COMPUTATION OF LINEAR RANK-WIDTH
Keywords: linear rank-width; rank-decomposition; path-decomposition;
vertex-minor
Internship atLimos, Clermont-Ferrand, supervised by Mamadou M. Kant´e.
Introduction. Rank-widthis a complexity measure introduced by Oum and Seymour [7, 6]. Rank-width is interesting for several reasons.
(1) It is equivalent to clique-width, a complexity measure introduced by Courcelle et al. [4], that generalises the well-known complexity measure tree-width introduced by Robertson and Seymour in their graph minors series.
(2) It is algorithmically more interesting than clique-width because we can recognise in polynomial time graphs of rank-width at most k (for fixed k)
(3) It shares with tree-width many structural properties (its is for in- stance related to the theory of matroids and is related to thevertex- minor relation [6]).
(4) . . .
While there exist several algorithms for computing the tree-width of some graph classes, a little is known about the computation of the rank-width of some graph classes. In particular, it is open whether one can compute the rank-width ofcircle graphsin polynomial time (circle graphs are graphs that are conjectured to play a central role in vertex-minor theory, a role similar to the one of planar graphs in the graph minor theory). Acircle graph is an intersection graph of chords in a circle.
In this internship, we are interested in a linearised version of rank-width, called linear rank-width (see for instance [5] for a definition). Indeed, the decomposition associated to rank-width is a tree, and for linear rank-width we impose that tree to be a caterpillar. Linear rank-width is to rank-width what path-width is to tree-width.
The goal. In [2] the authors have given a linear time algorithm to com- pute the linear rank-width and the linear clique-width of trees (a subclass of circle graphs). The goal of the internship is to implement both algorithms and to adapt them to the case of linear Boolean-width (Boolean-width is a complexity measure similar to rank-width, but the measure is with respect to Boolean algebra instead of F2; a definition can be found in [3]). If there is time we will look at the generation of trees that are obstructions to linear rank-width.
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2 COMPUTATION OF LINEAR RANK-WIDTH
All materials and needed explanations will be provided and for further information please feel free to contact me atkante@isima.fr. You can find the definition of rank-width and clique-width in my Phd Thesis available online [8, Chapter 1]. The papers in the references are available in the authors’ webpages. An implementation of an algorithm for approximating the rank-width of graphs of rank-width at most k (for fixed k) exists in SAGE [1].
References
[1] http://sagemath.org/doc/reference/sage/graphs/graph decompositions/rankwidth.html.
[2] Isolde Adler and Mamadou Moustapha Kant´e. Linear rank-width and linear clique- width of trees. To be submitted, 2013.
[3] Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Boolean-width of graphs.
Theor. Comput. Sci., 412(39):5187–5204, 2011.
[4] Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs.
Discrete Applied Mathematics, 101(1-3):77–114, 2000.
[5] Robert Ganian. Thread graphs, linear rank-width and their algorithmic applications.
In Costas S. Iliopoulos and William F. Smyth, editors,IWOCA, volume 6460 ofLecture Notes in Computer Science, pages 38–42. Springer, 2010.
[6] Sang il Oum. Rank-width and vertex-minors.J. Comb. Theory, Ser. B, 95(1):79–100, 2005.
[7] Sang il Oum and Paul D. Seymour. Approximating clique-width and branch-width.J.
Comb. Theory, Ser. B, 96(4):514–528, 2006.
[8] Mamadou Moustapha Kant´e. Graph Structurings: Some algorithmic applications. PhD thesis, Universit´e Bordeaux 1, 2008. Available at http://www.isima.fr/∼kante/articles/KanteThesis2008E.pdf.
