662 research outputs found
Permanents, Pfaffian orientations, and even directed circuits
Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in
such a way that the permanent of A equals the determinant of the modified
matrix? When does a real square matrix have the property that every real matrix
with the same sign pattern (that is, the corresponding entries either have the
same sign or are both zero) is nonsingular? When is a hypergraph with n
vertices and n hyperedges minimally nonbipartite? When does a bipartite graph
have a "Pfaffian orientation"? Given a digraph, does it have no directed
circuit of even length? Given a digraph, does it have a subdivision with no
even directed circuit?
It is known that all of the above problems are equivalent. We prove a
structural characterization of the feasible instances, which implies a
polynomial-time algorithm to solve all of the above problems. The structural
characterization says, roughly speaking, that a bipartite graph has a Pfaffian
orientation if and only if it can be obtained by piecing together (in a
specified way) planar bipartite graphs and one sporadic nonplanar bipartite
graph.Comment: 47 pages, published versio
Superpatterns and Universal Point Sets
An old open problem in graph drawing asks for the size of a universal point
set, a set of points that can be used as vertices for straight-line drawings of
all n-vertex planar graphs. We connect this problem to the theory of
permutation patterns, where another open problem concerns the size of
superpatterns, permutations that contain all patterns of a given size. We
generalize superpatterns to classes of permutations determined by forbidden
patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the
213-avoiding permutations, half the size of known superpatterns for
unconstrained permutations. We use our superpatterns to construct universal
point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16
factor. We prove that every proper subclass of the 213-avoiding permutations
has superpatterns of size O(n log^O(1) n), which we use to prove that the
planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA
On the Critical Behavior of D1-brane Theories
We study renormalization-group flow patterns in theories arising on D1-branes
in various supersymmetry-breaking backgrounds. We argue that the theory of N
D1-branes transverse to an orbifold space can be fine-tuned to flow to the
corresponding orbifold conformal field theory in the infrared, for particular
values of the couplings and theta angles which we determine using the discrete
symmetries of the model. By calculating various nonplanar contributions to the
scalar potential in the worldvolume theory, we show that fine-tuning is in fact
required at finite N, as would be generically expected. We further comment on
the presence of singular conformal field theories (such as those whose target
space includes a ``throat'' described by an exactly solvable CFT) in the
non-supersymmetric context. Throughout the analysis two applications are
considered: to gauge theory/gravity duality and to linear sigma model
techniques for studying worldsheet string theory.Comment: 23 pages in harvmac big, 8 figure
Finding k partially disjoint paths in a directed planar graph
The {\it partially disjoint paths problem} is: {\it given:} a directed graph,
vertices , and a set of pairs from
, {\it find:} for each a directed path
such that if then and are disjoint.
We show that for fixed , this problem is solvable in polynomial time if
the directed graph is planar. More generally, the problem is solvable in
polynomial time for directed graphs embedded on a fixed compact surface.
Moreover, one may specify for each edge a subset of
prescribing which of the paths are allowed to traverse this edge
On almost-planar graphs
A nonplanar graph G is called almost-planar if for every edge e of G, at least one of G\e and G/e is planar. In 1990, Gubser characterized 3-connected almost-planar graphs in his dissertation. However, his proof is so long that only a small portion of it was published. The main purpose of this paper is to provide a short proof of this result. We also discuss the structure of almost-planar graphs that are not 3-connected
Graph Theoretic Properties of the Zero-Divisor Graph of a Ring
Let R be a commutative ring with 1 ≠0, and let Z(R) denote the set of zero-divisors of R. One can associate with R a graph Γ(R) whose vertices are the nonzero zero-divisors of R. Two distinct vertices x and y are joined by an edge if and only if xy = 0 in R. Γ® is often called the zero-divisor graph of R. We determine which finite commutative rings yield a planar zero-divisor graph. Next, we investigate the structure of Γ(R) when Γ(R) is an infinite planar graph. Next, it is possible to extend the definition of the zero-divisor graph to a commutative semigroup. We investigate the problem of extending the definition of the zero-divisor graph to a noncommutative semigroup, and attempt to generalize results from the commutative ring setting. Finally, we investigate the structure of Γ(k1 × ∙ ∙ ∙ × kn) where each ki is a finite field. The appendices give planar embeddings of many families of zero-divisor graphs
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