Graph minors. V. Excluding a planar graph
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B
European Journal of Combinatorics
Graph minors. IV. Tree-width and well-quasi-ordering
Journal of Combinatorial Theory Series B
Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series B
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
A combinatorial strongly polynomial algorithm for minimizing submodular functions
Journal of the ACM (JACM)
Branch-width and well-quasi-ordering in matroids and graphs
Journal of Combinatorial Theory Series B
Polynomial Time Recognition of Clique-Width \le \leq 3 Graphs (Extended Abstract)
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
On the excluded minors for the matroids of branch-width k
Journal of Combinatorial Theory Series B
The interlace polynomial of a graph
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Vertex-minors, monadic second-order logic, and a conjecture by Seese
Journal of Combinatorial Theory Series B
The relative clique-width of a graph
Journal of Combinatorial Theory Series B
Digraph measures: Kelly decompositions, games, and orderings
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Vertex-minor reductions can simulate edge contractions
Discrete Applied Mathematics
Boundary classes of planar graphs
Combinatorics, Probability and Computing
Digraph measures: Kelly decompositions, games, and orderings
Theoretical Computer Science
Approximating rank-width and clique-width quickly
ACM Transactions on Algorithms (TALG)
Graph operations characterizing rank-width
Discrete Applied Mathematics
Recent developments on graphs of bounded clique-width
Discrete Applied Mathematics
TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width
Discrete Applied Mathematics
Finding branch-decompositions and rank-decompositions
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Dynamic distance hereditary graphs using split decomposition
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Theoretical Computer Science
F-rank-width of (edge-colored) graphs
CAI'11 Proceedings of the 4th international conference on Algebraic informatics
Graph reductions, binary rank, and pivots in gene assembly
Discrete Applied Mathematics
The group structure of pivot and loop complementation on graphs and set systems
European Journal of Combinatorics
Approximating rank-width and clique-width quickly
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Pivot and loop complementation on graphs and set systems
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Well-quasi-ordering of matrices under Schur complement and applications to directed graphs
European Journal of Combinatorics
Pivots, determinants, and perfect matchings of graphs
Theoretical Computer Science
Recognition of probe distance-hereditary graphs
Discrete Applied Mathematics
Clique-width and edge contraction
Information Processing Letters
Obstructions for linear rank-width at most 1
Discrete Applied Mathematics
Graphs of small rank-width are pivot-minors of graphs of small tree-width
Discrete Applied Mathematics
Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width
Fundamenta Informaticae - MFCS & CSL 2010 Satellite Workshops: Selected Papers
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The rank-width is a graph parameter related in terms of fixed functions to clique-width but more tractable. Clique-width has nice algorithmic properties, but no good "minor" relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartite graphs and minors of binary matroids, and as an application, we prove that bipartite graphs of sufficiently large rank-width contain certain bipartite graphs as vertex-minors. 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. Furthermore, we prove that a graph has rank-width at most 1 if and only if it is distance-hereditary.