Splitting Plane Graphs to Outerplanarity

TL;DR

This paper investigates the problem of transforming plane graphs into outerplanar graphs through vertex splitting, establishing complexity results and algorithms for specific graph classes.

Contribution

It introduces a novel approach linking outerplanarity vertex splitting to face covers and feedback vertex sets, with complexity and algorithmic results.

Findings

NP-completeness for plane biconnected graphs

Polynomial-time algorithm for maximal planar graphs

Bounds for certain maximal planar graph families

Abstract

Vertex splitting replaces a vertex by two copies and partitions its incident edges amongst the copies. This problem has been studied as a graph editing operation to achieve desired properties with as few splits as possible, most often planarity, for which the problem is NP-hard. Here we study how to minimize the number of splits to turn a plane graph into an outerplane one. We tackle this problem by establishing a direct connection between splitting a plane graph to outerplanarity, finding a connected face cover, and finding a feedback vertex set in its dual. We prove NP-completeness for plane biconnected graphs, while we show that a polynomial-time algorithm exists for maximal planar graphs. Finally, we provide upper and lower bounds for certain families of maximal planar graphs.