30,606 research outputs found
On fractional realizations of graph degree sequences
We introduce fractional realizations of a graph degree sequence and a closely
associated convex polytope. Simple graph realizations correspond to a subset of
the vertices of this polytope. We describe properties of the polytope vertices
and characterize degree sequences for which each polytope vertex corresponds to
a simple graph realization. These include the degree sequences of pseudo-split
graphs, and we characterize their realizations both in terms of forbidden
subgraphs and graph structure.Comment: 18 pages, 4 figure
Upward-closed hereditary families in the dominance order
The majorization relation orders the degree sequences of simple graphs into
posets called dominance orders. As shown by Hammer et al. and Merris, the
degree sequences of threshold and split graphs form upward-closed sets within
the dominance orders they belong to, i.e., any degree sequence majorizing a
split or threshold sequence must itself be split or threshold, respectively.
Motivated by the fact that threshold graphs and split graphs have
characterizations in terms of forbidden induced subgraphs, we define a class
of graphs to be dominance monotone if whenever no realization of
contains an element as an induced subgraph, and majorizes
, then no realization of induces an element of . We present
conditions necessary for a set of graphs to be dominance monotone, and we
identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
Finding weakly reversible realizations of chemical reaction networks using optimization
An algorithm is given in this paper for the computation of dynamically
equivalent weakly reversible realizations with the maximal number of reactions,
for chemical reaction networks (CRNs) with mass action kinetics. The original
problem statement can be traced back at least 30 years ago. The algorithm uses
standard linear and mixed integer linear programming, and it is based on
elementary graph theory and important former results on the dense realizations
of CRNs. The proposed method is also capable of determining if no dynamically
equivalent weakly reversible structure exists for a given reaction network with
a previously fixed complex set.Comment: 18 pages, 9 figure
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
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