13,311 research outputs found
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
Nested hierarchies in planar graphs
We construct a partial order relation which acts on the set of 3-cliques of a
maximal planar graph G and defines a unique hierarchy. We demonstrate that G is
the union of a set of special subgraphs, named `bubbles', that are themselves
maximal planar graphs. The graph G is retrieved by connecting these bubbles in
a tree structure where neighboring bubbles are joined together by a 3-clique.
Bubbles naturally provide the subdivision of G into communities and the tree
structure defines the hierarchical relations between these communities
Macroscopic network circulation for planar graphs
The analysis of networks, aimed at suitably defined functionality, often
focuses on partitions into subnetworks that capture desired features. Chief
among the relevant concepts is a 2-partition, that underlies the classical
Cheeger inequality, and highlights a constriction (bottleneck) that limits
accessibility between the respective parts of the network. In a similar spirit,
the purpose of the present work is to introduce a new concept of maximal global
circulation and to explore 3-partitions that expose this type of macroscopic
feature of networks. Herein, graph circulation is motivated by transportation
networks and probabilistic flows (Markov chains) on graphs. Our goal is to
quantify the large-scale imbalance of network flows and delineate key parts
that mediate such global features. While we introduce and propose these notions
in a general setting, in this paper, we only work out the case of planar
graphs. We explain that a scalar potential can be identified to encapsulate the
concept of circulation, quite similarly as in the case of the curl of planar
vector fields. Beyond planar graphs, in the general case, the problem to
determine global circulation remains at present a combinatorial problem
Mixed Linear Layouts of Planar Graphs
A -stack (respectively, -queue) layout of a graph consists of a total
order of the vertices, and a partition of the edges into sets of
non-crossing (non-nested) edges with respect to the vertex ordering. In 1992,
Heath and Rosenberg conjectured that every planar graph admits a mixed
-stack -queue layout in which every edge is assigned to a stack or to a
queue that use a common vertex ordering.
We disprove this conjecture by providing a planar graph that does not have
such a mixed layout. In addition, we study mixed layouts of graph subdivisions,
and show that every planar graph has a mixed subdivision with one division
vertex per edge.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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