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A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other partition, together with the edges, must be added to the drawing without any restriction on their positions. We assume that G is planar and study the problem of finding a planar straight-line drawing of G. We show that deciding whether G admits such a drawing is NP -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 in NP. 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. [ABSTRACT FROM AUTHOR]

A drawing of a graph is greedy if for each ordered pair of vertices u and v , there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. From an application perspective, greedy drawings are especially relevant to support routing schemes in ad hoc wireless networks. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and drawing succinctness. We introduce greedy rectilinear drawings , where edges are horizontal or vertical segments. These drawings have several properties that improve human readability and support network routing. We address the problem of testing whether a planar rectilinear representation , i.e., a plane graph with prescribed vertex angles, admits a greedy rectilinear drawing. We give a characterization, a linear-time testing algorithm, and a full generative scheme for universal greedy rectilinear representations, i.e., those for which every drawing is greedy. For general greedy rectilinear representations, we give a combinatorial characterization and, based on it, a polynomial-time testing and drawing algorithm for a meaningful subset of instances. [ABSTRACT FROM AUTHOR]