# Computing k-Modal Embeddings of Planar Digraphs

## TL;DR

This paper investigates the existence of k-modal embeddings in planar digraphs, a problem relevant to constrained graph embeddings and network visualization, by analyzing the combinatorial properties of such embeddings.

## Contribution

It introduces the k-Modality problem for planar digraphs and explores its complexity and properties, advancing understanding of constrained planar embeddings.

## Key findings

- Characterization of k-modal embeddings in planar digraphs
- Complexity results for the k-Modality problem
- Applications to constrained network visualization

## Abstract

Given a planar digraph $G$ and a positive even integer $k$, an embedding of $G$ in the plane is k-modal, if every vertex of $G$ is incident to at most $k$ pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the $k$-Modality problem, which asks for the existence of a $k$-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.