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Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, (GD) 2 , that can handle multiple readability criteria. (GD) 2 can optimize any criterion that can be described by a smooth function. If the criterion cannot be captured by a smooth function, a non-smooth function for the criterion is combined with another smooth function, or auto-differentiation tools are used for the optimization. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., vertex resolution, angular resolution, aspect ratio). We provide quantitative and qualitative evidence of the effectiveness of (GD) 2 with experimental data and a functional prototype: http://hdc.cs.arizona.edu/ ~mwli/graph-drawing/.

Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with $\kappa$ colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, $\kappa$)-colorings, in which the monochromatic components have diameter $1$, and (star, $\kappa$)-colorings, in which they have diameter $2$. We prove that the (edge, $3$)-coloring problem remains NP-complete even for graphs with maximum vertex-degree $6$, hence answering an open question posed by Cowen et al. [Cowen, Goddard, Jesurum, J. Graph Theory, 1997], and for planar graphs with maximum vertex-degree $7$, and we prove that the (star, $3$)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, $2$)-colorings and of (star, $2$)-colorings of partial $2$-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, $2$)-colorings.

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A Stick graph is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a "ground line," a line with slope −1.It is an open question to decide in polynomial time whether a given bipartite graph G with bipartition A∪B has a Stick representation where the vertices in A and B correspond to horizontal and vertical segments, respectively. We prove that G has a Stick representation if and only if there are orderings of A and B such that G's bipartite adjacency matrix with rows A and columns B excludes three small 'forbidden' submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of A and B permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of A is given, we present an O(|A| 3 |B| 3 )-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.