Non-planar square-orthogonal drawing with few-bend edges

We investigate square-orthogonal drawings of non-planar graphs with vertices represented as unit grid squares. We present quadratic-time algorithms to construct the square-orthogonal drawings of 5-graphs, 6-graphs, and 8-graphs such that each edge in the drawing contains at most two, two, and three bends, respectively. In particular, the novel analysis method we use to split a vertex so as to build some specific propagation channels in our algorithms is an interesting technique and may be of independent interest. Moreover, we show that the decision problem of determining whether an 8-graph has a square-orthogonal drawing without edge-bends is NP-complete.