Recognizing DAGs with Page-Number 2 is NP-complete

TL;DR

This paper proves that recognizing DAGs with page-number 2 is NP-complete, confirming a long-standing conjecture and extending the complexity results to special classes like planar graphs.

Contribution

It establishes the NP-completeness of recognizing DAGs with page-number 2, resolving a 20-year-old conjecture and extending complexity results to planar and st-planar graphs.

Findings

Recognition of DAGs with page-number 2 is NP-complete.

NP-completeness holds even for planar and st-planar graphs.

Confirms the conjecture by Heath and Pemmaraju from 1999.

Abstract

The page-number of a directed acyclic graph (a DAG, for short) is the minimum $k$ for which the DAG has a topological order and a $k$-coloring of its edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological order. In 1999, Heath and Pemmaraju conjectured that the recognition of DAGs with page-number $2$ is NP-complete and proved that recognizing DAGs with page-number $6$ is NP-complete [SIAM J. Computing, 1999]. Binucci et al. recently strengthened this result by proving that recognizing DAGs with page-number $k$ is NP-complete, for every $k\geq 3$ [SoCG 2019]. In this paper, we finally resolve Heath and Pemmaraju's conjecture in the affirmative. In particular, our NP-completeness result holds even for $st$-planar graphs and planar posets.