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M. Bekos, F. De Luca, W. Didimo, T. Mchedlidze, M. Nöllenburg, A. Symvonis, I. Tollis:
"Planar drawings of fixed-mobile bigraphs";
Theoretical Computer Science, 795 (2019), 408 - 419.

A fixed-mobile bigraphGis a bipartite graph such that the vertices of one partition set are given with fixedpositions in the plane and the mobilevertices of the other partition, together with the edges, must be added to the drawing without any restriction on their positions. We assume that Gis planar and study the problem of finding a planar straight-line drawing of G. We show that deciding whether Gadmits such a drawing isNP-hard in the general case. Under the assumption that each mobile vertex is placed in the convex hull of its neighbors, we are able to prove that the problem is also inNP. Moreover, if the intersection graph of these convex hulls is a path, a cycle or, more generally, a cactus, the problem is polynomial-time solvable through a dynamic programming approach. Finally, we describe linear-time testing algorithms when the fixed vertices are collinear or when they lie on a finite set of horizontal levels (lines) and no edge can intersect a level except at its fixed vertex.

http://dx.doi.org/10.1016/j.tcs.2019.07.025

https://publik.tuwien.ac.at/files/publik_284604.pdf