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A 1 -plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a 1-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a 1-plane graph G is a coloring of the edges of G with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: ( i ) Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. ( ii ) Deciding whether a NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. ( iii ) Deciding whether a 1-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed. [ABSTRACT FROM AUTHOR]

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]