# Dynamic survey on rank-width

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

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

### 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 (6k+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 3Ω(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(8k n9 logn). (Here, n is the number of vertices of the input graph.) The running time was later improved by Oum [30] to O(8k 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(c1k nc) 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(n8k+c), and if we apply it to rank-width, we get an algorithm of running time O(n8k+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(n2k+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 (6k−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 3·2k−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 2k/2−1. This also implies that there are graphs having rank-width at most k+1 and clique-width at least 2k/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 Kr minor with r > 2, then tree-width is at most 2O(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.
 Width Parameter Running time Paper Rank-width O*(2n) Oum [33] Linear rank-width O*(2n) forklore (trivial)
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.
Remark. Johansson [17] (also in his Ph.D. thesis [18]) claimed in a conference paper that NLC-width and clique-width of a random graph G(n,p) for a fixed 0 < p < 1 is Ω(n) almost surely. But we believe that the proof is incorrect.
In 2009, Marecek [25] studied the rank-width of G(n,1/2), though I believe that the first version of his paper on arxiv1 is incorrect; Later versions have different proofs.

## 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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### Footnotes:

1http://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