4,497 research outputs found
Spanning Properties of Theta-Theta Graphs
We study the spanning properties of Theta-Theta graphs. Similar in spirit
with the Yao-Yao graphs, Theta-Theta graphs partition the space around each
vertex into a set of k cones, for some fixed integer k > 1, and select at most
one edge per cone. The difference is in the way edges are selected. Yao-Yao
graphs select an edge of minimum length, whereas Theta-Theta graphs select an
edge of minimum orthogonal projection onto the cone bisector. It has been
established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio
11.67, for k' >= 6. In this paper we establish a first spanning ratio of
for Theta-Theta graphs, for the same values of . We also extend the class of
Theta-Theta spanners with parameter 6k', and establish a spanning ratio of
for k' >= 5. We surmise that these stronger results are mainly due to a
tighter analysis in this paper, rather than Theta-Theta being superior to
Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta
graphs decreases to 4.64 as k' increases to 8. These are the first results on
the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table
The Price of Order
We present tight bounds on the spanning ratio of a large family of ordered
-graphs. A -graph partitions the plane around each vertex into
disjoint cones, each having aperture . An ordered
-graph is constructed by inserting the vertices one by one and
connecting each vertex to the closest previously-inserted vertex in each cone.
We show that for any integer , ordered -graphs with
cones have a tight spanning ratio of . We also show that for any integer , ordered
-graphs with cones have a tight spanning ratio of . We provide lower bounds for ordered -graphs with and cones. For ordered -graphs with and
cones these lower bounds are strictly greater than the worst case spanning
ratios of their unordered counterparts. These are the first results showing
that ordered -graphs have worse spanning ratios than unordered
-graphs. Finally, we show that, unlike their unordered counterparts,
the ordered -graphs with 4, 5, and 6 cones are not spanners
On a family of strong geometric spanners that admit local routing strategies
We introduce a family of directed geometric graphs, denoted \paz, that
depend on two parameters and . For and , the \paz graph is a strong
-spanner, with . The out-degree of a node
in the \paz graph is at most . Moreover, we show that routing can be
achieved locally on \paz. Next, we show that all strong -spanners are also
-spanners of the unit disk graph. Simulations for various values of the
parameters and indicate that for random point sets, the
spanning ratio of \paz is better than the proven theoretical bounds
A signature invariant for knotted Klein graphs
We define some signature invariants for a class of knotted trivalent graphs
using branched covers. We relate them to classical signatures of knots and
links. Finally, we explain how to compute these invariants through the example
of Kinoshita's knotted theta graph.Comment: 23 pages, many figures. Comments welcome ! Historical inaccuracy
fixe
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