Upward Point Set Embeddability for Convex Point Sets is in $P$

TL;DR

This paper introduces a polynomial-time algorithm for testing upward planar embeddings of directed trees and outerplanar digraphs into convex point sets, enabling efficient embedding verification.

Contribution

It presents the first polynomial-time algorithm for upward embeddability of directed trees and extends it to outerplanar digraphs, advancing graph embedding theory.

Findings

Polynomial dynamic programming algorithm for directed trees

Extension of the algorithm to outerplanar digraphs

Efficient testing of upward planar embeddings into convex point sets

Abstract

In this paper, we present a polynomial dynamic programming algorithm that tests whether a $n$-vertex directed tree $T$ has an upward planar embedding into a convex point-set $S$ of size $n$. Further, we extend our approach to the class of outerplanar digraphs. This nontrivial and surprising result implies that any given digraph can be efficiently tested for an upward planar embedding into a given convex point set.