Planar Steiner Orientation is NP-complete

TL;DR

This paper proves that the Steiner Orientation problem remains NP-complete even when restricted to planar graphs, extending the understanding of its computational complexity in special graph classes.

Contribution

It establishes NP-completeness of Steiner Orientation specifically for planar graphs, a previously unresolved case in the problem's complexity landscape.

Findings

Steiner Orientation is NP-complete on planar graphs.

The problem remains hard even with planar restrictions.

Extends complexity results to a broader class of graphs.

Abstract

Many applications in graph theory are motivated by routing or flow problems. Among these problems is Steiner Orientation: given a mixed graph G (having directed and undirected edges) and a set T of k terminal pairs in G, is there an orientation of the undirected edges in G such that there is a directed path for every terminal pair in T ? This problem was shown to be NP -complete by Arkin and Hassin [1] and later W [1]-hard by Pilipczuk and Wahlstr\"om [7], parametrized by k. On the other hand, there is an XP algorithm by Cygan et al. [3] and a polynomial time algorithm for graphs without directed edges by Hassin and Megiddo [5]. Chitnis and Feldmann [2] showed W [1]-hardness of the problem for graphs of genus 1. We consider a further restriction to planar graphs and show NP -completeness.