Corpus ID: 201271493

Tree pivot-minors and linear rank-width

  title={Tree pivot-minors and linear rank-width},
  author={Konrad K. Dabrowski and F. Dross and Jisu Jeong and M. Kant{\'e} and O. Kwon and Sang-il Oum and D. Paulusma},
Treewidth and its linear variant path-width play a central role for the graph minor relation. Rank-width and linear rank-width do the same for the graph pivot-minor relation. Robertson and Seymour (1983) proved that for every tree T there exists a constant cT such that every graph of path-width at least cT contains T as a minor. Motivated by this result, we examine whether for every tree T there exists a constant dT such that every graph of linear rank-width at least dT contains T as a… Expand

Figures and Topics from this paper


Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width
It is proved that every graph of rank-width k is a pivot-minor of a graph of tree-width at most 2k, and it is shown that bipartite graphs ofRank- width at most 1 are exactly pivot- Minors of trees and bipartITE graphs of linear rank- Width 1 are precisely pivot-Minors of paths. Expand
Linear rank-width and linear clique-width of trees
Using the characterization of linear rank-width of forests, the set of minimal excluded acyclic vertex-minors for the class of graphs of linearrank-width at most k is determined. Expand
Rank-width and vertex-minors
  • Sang-il Oum
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. B
  • 2005
The main theorem of this paper is that for fixed k, there is a finite list of graphs such that a graph G has rank-width at most k if and only if no graph in the list is isomorphic to a vertex-minor of G. Expand
Thread Graphs, Linear Rank-Width and Their Algorithmic Applications
Rank-width is proposed, which allows the solution of many such hard problems on a less restricted graph classes which are NP-hard even on graphs of bounded tree-width or even on trees. Expand
Computing Small Pivot-Minors
It is proved that the Pivot-Minor problem, which asks if a given graph G contains a givenGraph H as a pivot-minor, is NP-complete, and gives a certifying polynomial-time algorithm for H -Pivot- minor for every graph H with|V(H)|\le 4\) except when H is not part of the input. Expand
Graph minors. I. Excluding a forest
It is proved that there is a numberk such that every graph with no minor isomorphic toH has path-width≆k, and this implies that ifP is any property of graphs such that some forest does not have propertyP, then the set of minor-minimal graphs without propertyP is finite. Expand
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
It is proved that for a fixed tree $T$, every distance-hereditary graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$, and it is conjectured that it is sufficient to prove this conjecture for prime graphs. Expand
Linear Rank-Width of Distance-Hereditary Graphs
It is shown that the linear rank-width of every \(n\)-vertex distance-hereditary graph can be computed in time \(\mathcal {O}(n^2\cdot \log (n)), and a linear layout witnessing the linearRank-width can be compute with the same time complexity. Expand
Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm
This paper shows that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank- width at most 1, can be computed in time, and introduces a notion of ‘limbs’ of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization. Expand
Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width
It is proved that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. Expand