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Given a graph $G$ and a vertex set $X$, the annotated treewidth tw$(G,X)$ of $X$ in $G$ is the maximum treewidth of an $X$-rooted minor of $G$, i.e., a minor $H$ where the model of each vertex of $H$ contains some vertex of $X$. That way, tw$(G,X)$ can be seen as a measure of the contribution of $X$ to the tree-decomposability of $G$. We introduce the logic CMSO/tw as the fragment of monadic second-order logic on graphs obtained by restricting set quantification to sets of bounded annotated treewidth. We prove the following Algorithmic Meta-Theorem (AMT): for every non-trivial minor-closed graph class, model checking for CMSO/tw formulas can be done in quadratic time. Our proof works for the more general CMSO/tw+dp logic, that is CMSO/tw enhanced by disjoint-path predicates. Our AMT can be seen as an extension of Courcelle's theorem to minor-closed graph classes where the bounded-treewidth condition in the input graph is replaced by the bounded-treewidth quantification in the formulas. Our results yield, as special cases, all known AMTs whose combinatorial restriction is non-trivial minor-closedness.

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest $(s,t)$-Path problem, which requires exponentially many queries to the oracle for finding an $n^{2/3-\varepsilon}$-approximation [Goel et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We show that for every $\varepsilon > 0$, obtaining a $(1+\varepsilon)$-approximation requires at least $n^{\Omega(\log 1/ \varepsilon)}$ queries to the oracle. When the function $f$ is integer-valued, our algorithm yields that a cycle of cost $\textsf{OPT}$ can be found in time $n^{\mathcal{O}(\log \textsf{OPT})}$. In particular, for $\textsf{OPT}=n^{\mathcal{O}(1)}$ this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that $n^{\mathcal{O}(\log n)}$ queries are required even when $\textsf{OPT} = \mathcal{O}(n)$., Comment: 17 pages, 1 figure. Accepted to SODA 2023

The disjoint paths logic, FOL+DP, is an extension of First-Order Logic (FOL) with the extra atomic predicate $\mathsf{dp}_k(x_1,y_1,\ldots,x_k,y_k),$ expressing the existence of internally vertex-disjoint paths between $x_i$ and $y_i,$ for $i\in\{1,\ldots, k\}$. This logic can express a wide variety of problems that escape the expressibility potential of FOL. We prove that for every proper minor-closed graph class, model-checking for FOL+DP can be done in quadratic time. We also introduce an extension of FOL+DP, namely the scattered disjoint paths logic, FOL+SDP, where we further consider the atomic predicate $s{\sf -sdp}_k(x_1,y_1,\ldots,x_k,y_k),$ demanding that the disjoint paths are within distance bigger than some fixed value $s$. Using the same technique we prove that model-checking for FOL+SDP can be done in quadratic time on classes of graphs with bounded Euler genus., Comment: An extended abstract of this paper appeared in the Proceedings of the 34th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2023)

Let ${\cal G}$ be a minor-closed graph class and let $G$ be an $n$-vertex graph. We say that $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. Our first result is an algorithm that decides whether $G$ is a $k$-apex of ${\cal G}$ in time $2^{{\sf poly}(k)}\cdot n^2$, where ${\sf poly}$ is a polynomial function depending on ${\cal G}$. This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020], whose running time was $2^{{\sf poly}(k)}\cdot n^3$. The elimination distance of $G$ to ${\cal G}$, denoted by ${\sf ed}_{\cal G}(G)$, is the minimum number of rounds required to reduce each connected component of $G$ to a graph in ${\cal G}$ by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] provided an FPT-algorithm, with parameter $k$, to decide whether ${\sf ed}_{\cal G}(G)\leq k$. However, its dependence on $k$ is not explicit. We extend the techniques used in the first algorithm to decide whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{2^{2^{{\sf poly}(k)}}}\cdot n^2$. This is the first algorithm for this problem with an explicit parametric dependence in $k$. In the special case where ${\cal G}$ excludes some apex-graph as a minor, we give two alternative algorithms, running in time $2^{2^{{\cal O}(k^2\log k)}}\cdot n^2$ and $2^{{\sf poly}(k)}\cdot n^3$ respectively, where $c$ and ${\sf poly}$ depend on ${\cal G}$. As a stepping stone for these algorithms, we provide an algorithm that decides whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{{\cal O}({\sf tw}\cdot k+{\sf tw}\log{\sf tw})}\cdot n$, where ${\sf tw}$ is the treewidth of $G$. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs ${\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}$., Comment: 75 pages. TheoretiCS journal article. Abstract abbreviated to fit arXiv limitation

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or $\mathcal{G}$-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates.

