1,248 research outputs found
Relating Graph Thickness to Planar Layers and Bend Complexity
The thickness of a graph with vertices is the minimum number of
planar subgraphs of whose union is . A polyline drawing of in
is a drawing of , where each vertex is mapped to a
point and each edge is mapped to a polygonal chain. Bend and layer complexities
are two important aesthetics of such a drawing. The bend complexity of
is the maximum number of bends per edge in , and the layer complexity
of is the minimum integer such that the set of polygonal chains in
can be partitioned into disjoint sets, where each set corresponds
to a planar polyline drawing. Let be a graph of thickness . By
F\'{a}ry's theorem, if , then can be drawn on a single layer with bend
complexity . A few extensions to higher thickness are known, e.g., if
(resp., ), then can be drawn on layers with bend complexity 2
(resp., ). However, allowing a higher number of layers may reduce the
bend complexity, e.g., complete graphs require layers to be drawn
using 0 bends per edge.
In this paper we present an elegant extension of F\'{a}ry's theorem to draw
graphs of thickness . We first prove that thickness- graphs can be
drawn on layers with bends per edge. We then develop another
technique to draw thickness- graphs on layers with bend complexity,
i.e., , where . Previously, the bend complexity was not known to be sublinear for
. Finally, we show that graphs with linear arboricity can be drawn on
layers with bend complexity .Comment: A preliminary version appeared at the 43rd International Colloquium
on Automata, Languages and Programming (ICALP 2016
Single failure resiliency in greedy routing
Using greedy routing, network nodes forward packets towards neighbors which are closer to their destination. This approach makes greedy routers significantly more memory-efficient than traditional IP-routers using longest-prefix matching. Greedy embeddings map network nodes to coordinates, such that greedy routing always leads to the destination. Prior works showed that using a spanning tree of the network topology, greedy embeddings can be found in different metric spaces for any graph. However, a single link/node failure might affect the greedy embedding and causes the packets to reach a dead end. In order to cope with network failures, existing greedy methods require large resources and cause significant loss in the quality of the routing (stretch loss). We propose efficient recovery techniques which require very limited resources with minor effect on the stretch. As the proposed techniques are protection, the switch-over takes place very fast. Low overhead, simplicity and scalability of the methods make them suitable for large-scale networks. The proposed schemes are validated on large topologies with properties similar to the Internet. The performances of the schemes are compared with an existing alternative referred as gravity pressure routing
On a Tree and a Path with no Geometric Simultaneous Embedding
Two graphs and admit a geometric simultaneous
embedding if there exists a set of points P and a bijection M: P -> V that
induce planar straight-line embeddings both for and for . While it
is known that two caterpillars always admit a geometric simultaneous embedding
and that two trees not always admit one, the question about a tree and a path
is still open and is often regarded as the most prominent open problem in this
area. We answer this question in the negative by providing a counterexample.
Additionally, since the counterexample uses disjoint edge sets for the two
graphs, we also negatively answer another open question, that is, whether it is
possible to simultaneously embed two edge-disjoint trees. As a final result, we
study the same problem when some constraints on the tree are imposed. Namely,
we show that a tree of depth 2 and a path always admit a geometric simultaneous
embedding. In fact, such a strong constraint is not so far from closing the gap
with the instances not admitting any solution, as the tree used in our
counterexample has depth 4.Comment: 42 pages, 33 figure
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