38,770 research outputs found
From isovists to visibility graphs: a methodology for the analysis of architectural space
An isovist, or viewshed, is the area in a spatial environment directly visible from a location within the space. Here we show how a set of isovists can be used to generate a graph of mutual visibility between locations. We demonstrate that this graph can also be constructed without reference to isovists and that we are in fact invoking the more general concept of a visibility graph. Using the visibility graph, we can extend both isovist and current graph-based analyses of architectural space to form a new methodology for the investigation of configurational relationships. The measurement of local and global characteristics of the graph, for each vertex or for the system as a whole, is of interest from an architectural perspective, allowing us to describe a configuration with reference to accessibility and visibility, to compare from location to location within a system, and to compare systems with different geometries. Finally we show that visibility graph properties may be closely related to manifestations of spatial perception, such as way-finding, movement, and space use
Bar 1-Visibility Graphs and their relation to other Nearly Planar Graphs
A graph is called a strong (resp. weak) bar 1-visibility graph if its
vertices can be represented as horizontal segments (bars) in the plane so that
its edges are all (resp. a subset of) the pairs of vertices whose bars have a
-thick vertical line connecting them that intersects at most one
other bar.
We explore the relation among weak (resp. strong) bar 1-visibility graphs and
other nearly planar graph classes. In particular, we study their relation to
1-planar graphs, which have a drawing with at most one crossing per edge;
quasi-planar graphs, which have a drawing with no three mutually crossing
edges; the squares of planar 1-flow networks, which are upward digraphs with
in- or out-degree at most one. Our main results are that 1-planar graphs and
the (undirected) squares of planar 1-flow networks are weak bar 1-visibility
graphs and that these are quasi-planar graphs
Edge Routing with Ordered Bundles
Edge bundling reduces the visual clutter in a drawing of a graph by uniting
the edges into bundles. We propose a method of edge bundling drawing each edge
of a bundle separately as in metro-maps and call our method ordered bundles. To
produce aesthetically looking edge routes it minimizes a cost function on the
edges. The cost function depends on the ink, required to draw the edges, the
edge lengths, widths and separations. The cost also penalizes for too many
edges passing through narrow channels by using the constrained Delaunay
triangulation. The method avoids unnecessary edge-node and edge-edge crossings.
To draw edges with the minimal number of crossings and separately within the
same bundle we develop an efficient algorithm solving a variant of the
metro-line crossing minimization problem. In general, the method creates clear
and smooth edge routes giving an overview of the global graph structure, while
still drawing each edge separately and thus enabling local analysis
On the connectivity of visibility graphs
The visibility graph of a finite set of points in the plane has the points as
vertices and an edge between two vertices if the line segment between them
contains no other points. This paper establishes bounds on the edge- and
vertex-connectivity of visibility graphs.
Unless all its vertices are collinear, a visibility graph has diameter at
most 2, and so it follows by a result of Plesn\'ik (1975) that its
edge-connectivity equals its minimum degree. We strengthen the result of
Plesn\'ik by showing that for any two vertices v and w in a graph of diameter
2, if deg(v) <= deg(w) then there exist deg(v) edge-disjoint vw-paths of length
at most 4. Furthermore, we find that in visibility graphs every minimum edge
cut is the set of edges incident to a vertex of minimum degree.
For vertex-connectivity, we prove that every visibility graph with n vertices
and at most l collinear vertices has connectivity at least (n-1)/(l-1), which
is tight. We also prove the qualitatively stronger result that the
vertex-connectivity is at least half the minimum degree. Finally, in the case
that l=4 we improve this bound to two thirds of the minimum degree.Comment: 16 pages, 8 figure
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