55 research outputs found
Directed pathos middle digraph of an arborescence
A directed pathos middle digraph of an arborescence Aᵣ, written Q = DPM(Aᵣ), is the digraph whose vertex set V (Q) = V (Aᵣ) ∪ A(Aᵣ) ∪ P(Aᵣ), where V (Aᵣ) is the vertex set, A(Aᵣ) is the arc set, and P(Aᵣ) is a directed pathos set of Aᵣ. The arc set A(Q) consists of the following arcs: ab such that a, b ∈ A(Aᵣ) and the head of a coincides with the tail of b; for every v ∈ V (Aᵣ), all arcs a1v, va2; for which v is a head of the arc a1 and tail of the arc a2 in Aᵣ; P a such that a ∈ A(Aᵣ) and P ∈ P(Aᵣ) and the arc a lies on the directed path P; PᵢiPj such that Pi, Pj ∈ P(Aᵣ) and it is possible to reach the head of Pj from the tail of Pi through a common vertex, but it is possible to reach the head of Pi from the tail of Pj . The problem of reconstructing an arborescence from its directed pathos middle digraph is presented. The characterization of digraphs whose DPM(Aᵣ) are planar; outerplanar; maximal outerplanar; and minimally non-outerplanar is studied.Publisher's Versio
Reconstructing Generalized Staircase Polygons with Uniform Step Length
Visibility graph reconstruction, which asks us to construct a polygon that
has a given visibility graph, is a fundamental problem with unknown complexity
(although visibility graph recognition is known to be in PSPACE). We show that
two classes of uniform step length polygons can be reconstructed efficiently by
finding and removing rectangles formed between consecutive convex boundary
vertices called tabs. In particular, we give an -time reconstruction
algorithm for orthogonally convex polygons, where and are the number of
vertices and edges in the visibility graph, respectively. We further show that
reconstructing a monotone chain of staircases (a histogram) is fixed-parameter
tractable, when parameterized on the number of tabs, and polynomially solvable
in time under reasonable alignment restrictions.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Reconstructing Geometric Structures from Combinatorial and Metric Information
In this dissertation, we address three reconstruction problems. First, we address the problem of reconstructing a Delaunay triangulation from a maximal planar graph. A maximal planar graph G is Delaunay realizable if there exists a realization of G as a Delaunay triangulation on the plane. Several classes of graphs with particular graph-theoretic properties are known to be Delaunay realizable. One such class of graphs is outerplanar graph. In this dissertation, we present a new proof that an outerplanar graph is Delaunay realizable.
Given a convex polyhedron P and a point s on the surface (the source), the ridge tree or cut locus is a collection of points with multiple shortest paths from s on the surface of P. If we compute the shortest paths from s to all polyhedral vertices of P and cut the surface along these paths, we obtain a planar polygon called the shortest path star (sp-star) unfolding. It is known that for any convex polyhedron and a source point, the ridge tree is contained in the sp-star unfolding polygon [8]. Given a combinatorial structure of a ridge tree, we show how to construct the ridge tree and the sp-star unfolding in which it lies. In this process, we address several problems concerning the existence of sp-star unfoldings on specified source point sets.
Finally, we introduce and study a new variant of the sp-star unfolding called (geodesic) star unfolding. In this unfolding, we cut the surface of the convex polyhedron along a set of non-crossing geodesics (not-necessarily the shortest). We study its properties and address its realization problem. Finally, we consider the following problem: given a geodesic star unfolding of some convex polyhedron and a source point, how can we derive the sp-star unfolding of the same polyhedron and the source point? We introduce a new algorithmic operation and perform experiments using that operation on a large number of geodesic star unfolding polygons. Experimental data provides strong evidence that the successive applications of this operation on geodesic star unfoldings will lead us to the sp-star unfolding
Right-angled Artin groups and the cohomology basis graph
Let be a finite graph and let be the corresponding
right-angled Artin group. From an arbitrary basis of
over an arbitrary field, we construct a natural
graph from the cup product, called the \emph{cohomology
basis graph}. We show that always contains as a
subgraph. This provides an effective way to reconstruct the defining graph
from the cohomology of , to characterize the planarity of
the defining graph from the algebra of , and to recover many other
natural graph-theoretic invariants. We also investigate the behavior of the
cohomology basis graph under passage to elementary subminors, and show that it
is not well-behaved under edge contraction.Comment: 17 page
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
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