120 research outputs found
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Adjacency Graphs of Polyhedral Surfaces
We study whether a given graph can be realized as an adjacency graph of the
polygonal cells of a polyhedral surface in . We show that every
graph is realizable as a polyhedral surface with arbitrary polygonal cells, and
that this is not true if we require the cells to be convex. In particular, if
the given graph contains , , or any nonplanar -tree as a
subgraph, no such realization exists. On the other hand, all planar graphs,
, and can be realized with convex cells. The same holds for
any subdivision of any graph where each edge is subdivided at least once, and,
by a result from McMullen et al. (1983), for any hypercube.
Our results have implications on the maximum density of graphs describing
polyhedral surfaces with convex cells: The realizability of hypercubes shows
that the maximum number of edges over all realizable -vertex graphs is in
. From the non-realizability of , we obtain that
any realizable -vertex graph has edges. As such, these graphs
can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
Manhattan orbifolds
We investigate a class of metrics for 2-manifolds in which, except for a
discrete set of singular points, the metric is locally isometric to an L_1 (or
equivalently L_infinity) metric, and show that with certain additional
conditions such metrics are injective. We use this construction to find the
tight span of squaregraphs and related graphs, and we find an injective metric
that approximates the distances in the hyperbolic plane analogously to the way
the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised
since the previous version, and a new example has been adde
The DFS-heuristic for orthogonal graph drawingââSome of these result were published in the author's PhD thesis at Rutgers University; the author would like to thank her advisor, Prof. Endre Boros, for much helpful input. The results in Section 5 have been presented at the 8th Canadian Conference on Computational Geometry, Ottawa, 1996, see [1].
AbstractIn this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of mân+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of mâ2 bends
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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Force-Directed Graph Drawing and Aesthetics Measurement in a Non-Strict Pure Functional Programming Language
Non-strict pure functional programming often requires redesigning algorithms and data structures to work more effectively under new constraints of non-strict evaluation and immutable state. Graph drawing algorithms, while numerous and broadly studied, have no presence in the non-strict pure functional programming model. Additionally, there is currently no freely licensed standalone toolkit used to quantitatively analyze aesthetics of graph drawings. This thesis addresses two previously unexplored questions. Can a force-directed graph drawing algorithm be implemented in a non-strict functional language, such as Haskell, and still be practically usable? Can an easily extensible aesthetic measuring tool be implemented in a language such as Haskell and still be practically usable? The focus of the thesis is on implementing one of the simplest force-directed algorithms, that of Fruchterman and Reingold, and comparing its resulting aesthetics to those of a well-known C++ implementation of the same algorithm
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