2,003 research outputs found
On quadratic orbital networks
These are some informal remarks on quadratic orbital networks over finite
fields. We discuss connectivity, Euler characteristic, number of cliques,
planarity, diameter and inductive dimension. We find a non-trivial disconnected
graph for d=3. We prove that for d=1 generators, the Euler characteristic is
always non-negative and for d=2 and large enough p the Euler characteristic is
negative. While for d=1, all networks are planar, we suspect that for d larger
or equal to 2 and large enough prime p, all networks are non-planar. As a
consequence on bounds for the number of complete sub graphs of a fixed
dimension, the inductive dimension of all these networks goes 1 as p goes to
infinity.Comment: 13 figures 15 page
A Note on the Practicality of Maximal Planar Subgraph Algorithms
Given a graph , the NP-hard Maximum Planar Subgraph problem (MPS) asks for
a planar subgraph of with the maximum number of edges. There are several
heuristic, approximative, and exact algorithms to tackle the problem, but---to
the best of our knowledge---they have never been compared competitively in
practice. We report on an exploratory study on the relative merits of the
diverse approaches, focusing on practical runtime, solution quality, and
implementation complexity. Surprisingly, a seemingly only theoretically strong
approximation forms the building block of the strongest choice.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
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