18,260 research outputs found
Edge Partitions of Optimal -plane and -plane Graphs
A topological graph is a graph drawn in the plane. A topological graph is
-plane, , if each edge is crossed at most times. We study the
problem of partitioning the edges of a -plane graph such that each partite
set forms a graph with a simpler structure. While this problem has been studied
for , we focus on optimal -plane and -plane graphs, which are
-plane and -plane graphs with maximum density. We prove the following
results. (i) It is not possible to partition the edges of a simple optimal
-plane graph into a -plane graph and a forest, while (ii) an edge
partition formed by a -plane graph and two plane forests always exists and
can be computed in linear time. (iii) We describe efficient algorithms to
partition the edges of a simple optimal -plane graph into a -plane graph
and a plane graph with maximum vertex degree , or with maximum vertex
degree if the optimal -plane graph is such that its crossing-free edges
form a graph with no separating triangles. (iv) We exhibit an infinite family
of simple optimal -plane graphs such that in any edge partition composed of
a -plane graph and a plane graph, the plane graph has maximum vertex degree
at least and the -plane graph has maximum vertex degree at least .
(v) We show that every optimal -plane graph whose crossing-free edges form a
biconnected graph can be decomposed, in linear time, into a -plane graph and
two plane forests
Separators in Region Intersection Graphs
For undirected graphs G=(V,E) and G_0=(V_0,E_0), say that G is a region intersection graph over G_0 if there is a family of connected subsets {R_u subseteq V_0 : u in V} of G_0 such that {u,v} in E iff R_u cap R_v neq emptyset.
We show if G_0 excludes the complete graph K_h as a minor for some h geq 1, then every region intersection graph G over G_0 with m edges has a balanced separator with at most c_h sqrt{m} nodes, where c_h is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning.
A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O(sqrt{m}). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), and improves over the O(sqrt{m} log m) bound of Matousek (2013)
Tree-width of hypergraphs and surface duality
In Graph Minors III, Robertson and Seymour write: "It seems that the
tree-width of a planar graph and the tree-width of its geometric dual are
approximately equal - indeed, we have convinced ourselves that they differ by
at most one". They never gave a proof of this. In this paper, we prove a
generalisation of this statement to embedding of hypergraphs on general
surfaces, and we prove that our bound is tight
On the Pauli graphs of N-qudits
A comprehensive graph theoretical and finite geometrical study of the
commutation relations between the generalized Pauli operators of N-qudits is
performed in which vertices/points correspond to the operators and edges/lines
join commuting pairs of them. As per two-qubits, all basic properties and
partitionings of the corresponding Pauli graph are embodied in the geometry of
the generalized quadrangle of order two. Here, one identifies the operators
with the points of the quadrangle and groups of maximally commuting subsets of
the operators with the lines of the quadrangle. The three basic partitionings
are (a) a pencil of lines and a cube, (b) a Mermin's array and a bipartite-part
and (c) a maximum independent set and the Petersen graph. These factorizations
stem naturally from the existence of three distinct geometric hyperplanes of
the quadrangle, namely a set of points collinear with a given point, a grid and
an ovoid, which answer to three distinguished subsets of the Pauli graph,
namely a set of six operators commuting with a given one, a Mermin's square,
and set of five mutually non-commuting operators, respectively. The generalized
Pauli graph for multiple qubits is found to follow from symplectic polar spaces
of order two, where maximal totally isotropic subspaces stand for maximal
subsets of mutually commuting operators. The substructure of the (strongly
regular) N-qubit Pauli graph is shown to be pseudo-geometric, i. e., isomorphic
to a graph of a partial geometry. Finally, the (not strongly regular) Pauli
graph of a two-qutrit system is introduced; here it turns out more convenient
to deal with its dual in order to see all the parallels with the two-qubit case
and its surmised relation with the generalized quadrangle Q(4, 3), the dual
ofW(3).Comment: 17 pages. Expanded section on two-qutrits, Quantum Information and
Computation (2007) accept\'
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