinený and Oum [14] constructed an algorithm to decide whether rank-width is at most k
and find a rank-decomposition of width at most k, if it exists, in time O(f(k)n³).
Here, f(k) is growing very fast with k, because the algorithm uses the monadic second-order logic as well as the list of forbidden minors for branch-width at most k for matroids representable over a fixed finite field.
Geelen, Gerards, Robertson, and Whittle [11] proved that the forbidden minors for matroids of branch-width at most k have at most (6^k−1)/5 elements. This implies that we can construct an explicit algorithm for testing rank-width at most k and constructing a rank-decomposition of width at most k if it exists. (If there was no upper bound, then it may be impossible to enumerate all forbidden minors.)

One can also decide whether the linear rank-width is at most k in time O(n³) by using the well-quasi-ordering theorem [31] and monadic second-order logic formula to test vertex-minors [6]. But it is not known how to find the list of forbidden vertex-minors.

Wahlström [39] showed that deciding whether clique-width is at most k and finding a k-expression can be done in time O^*((2k+1)ⁿ).

Espelage, Gurski, and Wanke [7] constructed an algorithm to decide whether a graph has clique-width at most k for graphs of bounded tree-width.

7  Relation to Tree-width

Kante [19] showed that rank-width is at most 4k+2 if the tree-width is k.
Later,
Oum [32] showed that rank-width is at most k+1 if tree-width is k.
In fact, it is shown that

if G has branch-width k,
then the incidence graph of G has rank-width k or k−1,
unless k=0.

Corneil and Rotics [3] showed that the clique-width is at most 3·2^k−1 if the tree-width is k.
Moreover, they proved that for each k, there is a graph G having tree-width k and clique-width at least 2^{k/2−1}.
This also implies that there are graphs having rank-width at most k+1 and clique-width at least 2^{k/2−1} for each k.

Kwon and Oum [22] proved that every graph of rank-width k
is a pivot-minor of a graph of tree-width at most 2k.
They also proved that every graph of linear rank-width k is a pivot-minor of a graph of path-width k+1.
In other words,
a set I of graphs has bounded rank-width
if and only if it is a set of pivot-minors of graphs of bounded tree-width.

Fomin, Oum, and Thilikos [10] showed that when graphs are planar, or H-minor-free, then
having bounded tree-width is equivalent to having bounded rank-width.
For instance, if a graph G is planar and has rank-width k,
then tree-width is at most 72k−2.
If G of rank-width k
has no K_r minor with r > 2, then tree-width is at most 2^O(rloglogr)k.
This is already proven in Courcelle and Olariu [5] without explicit bounds because they use logical tools.

8  Exact Algorithms

There are small number of papers dealing with computing exact value of rank-width or related width parameters.

The running time to compute clique-width exactly seems open.

Müller and Urner [27] proved that NLC-width can be computed in time O(3ⁿ n) for an n-vertex graph.
When they gave a talk at GROW2007 about this result, they further claimed that clique-width can be computed in time O^*(3ⁿ) by finding a polynomial-time algorithm to compute relative clique-width [24] but later Ruth Urner emailed me that there was a mistake.

Branch-width of graphs can be computed in time O^*((2√3)ⁿ), shown by Fomin, Mazoit, and Todinca [9].

9  Random graphs

Lee, Lee, and Oum [23] showed that asymptotically almost surely the Erdös-Rényi random graph G(n,p) has rank-width n/3−O(1) if p is a constant between 0 and 1.
Furthermore, [1/(n)] << p ≤ [1/2], then the rank-width is [(n)/3]−o(n),
and if p = c/n and c > 1, then rank-width is at least r n for some r = r(c).

Since the clique-width is at least rank-width, this also gives a lower bound for clique-width.

Adler, Bui-Xuan, Rabinovich, Renault, Telle, and Vatshelle [1] claimed that boolean-width of G(n,p) for fixed 0 < p < 1 is Θ(log² n) asymptotically almost surely.

10  Explicit graphs

Jelínek [15] proved that an n×n grid has rank-width n−1.

11  Software

Philipp Klau Krause implemented a simple dynamic programming algorithm
to compute the rank-width of a graph².
This software is now included in the open source mathematics software
package called SAGE; see the
manual³.

Apparently Friedmanský wrote Master’s thesis on “implementation of
optimization of a graph algorithm for computing rank-width”⁴ under the
supervision of Robert Ganian.

Acknowledgment

I would like to thank careful readers who kindly emailed me
suggestions for this survey.

• August 2013: Jisu Jeong found a typo.
   
• August 2013: Chiheon Kim pointed out a mistake on the statement
  on the consequence of a theorem by Corneil and Rotics.
   
• July 2013: J. Marecek explained his paper on arxiv.
   
• July 2013: F. Gurski provided a journal version of his paper
  with E. Wanke cited
  in the survey.
   

References

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Isolde Adler, Binh-Minh Bui-Xuan, Yuri Rabinovich, Gabriel Renault, Jan Arne
Telle, and Martin Vatshelle.
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Fedor V. Fomin, Sang-il Oum, and Dimitrios M. Thilikos.
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Öjvind Johansson.
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PhD thesis, Royal Institute of Technology, 2001.

 

[19]

Mamadou Moustapha Kanté.
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[21]

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[22]

O-joung Kwon and Sang-il Oum.
Graphs of small rank-width are pivot-minors of graphs of small
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Choongbum Lee, Joonkyung Lee, and Sang-il Oum.
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Jakub Marecek.
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Sang-il Oum.
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Footnotes:

¹https://arxiv.org/pdf/0908.1772v1.pdf

²http://pholia.tdi.informatik.uni-frankfurt.de/~philipp/software/rw.shtml

³http://www.sagemath.org/doc/reference/graphs/sage/graphs/graph_decompositions/rankwidth.html

⁴http://is.muni.cz/th/172614/fi_m/thesis.pdf