On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

TL;DR

This paper advances the understanding of the computational complexity of finding orthogonal graph drawings with minimal bends, showing fixed-parameter tractability under less restrictive parameters, thus improving previous results.

Contribution

It proves that the problem remains fixed-parameter tractable when parameterized by the number of degree-two vertices plus the number of bends, refining earlier complexity bounds.

Findings

The problem is fixed-parameter tractable with respect to degree-two vertices and bends.

Rectilinear planarity testing is FPT when parameterized by degree-two vertices.

Improves previous FPT results by reducing the parameter set.

Abstract

Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number of bends, the number of vertices of degree at most two, and the treewidth of the input graph (Di Giacomo et al., 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by the number of vertices of degree at most two plus the number of bends. As a consequence, rectilinear planarity testing lies in \FPT~parameterized by the number of vertices of degree at most two.