Extending Nearly Complete 1-Planar Drawings in Polynomial Time

TL;DR

This paper presents a polynomial-time algorithm for extending partial 1-planar graph drawings to full graphs when the partial subgraph is obtained by removing a bounded number of vertices and edges, advancing graph drawing theory.

Contribution

It introduces the first polynomial-time solution for extending partial 1-planar drawings under bounded modifications, building on previous work on planarity extension.

Findings

Polynomial-time decision algorithm developed

Extends previous planarity extension results to 1-planar graphs

Addresses an open problem from prior research

Abstract

The problem of extending partial geometric graph representations such as plane graphs has received considerable attention in recent years. In particular, given a graph $G$, a connected subgraph $H$ of $G$ and a drawing $\mathcal{H}$ of $H$, the extension problem asks whether $\mathcal{H}$ can be extended into a drawing of $G$ while maintaining some desired property of the drawing (e.g., planarity). In their breakthrough result, Angelini et al. [ACM TALG 2015] showed that the extension problem is polynomial-time solvable when the aim is to preserve planarity. Very recently we considered this problem for partial 1-planar drawings [ICALP 2020], which are drawings in the plane that allow each edge to have at most one crossing. The most important question identified and left open in that work is whether the problem can be solved in polynomial time when $H$ can be obtained from $G$ by deleting a bounded number of vertices and edges. In this work, we answer this question positively by providing a constructive polynomial-time decision algorithm.