Giannos Stamoulis

Petr A. Golovach, Stavros G. Kolliopoulos, Giannos Stamoulis et al.

The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists of decomposing any surface-embedded graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the $\mathcal{F}$-Minor-Deletion problem.

Ignasi Sau, Nicole Schirrmacher, Sebastian Siebertz et al.

Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model checking on graph classes excluding a fixed topological minor, comparable results for richer fragments of CMSO were previously unknown. We further develop the framework of Sau, Stamoulis, and Thilikos [SODA 2025] for fragmenting CMSO via annotated graph parameters, which restrict set quantification to vertex sets satisfying bounded structural conditions. Following this approach, we identify a fragment of CMSO, namely the one defined by allowing quantification only over sets having what we call low monodimensionality, that generalizes several previously-known logics and we show that model checking for this fragment, enhanced with the disjoint-paths predicate, is fixed-parameter tractable on topological-minor-free graph classes. Such classes essentially delimit the tractability for this logic on subgraph-closed classes. As a consequence, our results lift several known algorithmic meta-theorems beyond first-order logic to the topological-minor-free setting.