The Parameterized Complexity of Extending Stack Layouts

TL;DR

This paper explores the computational complexity of extending partial stack layouts of graphs, revealing a nuanced landscape with various tractable and intractable cases through parameterized complexity analysis.

Contribution

It provides a detailed complexity classification of the stack layout extension problem, identifying conditions under which it is tractable or hard.

Findings

Identification of paraNP-hard and W[1]-hard cases.

Discovery of XP-tractable and fixed-parameter tractable fragments.

Comprehensive complexity landscape of the extension problem.

Abstract

An $\ell$-page stack layout (also known as an $\ell$-page book embedding) of a graph is a linear order of the vertex set together with a partition of the edge set into $\ell$ stacks (or pages), such that the endpoints of no two edges on the same stack alternate. We study the problem of extending a given partial $\ell$-page stack layout into a complete one, which can be seen as a natural generalization of the classical NP-hard problem of computing a stack layout of an input graph from scratch. Given the inherent intractability of the problem, we focus on identifying tractable fragments through the refined lens of parameterized complexity analysis. Our results paint a detailed and surprisingly rich complexity-theoretic landscape of the problem which includes the identification of paraNP-hard, W[1]-hard and XP-tractable, as well as fixed-parameter tractable fragments of stack layout extension via a natural sequence of parameterizations.