Orthogonal Graph Drawing with Inflexible Edges

Abstract

We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge $e$ a natural number ${{\mathrm{flex}}}(e)$, its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge $e$ has at most ${{\mathrm{flex}}}(e)$ bends. It is known that FlexDraw is NP-hard if ${{\mathrm{flex}}}(e) = 0$ for every edge $e$ [7]. On the other hand, FlexDraw can be solved efficiently if ${{\mathrm{flex}}}(e) \ge 1$ [2] and is trivial if ${{\mathrm{flex}}}(e) \ge 2$ [1] for every edge $e$.

To close the gap between the NP-hardness for ${{\mathrm{flex}}}(e) = 0$ and the efficient algorithm for ${{\mathrm{flex}}}(e) \ge 1$, we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility $0$). We show that for any $\varepsilon > 0$ FlexDraw is NP-complete for instances with $O(n^\varepsilon )$ inflexible edges with pairwise distance $\Omega (n^{1-\varepsilon })$ (including the case where they induce a matching). On the other hand, we give an FPT-algorithm with running time $O(2^k\cdot n \cdot T_{{{\mathrm{flow}}}}(n))$, where $T_{{{\mathrm{flow}}}}(n)$ is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and $k$ is the number of inflexible edges having at least one endpoint of degree 4.

Partially supported by grant WA 654/21-1 of the German Research Foundation (DFG).