Small Point-Sets Supporting Graph Stories

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 ω, 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 drawing the entire sequence by mapping the vertices only to ω+ 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 its size; (ii) the problem is NP-hard and is FPT when parameterized by ω+ k ; (iii) there are families of graph stories that can be drawn with k= 0 for any ω, while for k= 0 and small values of ω there are families of graph stories that can be drawn and others that cannot; (iv) there are families of graph stories that cannot be drawn for any fixed k and families of graph stories that require at least a certain k.