1,685 research outputs found
Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width
We prove that every graph of rank-width is a pivot-minor of a graph of
tree-width at most . We also prove that graphs of rank-width at most 1,
equivalently distance-hereditary graphs, are exactly vertex-minors of trees,
and graphs of linear rank-width at most 1 are precisely vertex-minors of paths.
In addition, we show that bipartite graphs of rank-width at most 1 are exactly
pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are
precisely pivot-minors of paths.Comment: 16 pages, 7 figure
Rank-width of Random Graphs
Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs
introduced by Oum and Seymour (2006).
We investigate the asymptotic behavior of rank-width of a random graph
G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant,
then rw(G(n,p)) = \lceil n/3 \rceil-O(1), (ii) if 1/n<< p <1/2, then
rw(G(n,p))= \lceil n/3\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p))
> r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2.
As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n
for each c>1, answering a question of Gao (2006).Comment: 10 page
Perfect Matchings in Claw-free Cubic Graphs
Lovasz and Plummer conjectured that there exists a fixed positive constant c
such that every cubic n-vertex graph with no cutedge has at least 2^(cn)
perfect matchings. Their conjecture has been verified for bipartite graphs by
Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every
claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect
matchings, thus verifying the conjecture for claw-free graphs.Comment: 6 pages, 2 figure
Income Distribution and Poverty in a CGE Framework: A Proposed Methodology
The paper discusses methodologies addressing income distribution and poverty in a Computable General Equilibrium (CGE) model framework, by describing how to link CGE results with household survey data to analyze income distribution and poverty implications. The most basic approach is simply to fit the household income/expenditure to the survey data by suitable parametric distribution functions. The post-simulation poverty indices can be estimated by either assuming that the income of each individual household within the group moves proportionally with the group's mean income, or by our proposed elasticity method. In our proposed method, we use the elasticity estimated from existing surveys to calculate the change in expenditure of each subgroup category in response to change in the household category's mean consumption, supplied by the core model's simulation, to derive post-simulation poverty indices. Our approach may better capture intra-group income distribution of households and moderate gains or losses in welfare from economic growths.Computable General Equilibrium, Income Distribution, Poverty.
Rank-width and Well-quasi-ordering of Skew-Symmetric or Symmetric Matrices
We prove that every infinite sequence of skew-symmetric or symmetric matrices
M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such
that M_i is isomorphic to a principal submatrix of the Schur complement of a
nonsingular principal submatrix in M_j, if those matrices have bounded
rank-width. This generalizes three theorems on well-quasi-ordering of graphs or
matroids admitting good tree-like decompositions; (1) Robertson and Seymour's
theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's
theorem for matroids representable over a fixed finite field having bounded
branch-width, and (3) Oum's theorem for graphs of bounded rank-width with
respect to pivot-minors.Comment: 43 page
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems
to derive tangle-type duality theorems for width-parameters in graphs and
matroids. We further derive a duality theorem for the existence of clusters in
large data sets.
Our applications to graphs include new, tangle-type, duality theorems for
tree-width, path-width, and tree-decompositions of small adhesion. Conversely,
we show that carving width is dual to edge-tangles. For matroids we obtain a
duality theorem for tree-width.
Our results can be used to derive short proofs of all the classical duality
theorems for width parameters in graph minor theory, such as path-width,
tree-width, branch-width and rank-width.Comment: arXiv admin note: text overlap with arXiv:1406.379
Tangle-tree duality in abstract separation systems
We prove a general width duality theorem for combinatorial structures with
well-defined notions of cohesion and separation. These might be graphs and
matroids, but can be much more general or quite different. The theorem asserts
a duality between the existence of high cohesiveness somewhere local and a
global overall tree structure.
We describe cohesive substructures in a unified way in the format of tangles:
as orientations of low-order separations satisfying certain consistency axioms.
These axioms can be expressed without reference to the underlying structure,
such as a graph or matroid, but just in terms of the poset of the separations
themselves. This makes it possible to identify tangles, and apply our
tangle-tree duality theorem, in very diverse settings.
Our result implies all the classical duality theorems for width parameters in
graph minor theory, such as path-width, tree-width, branch-width or rank-width.
It yields new, tangle-type, duality theorems for tree-width and path-width. It
implies the existence of width parameters dual to cohesive substructures such
as -blocks, edge-tangles, or given subsets of tangles, for which no width
duality theorems were previously known.
Abstract separation systems can be found also in structures quite unlike
graphs and matroids. For example, our theorem can be applied to image analysis
by capturing the regions of an image as tangles of separations defined as
natural partitions of its set of pixels. It can be applied in big data contexts
by capturing clusters as tangles. It can be applied in the social sciences,
e.g. by capturing as tangles the few typical mindsets of individuals found by a
survey. It could also be applied in pure mathematics, e.g. to separations of
compact manifolds.Comment: We have expanded Section 2 on terminology for better readability,
adding explanatory text, examples, and figures. This paper replaces the first
half of our earlier paper arXiv:1406.379
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