11,836 research outputs found
Unit Grid Intersection Graphs: Recognition and Properties
It has been known since 1991 that the problem of recognizing grid
intersection graphs is NP-complete. Here we use a modified argument of the
above result to show that even if we restrict to the class of unit grid
intersection graphs (UGIGs), the recognition remains hard, as well as for all
graph classes contained inbetween. The result holds even when considering only
graphs with arbitrarily large girth. Furthermore, we ask the question of
representing UGIGs on grids of minimal size. We show that the UGIGs that can be
represented in a square of side length 1+epsilon, for a positive epsilon no
greater than 1, are exactly the orthogonal ray graphs, and that there exist
families of trees that need an arbitrarily large grid
Interval-Like Graphs and Digraphs
We unify several seemingly different graph and digraph classes under one umbrella. These classes are all, broadly speaking, different generalizations of interval graphs, and include, in addition to interval graphs, adjusted interval digraphs, threshold graphs, complements of threshold tolerance graphs (known as `co-TT\u27 graphs), bipartite interval containment graphs, bipartite co-circular arc graphs, and two-directional orthogonal ray graphs. (The last three classes coincide, but have been investigated in different contexts.) This common view is made possible by introducing reflexive relationships (loops) into the analysis. We also show that all the above classes are united by a common ordering characterization, the existence of a min ordering. We propose a common generalization of all these graph and digraph classes, namely signed-interval digraphs, and show that they are precisely the digraphs that are characterized by the existence of a min ordering. We also offer an alternative geometric characterization of these digraphs. For most of the above graph and digraph classes, we show that they are exactly those signed-interval digraphs that satisfy a suitable natural restriction on the digraph, like having a loop on every vertex, or having a symmetric edge-set, or being bipartite. For instance, co-TT graphs are precisely those signed-interval digraphs that have each edge symmetric. We also offer some discussion of future work on recognition algorithms and characterizations
Radiographic analysis of sedimentary structures and depositional histories in Apollo 15 cores
Radiographs of the Apollo 15 deepdrill drive tubes were analyzed on an SDS electronic enhancer to determine sedimentary structures in the core samples. The data obtained were compared with all other Apollo mission radiographs and used to make inferences on the character of sedimentary depositional processes on the lunar surface
Empty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in
diagonally opposite corners of empty axis-aligned rectangles. The maximum
number of edges of such a graph on points is shown to be 1/4 n^2 +n -2.
This number also has other interpretations:
* It is the maximum number of edges of a graph of dimension
\bbetween{3}{4}, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}.
* It is the number of 1-faces in a special Scarf complex.
The last of these interpretations allows to deduce the maximum number of
empty axis-aligned rectangles spanned by 4-element subsets of a set of
points. Moreover, it follows that the extremal point sets for the two problems
coincide.
We investigate the maximum number of of edges of a graph of dimension
, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be .
Box graphs are defined as the 3-dimensional analog of rectangle graphs. The
maximum number of edges of such a graph on points is shown to be
Graph isomorphism completeness for trapezoid graphs
The complexity of the graph isomorphism problem for trapezoid graphs has been
open over a decade. This paper shows that the problem is GI-complete. More
precisely, we show that the graph isomorphism problem is GI-complete for
comparability graphs of partially ordered sets with interval dimension 2 and
height 3. In contrast, the problem is known to be solvable in polynomial time
for comparability graphs of partially ordered sets with interval dimension at
most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure
Twistor theory on a finite graph
We show how the description of a shear-free ray congruence in Minkowski space
as an evolving family of semi-conformal mappings can naturally be formulated on
a finite graph. For this, we introduce the notion of holomorphic function on a
graph. On a regular coloured graph of degree three, we recover the space-time
picture. In the spirit of twistor theory, where a light ray is the more
fundamental object from which space-time points should be derived, the line
graph, whose points are the edges of the original graph, should be considered
as the basic object. The Penrose twistor correspondence is discussed in this
context
Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints
We introduce a new structure for a set of points in the plane and an angle
, which is similar in flavor to a bounded-degree MST. We name this
structure -MST. Let be a set of points in the plane and let be an angle. An -ST of is a spanning tree of the
complete Euclidean graph induced by , with the additional property that for
each point , the smallest angle around containing all the edges
adjacent to is at most . An -MST of is then an
-ST of of minimum weight. For , an -ST does
not always exist, and, for , it always exists. In this paper,
we study the problem of computing an -MST for several common values of
.
Motivated by wireless networks, we formulate the problem in terms of
directional antennas. With each point , we associate a wedge of
angle and apex . The goal is to assign an orientation and a radius
to each wedge , such that the resulting graph is connected and its
MST is an -MST. (We draw an edge between and if , , and .) Unsurprisingly, the problem of computing an
-MST is NP-hard, at least for and . We
present constant-factor approximation algorithms for .
One of our major results is a surprising theorem for ,
which, besides being interesting from a geometric point of view, has important
applications. For example, the theorem guarantees that given any set of
points in the plane and any partitioning of the points into triplets,
one can orient the wedges of each triplet {\em independently}, such that the
graph induced by is connected. We apply the theorem to the {\em antenna
conversion} problem
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