25,215 research outputs found
Separation dimension of bounded degree graphs
The 'separation dimension' of a graph is the smallest natural number
for which the vertices of can be embedded in such that any
pair of disjoint edges in can be separated by a hyperplane normal to one of
the axes. Equivalently, it is the smallest possible cardinality of a family
of total orders of the vertices of such that for any two
disjoint edges of , there exists at least one total order in
in which all the vertices in one edge precede those in the other. In general,
the maximum separation dimension of a graph on vertices is . In this article, we focus on bounded degree graphs and show that the
separation dimension of a graph with maximum degree is at most
. We also demonstrate that the above bound is nearly
tight by showing that, for every , almost all -regular graphs have
separation dimension at least .Comment: One result proved in this paper is also present in arXiv:1212.675
Layout of Graphs with Bounded Tree-Width
A \emph{queue layout} of a graph consists of a total order of the vertices,
and a partition of the edges into \emph{queues}, such that no two edges in the
same queue are nested. The minimum number of queues in a queue layout of a
graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line
grid) drawing} of a graph represents the vertices by points in
and the edges by non-crossing line-segments. This paper contributes three main
results:
(1) It is proved that the minimum volume of a certain type of
three-dimensional drawing of a graph is closely related to the queue-number
of . In particular, if is an -vertex member of a proper minor-closed
family of graphs (such as a planar graph), then has a drawing if and only if has O(1) queue-number.
(2) It is proved that queue-number is bounded by tree-width, thus resolving
an open problem due to Ganley and Heath (2001), and disproving a conjecture of
Pemmaraju (1992). This result provides renewed hope for the positive resolution
of a number of open problems in the theory of queue layouts.
(3) It is proved that graphs of bounded tree-width have three-dimensional
drawings with O(n) volume. This is the most general family of graphs known to
admit three-dimensional drawings with O(n) volume.
The proofs depend upon our results regarding \emph{track layouts} and
\emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October
2002. This paper incorporates the following conference papers: (1) Dujmovic',
Morin & Wood. Path-width and three-dimensional straight-line grid drawings of
graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts,
tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS
2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of
-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200
Equistarable graphs and counterexamples to three conjectures on equistable graphs
Equistable graphs are graphs admitting positive weights on vertices such that
a subset of vertices is a maximal stable set if and only if it is of total
weight . In , Mahadev et al.~introduced a subclass of equistable
graphs, called strongly equistable graphs, as graphs such that for every and every non-empty subset of vertices that is not a maximal stable set,
there exist positive vertex weights such that every maximal stable set is of
total weight and the total weight of does not equal . Mahadev et al.
conjectured that every equistable graph is strongly equistable. General
partition graphs are the intersection graphs of set systems over a finite
ground set such that every maximal stable set of the graph corresponds to a
partition of . In , Orlin proved that every general partition graph is
equistable, and conjectured that the converse holds as well.
Orlin's conjecture, if true, would imply the conjecture due to Mahadev,
Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's
conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed
by Miklavi\v{c} and Milani\v{c} in , and states that every equistable
graph has a clique intersecting all maximal stable sets. The above conjectures
have been verified for several graph classes. We introduce the notion of
equistarable graphs and based on it construct counterexamples to all three
conjectures within the class of complements of line graphs of triangle-free
graphs
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
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