6,770 research outputs found
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Poisson Hail on a Hot Ground
We consider a queue where the server is the Euclidean space, and the
customers are random closed sets (RACS) of the Euclidean space. These RACS
arrive according to a Poisson rain and each of them has a random service time
(in the case of hail falling on the Euclidean plane, this is the height of the
hailstone, whereas the RACS is its footprint). The Euclidean space serves
customers at speed 1. The service discipline is a hard exclusion rule: no two
intersecting RACS can be served simultaneously and service is in the First In
First Out order: only the hailstones in contact with the ground melt at speed
1, whereas the other ones are queued; a tagged RACS waits until all RACS
arrived before it and intersecting it have fully melted before starting its own
melting. We give the evolution equations for this queue. We prove that it is
stable for a sufficiently small arrival intensity, provided the typical
diameter of the RACS and the typical service time have finite exponential
moments. We also discuss the percolation properties of the stationary regime of
the RACS in the queue.Comment: 26 page
Multi-triangulations as complexes of star polygons
Maximal -crossing-free graphs on a planar point set in convex
position, that is, -triangulations, have received attention in recent
literature, with motivation coming from several interpretations of them.
We introduce a new way of looking at -triangulations, namely as complexes
of star polygons. With this tool we give new, direct, proofs of the fundamental
properties of -triangulations, as well as some new results. This
interpretation also opens-up new avenues of research, that we briefly explore
in the last section.Comment: 40 pages, 24 figures; added references, update Section
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