On the Upward Book Thickness Problem: Combinatorial and Complexity Results

TL;DR

This paper investigates the upward book thickness of certain classes of outerplanar DAGs, proving boundedness for specific subfamilies and analyzing the computational complexity of related decision problems.

Contribution

It confirms the conjecture for particular subclasses of upward outerplanar graphs and establishes complexity results, including NP-hardness and fixed-parameter tractability.

Findings

Bounded upward book thickness for subfamilies like internally-triangulated outerpaths and cacti.

NP-hardness of deciding upward book thickness for graphs with bounded domination number.

FPT results for the problem parameterized by vertex cover number.

Abstract

A long-standing conjecture by Heath, Pemmaraju, and Trenk states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are $at$-outerplanar graphs. On the complexity side, it is known that deciding whether a graph has upward book thickness $k$ is NP-hard for any fixed $k \ge 3$. We show that the problem, for any $k \ge 5$, remains NP-hard for graphs whose domination number is $O(k)$, but it is FPT in the vertex cover number.