Extending Orthogonal Planar Graph Drawings is Fixed-Parameter Tractable

TL;DR

This paper proves that extending partial orthogonal graph drawings with minimal bends is fixed-parameter tractable when parameterized by the size of the missing subgraph, using novel geometric and combinatorial techniques.

Contribution

It introduces a fixed-parameter algorithm for the bend-minimal orthogonal drawing extension problem, combining new graph representations and bounds on treewidth.

Findings

The problem is fixed-parameter tractable with respect to the missing subgraph size.

New graph representation for bend-equivalent regions was developed.

Bounded treewidth of an auxiliary graph was established.

Abstract

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar connected graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be \NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.