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{{\phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${\phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and ${\phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence ${\phi}$, the Graph $\boxtimes$-Modification to Planarity and ${\phi}$ is solvable in $f(k,|{\phi}|)\cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman's Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems.

We introduce the block elimination distance as a measure of how close a graph is to some particular graph class. Formally, given a graph class ${\cal G}$, the class ${\cal B}({\cal G})$ contains all graphs whose blocks belong to ${\cal G}$ and the class ${\cal A}({\cal G})$ contains all graphs where the removal of a vertex creates a graph in ${\cal G}$. Given a hereditary graph class ${\cal G}$, we recursively define ${\cal G}^{(k)}$ so that ${\cal G}^{(0)}={\cal B}({\cal G})$ and, if $k\geq 1$, ${\cal G}^{(k)}={\cal B}({\cal A}({\cal G}^{(k-1)}))$. The block elimination distance of a graph $G$ to a graph class ${\cal G}$ is the minimum $k$ such that $G\in{\cal G}^{(k)}$ and can be seen as an analog of the elimination distance parameter, with the difference that connectivity is now replaced by biconnectivity. We show that, for every non-trivial hereditary class ${\cal G}$, the problem of deciding whether $G\in{\cal G}^{(k)}$ is NP-complete. We focus on the case where ${\cal G}$ is minor-closed and we study the minor obstruction set of ${\cal G}^{(k)}$. We prove that the size of the obstructions of ${\cal G}^{(k)}$ is upper bounded by some explicit function of $k$ and the maximum size of a minor obstruction of ${\cal G}$. This implies that the problem of deciding whether $G\in{\cal G}^{(k)}$ is constructively fixed parameter tractable, when parameterized by $k$. Our results are based on a structural characterization of the obstructions of ${\cal B}({\cal G})$, relatively to the obstructions of ${\cal G}$. We give two graph operations that generate members of ${\cal G}^{(k)}$ from members of ${\cal G}^{(k-1)}$ and we prove that this set of operations is complete for the class ${\cal O}$ of outerplanar graphs. This yields the identification of all members ${\cal O}\cap{\cal G}^{(k)}$, for every $k\in\mathbb{N}$ and every non-trivial minor-closed graph class ${\cal G}$.

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k (\mathcal{G})$ the set of all graphs that are $k$-apices of $\mathcal{G}.$ We prove that every graph in the obstruction set of $\mathcal{A}_k (\mathcal{G}),$ i.e., the minor-minimal set of graphs not belonging to $\mathcal{A}_k (\mathcal{G}),$ has size at most $2^{2^{2^{2^{\mathsf{poly}(k)}}}},$ where $\mathsf{poly}$ is a polynomial function whose degree depends on the size of the minor-obstructions of $\mathcal{G}.$ This bound drops to $2^{2^{\mathsf{poly}(k)}}$ when $\mathcal{G}$ excludes some apex graph as a minor., Comment: 48 pages and 12 figures. arXiv admin note: text overlap with arXiv:2004.12692

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications., Comment: arXiv admin note: text overlap with arXiv:2004.12692

Let ${\cal G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. We denote by ${\cal A}_k ({\cal G})$ the set of all graphs that are $k$-apices of ${\cal G}.$ In the first paper of this series we obtained upper bounds on the size of the graphs in the minor-obstruction set of ${\cal A}_k ({\cal G})$, i.e., the minor-minimal set of graphs not belonging to ${\cal A}_k ({\cal G}).$ In this article we provide an algorithm that, given a graph $G$ on $n$ vertices, runs in $2^{{\sf poly}(k)}\cdot n^3$-time and either returns a set $S$ certifying that $G \in {\cal A}_k ({\cal G})$, or reports that $G \notin {\cal A}_k ({\cal G})$. Here ${\sf poly}$ is a polynomial function whose degree depends on the maximum size of a minor-obstruction of ${\cal G}.$ In the special case where ${\cal G}$ excludes some apex graph as a minor, we give an alternative algorithm running in $2^{{\sf poly}(k)}\cdot n^2$-time., Comment: 37 pages, 3 figures

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., Comment: Accepted for publication to SODA 2025