12,270 research outputs found
Characterizing Block Graphs in Terms of their Vertex-Induced Partitions
Given a finite connected simple graph with vertex set and edge
set , we will show that
the (necessarily unique) smallest block graph with vertex set whose
edge set contains is uniquely determined by the -indexed family of the various partitions
of the set into the set of connected components of the
graph ,
the edge set of this block graph coincides with set of all -subsets
of for which and are, for all , contained
in the same connected component of ,
and an arbitrary -indexed family of
partitions of the set is of the form for some
connected simple graph with vertex set as above if and only if,
for any two distinct elements , the union of the set in
that contains and the set in that contains coincides with
the set , and holds for all .
As well as being of inherent interest to the theory of block graphs, these
facts are also useful in the analysis of compatible decompositions and block
realizations of finite metric spaces
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map
We introduce a new algorithm to construct travel time distances between a
point in the interior of a Riemannian manifold and points on the boundary of
the manifold, and describe a numerical implementation of the algorithm. It is
known that the travel time distances for all interior points determine the
Riemannian manifold in a stable manner. We do not assume that there are sources
or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map,
or its restriction, as our data. Our algorithm is a variant of the Boundary
Control method, and to our knowledge, this is the first numerical
implementation of the method in a geometric setting
When the path is never shortest: a reality check on shortest path biocomputation
Shortest path problems are a touchstone for evaluating the computing
performance and functional range of novel computing substrates. Much has been
published in recent years regarding the use of biocomputers to solve minimal
path problems such as route optimisation and labyrinth navigation, but their
outputs are typically difficult to reproduce and somewhat abstract in nature,
suggesting that both experimental design and analysis in the field require
standardising. This chapter details laboratory experimental data which probe
the path finding process in two single-celled protistic model organisms,
Physarum polycephalum and Paramecium caudatum, comprising a shortest path
problem and labyrinth navigation, respectively. The results presented
illustrate several of the key difficulties that are encountered in categorising
biological behaviours in the language of computing, including biological
variability, non-halting operations and adverse reactions to experimental
stimuli. It is concluded that neither organism examined are able to efficiently
or reproducibly solve shortest path problems in the specific experimental
conditions that were tested. Data presented are contextualised with biological
theory and design principles for maximising the usefulness of experimental
biocomputer prototypes.Comment: To appear in: Adamatzky, A (Ed.) Shortest path solvers. From software
to wetware. Springer, 201
Decision problems for 3-manifolds and their fundamental groups
We survey the status of some decision problems for 3-manifolds and their
fundamental groups. This includes the classical decision problems for finitely
presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and
also the Homeomorphism Problem for 3-manifolds and the Membership Problem for
3-manifold groups.Comment: 31 pages, final versio
To , or not to : Recent developments and comparisons of regularization schemes
We give an introduction to several regularization schemes that deal with
ultraviolet and infrared singularities appearing in higher-order computations
in quantum field theories. Comparing the computation of simple quantities in
the various schemes, we point out similarities and differences between them.Comment: 61 pages, 12 figures; version sent to EPJC, references update
Rank of divisors on hyperelliptic curves and graphs under specialization
Let be a hyperelliptic vertex-weighted graph of genus . We give a characterization of for which there exists a smooth
projective curve of genus over a complete discrete valuation field with
reduction graph such that the ranks of any divisors are preserved
under specialization. We explain, for a given vertex-weighted graph in general, how the existence of such relates the Riemann--Roch
formulae for and , and also how the existence of such is
related to a conjecture of Caporaso.Comment: 34 pages. The proof of Theorem 1.13 has been significantly simplifie
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