By Shi M., Yuan X., Cai M.
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Additional resources for (3,k)-Factor-Critical Graphs and Toughness
The crossing minimization problem is NP-hard even when restricted to 2-layer instances. Nevertheless, it has been attacked with a branch-and-cut algorithm that produces an optimum solution in exponential running time, see Healy and Kuusik . , Elf et al. ) However, such methods have not yet reached the maturity and practical efficiency to be used in software systems for automatic graph drawing. Instead, most systems apply a layer by layer sweep as follows: Starting from some initial permutation of the vertices on each layer, such heuristics consider pairs of layers (L fixed , Lfree) = (Ll' L2), (L2' L 3 ), ...
Mannino, C. (1994) Upward drawings of triconnected digraphs. Algorithmica 6 (12), 476-497 6. , Tamassia, R. (1998) Optimal upward planarity testing of single-source digraphs. SIAM Journal on Computing 27, 132-169 7. , Lueker, G. (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13, 335-379 8. Brandenburg, F. , Rohrer, C. (1996) An experimental comparison of force-directed and randomized graph drawing algorithms.
A typical force-directed layout. It should be stressed that the title of this section is a concession to the short tradition of automatic graph drawing and the title "Drawing on Physical Analogies" of a recommended survey of Brandes  would be much more appropriate, because what follows has little to do with what physicists do when they study ground states of physical systems and relaxation dynamics. Two-Dimensional Force-Directed Layout of Undirected Graphs. For ease of notation, we identify vertices and edges with their physical counterparts.
(3,k)-Factor-Critical Graphs and Toughness by Shi M., Yuan X., Cai M.