118,098 research outputs found
Graph connectivity and universal rigidity of bar frameworks
Let be a graph on nodes. In this note, we prove that if is
-vertex connected, , then there exists a
configuration in general position in such that the bar framework
is universally rigid. The proof is constructive and is based on a
theorem by Lovasz et al concerning orthogonal representations and connectivity
of graphs [12,13].Comment: updated versio
Rectilinear Planarity of Partial 2-Trees
A graph is rectilinear planar if it admits a planar orthogonal drawing
without bends. While testing rectilinear planarity is NP-hard in general (Garg
and Tamassia, 2001), it is a long-standing open problem to establish a tight
upper bound on its complexity for partial 2-trees, i.e., graphs whose
biconnected components are series-parallel. We describe a new O(n^2)-time
algorithm to test rectilinear planarity of partial 2-trees, which improves over
the current best bound of O(n^3 \log n) (Di Giacomo et al., 2022). Moreover,
for partial 2-trees where no two parallel-components in a biconnected component
share a pole, we are able to achieve optimal O(n)-time complexity. Our
algorithms are based on an extensive study and a deeper understanding of the
notion of orthogonal spirality, introduced several years ago (Di Battista et
al, 1998) to describe how much an orthogonal drawing of a subgraph is rolled-up
in an orthogonal drawing of the graph.Comment: arXiv admin note: substantial text overlap with arXiv:2110.00548
Appears in the Proceedings of the 30th International Symposium on Graph
Drawing and Network Visualization (GD 2022
Orthogonal Graph Drawing with Inflexible Edges
We consider the problem of creating plane orthogonal drawings of 4-planar
graphs (planar graphs with maximum degree 4) with constraints on the number of
bends per edge. More precisely, we have a flexibility function assigning to
each edge a natural number , its flexibility. The problem
FlexDraw asks whether there exists an orthogonal drawing such that each edge
has at most bends. It is known that FlexDraw is NP-hard
if for every edge . On the other hand, FlexDraw can
be solved efficiently if and is trivial if
for every edge .
To close the gap between the NP-hardness for and the
efficient algorithm for , we investigate the
computational complexity of FlexDraw in case only few edges are inflexible
(i.e., have flexibility~). We show that for any FlexDraw
is NP-complete for instances with inflexible edges with
pairwise distance (including the case where they
induce a matching). On the other hand, we give an FPT-algorithm with running
time , where
is the time necessary to compute a maximum flow in a planar flow network with
multiple sources and sinks, and is the number of inflexible edges having at
least one endpoint of degree 4.Comment: 23 pages, 5 figure
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