Small Point-Sets Supporting Graph Stories

Abstract

In a graph story the vertices enter a graph one at a time and each vertex persists in the graph for a fixed amount of time $\omega$, called viewing window. At any time, the user can see only the drawing of the graph induced by the vertices in the viewing window and this determines a sequence of drawings. For readability, we require that all the drawings of the sequence are planar. For preserving the user's mental map we require that when a vertex or an edge is drawn, it has the same drawing for its entire life. We study the problem of computing the whole sequence of drawings by mapping the vertices only to $\omega+k$ given points, where $k$ is as small as possible. We show that: $(i)$ The problem does not depend on the specific set of points but only on the size of the point set; $(ii)$ the problem is NP-hard (even when $k$ is a given constant) and it is FPT when parameterized by $\omega+k$; $(iii)$ for $k=0$ there are families of graph stories that can be drawn independent of $\omega$, but also families that cannot be drawn even when $\omega$ is small; $(iv)$ there are families of graph stories that cannot be drawn for any fixed $k$ and families of graph stories that can be realized only when $k$ is larger than a certain threshold.