12,320 research outputs found
Reconstruction of Planar Domains from Partial Integral Measurements
We consider the problem of reconstruction of planar domains from their
moments. Specifically, we consider domains with boundary which can be
represented by a union of a finite number of pieces whose graphs are solutions
of a linear differential equation with polynomial coefficients. This includes
domains with piecewise-algebraic and, in particular, piecewise-polynomial
boundaries. Our approach is based on one-dimensional reconstruction method of
[Bat]* and a kind of "separation of variables" which reduces the planar problem
to two one-dimensional problems, one of them parametric. Several explicit
examples of reconstruction are given.
Another main topic of the paper concerns "invisible sets" for various types
of incomplete moment measurements. We suggest a certain point of view which
stresses remarkable similarity between several apparently unrelated problems.
In particular, we discuss zero quadrature domains (invisible for harmonic
polynomials), invisibility for powers of a given polynomial, and invisibility
for complex moments (Wermer's theorem and further developments). The common
property we would like to stress is a "rigidity" and symmetry of the invisible
objects.
* D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse
Problems 25 (2009) 105001Comment: Proceedings of Complex Analysis and Dynamical Systems V, 201
An algebraic formulation of the graph reconstruction conjecture
The graph reconstruction conjecture asserts that every finite simple graph on
at least three vertices can be reconstructed up to isomorphism from its deck -
the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important
tool in graph reconstruction. Roughly speaking, given the deck of a graph
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on
vertices, and let be the number of distinct decks that can be
constructed from these graphs. Then the difference measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for -vertex graphs if and only if
.
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if is a matrix of covering numbers of graphs by sequences of
graphs, then . In particular, all
-vertex graphs are reconstructible if one such matrix has rank . To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix of covering numbers
satisfies .Comment: 12 pages, 2 figure
Graphs that are not pairwise compatible: A new proof technique (extended abstract)
A graph G = (V,E) is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dminand dmax, dmin≤ dmax, such that each node u∈V is uniquely associated to a leaf of T and there is an edge (u, v) ∈ E if and only if dmin≤ dT(u, v) ≤ dmax, where dT(u, v) is the sum of the weights of the edges on the unique path PT(u, v) from u to v in T. Understanding which graph classes lie inside and which ones outside the PCG class is an important issue. Despite numerous efforts, a complete characterization of the PCG class is not known yet. In this paper we propose a new proof technique that allows us to show that some interesting classes of graphs have empty intersection with PCG. We demonstrate our technique by showing many graph classes that do not lie in PCG. As a side effect, we show a not pairwise compatibility planar graph with 8 nodes (i.e. C28), so improving the previously known result concerning the smallest planar graph known not to be PCG
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