Dimitrios M. Thilikos

Evangelos Protopapas, Dimitrios M. Thilikos, Sebastian Wiederrecht

We introduce the notion of colorful minors, which generalizes the classical concept of rooted minors in graphs. A $q$-colorful graph is defined as a pair $(G, χ),$ where $G$ is a graph and $χ$ assigns to each vertex a (possibly empty) subset of at most $q$ colors. The colorful minor relation enhances the classical minor relation by merging color sets at contracted edges and allowing the removal of colors from vertices. This framework naturally models algorithmic problems involving graphs with (possibly overlapping) annotated vertex sets. We develop a structural theory for colorful minors by establishing three core theorems characterizing $\mathcal{H}$-colorful minor-free graphs, where $\mathcal{H}$ consists either of a clique or a grid with all vertices assigned all colors, or of grids with colors segregated and ordered on the outer face. Our results reveal that when exclusion is imposed not only on graphs but also to the way colors are distributed in them, a more refined structural landscape appears. Leveraging our structural insights, we provide a complete classification -- parameterized by the number $q$ of colors -- of all colorful graphs that exhibit the Erdős-Pósa property with respect to colorful minors. On the algorithmic side, we deduce that colorful minor testing is fixed-parameter tractable. Together with the fact that the colorful minor relation forms a well-quasi-order, this implies that every colorful minor-monotone parameter on colorful graphs admits a fixed-parameter algorithm. Furthermore, we derive two algorithmic meta-theorems (AMTs) whose structural conditions are linked to extensions of treewidth and Hadwiger number on colorful graphs. Our results suggest how known AMTs can be extended to incorporate not only the structure of the input graph but also the way the colored vertices are distributed in it.

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.

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos et al.

The Local Structure Theorem (LST) for Graph Minors roughly states that for every $H$-minor-free graph $G$ that contains a sufficiently large wall $W$, there is a small vertex subset $A,$ whose removal yields a graph that admits an "almost embedding" $δ$ on a surface $Σ$ on which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $\mathcal{H}$ whose vertex set is a subset of the vertex set of $G - A$ and an embedding of $\mathcal{H}$ on $Σ$ such that the drawing of each hyperedge of $\mathcal{H}$ corresponds to a cell of $δ,$ the boundary of each cell intersects only the vertices of the corresponding hyperedge, and all remaining vertices and edges of $G - A$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while "most" of the wall $W$ is disjoint from the vortices and is "grounded" in the embedding $δ$. Suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: $G - A$ contains a minor model of a large grid $Γ$ such that, for every color $α\in C$, the model of each vertex of $Γ$ contains the subgraph drawn within an $α$-colored cell. Moreover, $Γ$ can be chosen in a way that is "well-connected" to the original wall $W$.

Dario Cavallaro, Maximilian Gorsky, Stephan Kreutzer et al.

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

Rishikesh Gajjala, Saurabh Ray, Dimitrios M. Thilikos

We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.

Christophe Paul, Evangelos Protopapas, Dimitrios M. Thilikos

In a recent work, we introduced a parametric framework for obtaining obstruction characterizations of graph parameters with respect to a quasi-ordering $\leqslant$ on graphs. Towards this, we proposed the concepts of class obstruction, parametric obstruction, and universal obstruction as combinatorial objects that determine the approximate behaviour of a graph parameter. In this work, we explore its potential as a unifying framework for classifying graph parameters. Under this framework, we survey existing graph-theoretic results on many known graph parameters. Additionally, we provide some unifying results on their classification.

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.

Stratis Limnios, George Dasoulas, Dimitrios M. Thilikos et al.

Neural networks are the pinnacle of Artificial Intelligence, as in recent years we witnessed many novel architectures, learning and optimization techniques for deep learning. Capitalizing on the fact that neural networks inherently constitute multipartite graphs among neuron layers, we aim to analyze directly their structure to extract meaningful information that can improve the learning process. To our knowledge graph mining techniques for enhancing learning in neural networks have not been thoroughly investigated. In this paper we propose an adapted version of the k-core structure for the complete weighted multipartite graph extracted from a deep learning architecture. As a multipartite graph is a combination of bipartite graphs, that are in turn the incidence graphs of hypergraphs, we design k-hypercore decomposition, the hypergraph analogue of k-core degeneracy. We applied k-hypercore to several neural network architectures, more specifically to convolutional neural networks and multilayer perceptrons for image recognition tasks after a very short pretraining. Then we used the information provided by the hypercore numbers of the neurons to re-initialize the weights of the neural network, thus biasing the gradient optimization scheme. Extensive experiments proved that k-hypercore outperforms the state-of-the-art initialization methods.