Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs

TL;DR

This paper investigates crossing-free acyclic Hamiltonian path completions in embedded upward planar digraphs, providing linear-time algorithms for specific classes and establishing conditions for the existence of such completions.

Contribution

It introduces new linear-time algorithms for finding crossing-free acyclic HP-completions in certain upward planar digraphs and characterizes their properties.

Findings

Existence of crossing-free HP-completion sets with at most two edges per face.

Linear-time algorithm for N-free upward planar digraphs.

Efficient testing for crossing-free acyclic HP-completion in width-k embedded planar st-digraphs.

Abstract

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph $G$, determine whether there exists an upward 2-page book embedding of $G$ preserving the given planar embedding. Given an embedded $st$-digraph $G$ which has a crossing-free HP-completion set, we show that there always exists a crossing-free HP-completion set with at most two edges per face of $G$. For an embedded $N$-free upward planar digraph $G$, we show that there always exists a crossing-free acyclic HP-completion set for $G$ which, moreover, can be computed in linear time. For a width-$k$ embedded planar $st$-digraph $G$, we show that we can be efficiently test whether $G$ admits a crossing-free acyclic HP-completion set.