# Sketched Representations and Orthogonal Planarity of Bounded Treewidth Graphs

## TL;DR

This paper presents a polynomial-time algorithm for orthogonal graph drawing with bounded treewidth, introducing sketched representations and extending to related problems, improving existing complexity bounds for series-parallel graphs.

## Contribution

The paper introduces a novel FPT algorithm for OrthogonalPlanarity on bounded treewidth graphs using sketched orthogonal representations, extending to HV-Planarity and FlexDraw.

## Key findings

- OrthogonalPlanarity is solvable in polynomial time for bounded treewidth graphs.
- Series-parallel graphs can be decided in O(n^3 log n) time for OrthogonalPlanarity and HV-Planarity.
- The approach improves previous complexity bounds from O(n^4) to O(n^3 log n).

## Abstract

Given a planar graph $G$ and an integer $b$, OrthogonalPlanarity is the problem of deciding whether $G$ admits an orthogonal drawing with at most $b$ bends in total. We show that OrthogonalPlanarity can be solved in polynomial time if $G$ has bounded treewidth. Our proof is based on an FPT algorithm whose parameters are the number of bends, the treewidth and the number of degree-2 vertices of $G$. This result is based on the concept of sketched orthogonal representation that synthetically describes a family of equivalent orthogonal representations. Our approach can be extended to related problems such as HV-Planarity and FlexDraw. In particular, both OrthogonalPlanarity and HV-Planarity can be decided in $O(n^3 \log n)$ time for series-parallel graphs, which improves over the previously known $O(n^4)$ bounds.