524,414 research outputs found
-Stars or On Extending a Drawing of a Connected Subgraph
We consider the problem of extending the drawing of a subgraph of a given
plane graph to a drawing of the entire graph using straight-line and polyline
edges. We define the notion of star complexity of a polygon and show that a
drawing of an induced connected subgraph can be extended with at
most bends per edge, where is the
largest star complexity of a face of and is the size of the
largest face of . This result significantly improves the previously known
upper bound of [5] for the case where is connected. We also show
that our bound is worst case optimal up to a small additive constant.
Additionally, we provide an indication of complexity of the problem of testing
whether a star-shaped inner face can be extended to a straight-line drawing of
the graph; this is in contrast to the fact that the same problem is solvable in
linear time for the case of star-shaped outer face [9] and convex inner face
[13].Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Aspherical gravitational monopoles
We show how to construct non-spherically-symmetric extended bodies of uniform
density behaving exactly as pointlike masses. These ``gravitational monopoles''
have the following equivalent properties: (i) they generate, outside them, a
spherically-symmetric gravitational potential ; (ii) their
interaction energy with an external gravitational potential is ; and (iii) all their multipole moments (of order ) with
respect to their center of mass vanish identically. The method applies for
any number of space dimensions. The free parameters entering the construction
are: (1) an arbitrary surface bounding a connected open subset
of ; (2) the arbitrary choice of the center of mass within
; and (3) the total volume of the body. An extension of the method
allows one to construct homogeneous bodies which are gravitationally equivalent
(in the sense of having exactly the same multipole moments) to any given body.Comment: 55 pages, Latex , submitted to Nucl.Phys.
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
In the companion paper [Linear rank-width of distance-hereditary graphs I. A
polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a
characterization of the linear rank-width of distance-hereditary graphs, from
which we derived an algorithm to compute it in polynomial time. In this paper,
we investigate structural properties of distance-hereditary graphs based on
this characterization.
First, we prove that for a fixed tree , every distance-hereditary graph of
sufficiently large linear rank-width contains a vertex-minor isomorphic to .
We extend this property to bigger graph classes, namely, classes of graphs
whose prime induced subgraphs have bounded linear rank-width. Here, prime
graphs are graphs containing no splits. We conjecture that for every tree ,
every graph of sufficiently large linear rank-width contains a vertex-minor
isomorphic to . Our result implies that it is sufficient to prove this
conjecture for prime graphs.
For a class of graphs closed under taking vertex-minors, a graph
is called a vertex-minor obstruction for if but all of
its proper vertex-minors are contained in . Secondly, we provide, for
each , a set of distance-hereditary graphs that contains all
distance-hereditary vertex-minor obstructions for graphs of linear rank-width
at most . Also, we give a simpler way to obtain the known vertex-minor
obstructions for graphs of linear rank-width at most .Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1
Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity
Investigations into and around a 30-year old conjecture of Gregory Margulis
and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.Comment: 50 page
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