Graph Stories in Small Area

Abstract

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 $2W \times 2W$ and on an $(8W+1) \times (8W+1)$ grid, respectively, where $W$ 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 $W$.

A Tipping Point for the Planarity of Small and Medium Sized Graphs

“…Several popular Graph Drawing algorithms devised to draw graphs of smallmedium size assume that the graph to be drawn is planar both in the static setting [17,10,16] and in the dynamic one [2,7,4]. Hence, to assess the practical applicability of such algorithms it is crucial to study the probability that a smallmedium sized graph (say of about 100-200 vertices) is planar.…”