5 research outputs found
A Tipping Point for the Planarity of Small and Medium Sized Graphs
This paper presents an empirical study of the relationship between the
density of small-medium sized random graphs and their planarity. It is well
known that, when the number of vertices tends to infinite, there is a sharp
transition between planarity and non-planarity for edge density d=0.5. However,
this asymptotic property does not clarify what happens for graphs of reduced
size. We show that an unexpectedly sharp transition is also exhibited by small
and medium sized graphs. Also, we show that the same "tipping point" behavior
can be observed for some restrictions or relaxations of planarity (we
considered outerplanarity and near-planarity, respectively).Comment: Appears in the Proceedings of the 28th International Symposium on
Graph Drawing and Network Visualization (GD 2020
Graph Stories in Small Area
We study the problem of drawing a dynamic graph, where each vertex appears in
the graph at a certain time and remains in the graph for a fixed amount of
time, called the window size. This defines a graph story, i.e., a sequence of
subgraphs, each induced by the vertices that are in the graph at the same time.
The drawing of a graph story is a sequence of drawings of such subgraphs. To
support readability, we require that each drawing is straight-line and planar
and that each vertex maintains its placement in all the drawings. Ideally, the
area of the drawing of each subgraph should be a function of the window size,
rather than a function of the size of the entire graph, which could be too
large. We show that the graph stories of paths and trees can be drawn on a and on an grid, respectively, where
is the window size. These results are constructive and yield linear-time
algorithms. Further, we show that there exist graph stories of planar graphs
whose subgraphs cannot be drawn within an area that is only a function of