The Parameterized Complexity of Extending Stack Layouts

An $\mathrm{\ell }$-page stack layout (also known as an $\mathrm{\ell }$-page book embedding) of a graph is a linear order of the vertex set together with a partition of the edge set into $\mathrm{\ell }$ 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 $\mathrm{\ell }$-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.