The Stub Resolution of 1Planar Graphs. The resolution of a drawing plays a crucial role when defining criteria for its quality. In the past, grid resolution, edgelength resolution, angular resolution and crossing resolution have been investigated. In this paper, we investigate the stub resolution, a recently introduced criterion for nonplanar drawings. A crossed edge is divided into parts, called stubs, which should not be too short for the sake of readability. Thus, the stub resolution of a drawing is defined as the minimum ratio between the length of a stub and the length of the entire edge, over all the edges of the drawing. We consider 1planar graphs and we explore scenarios in which near optimal stub resolution, i.e. arbitrarily close to (frac12), can be obtained in drawings with zero, one, or two bends per edge, as well as further resolution criteria, such as angular and crossing resolution. In particular, our main contributions are as follows: (i) Every 1planar graph with independent crossing edges has a straightline drawing with near optimal stub resolution; (ii) Every 1planar graph has a 1bend drawing with near optimal stub resolution.

On the Edgelength Ratio of Outerplanar Graphs We show that any outerplanar graph admits a planar straightline drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any (epsilon u003e 0) there are outerplanar graphs that cannot be drawn with an edgelength ratio smaller than (2  epsilon ). We also show that every bipartite outerplanar graph has a planar straightline drawing with edgelength ratio 1, and that, for any (k ge 1), there exists an outerplanar graph with a given combinatorial embedding such that any planar straightline drawing has edgelength ratio greater than k.