CIAC 2015: Algorithms and Complexity pp 61-73

# Orthogonal Graph Drawing with Inflexible Edges

• Thomas Bläsius
• Sebastian Lehmann
• Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)

## 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.

## Keywords

Cost Function Outer Face Series Composition Orthogonal Representation Critical Edge
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Thomas Bläsius
• 1
• Sebastian Lehmann
• 1
• Ignaz Rutter
• 1
1. 1.Faculty of InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany