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Let $\mathcal{H}$ be a graph class and $k\in\mathbb{N}$. We say a graph $G$ admits a \emph{$k$-identification to $\mathcal{H}$} if there is a partition $\mathcal{P}$ of some set $X\subseteq V(G)$ of size at most $k$ such that after identifying each part in $\mathcal{P}$ to a single vertex, the resulting graph belongs to $\mathcal{H}$. The graph parameter ${\sf id}_{\mathcal{H}}$ is defined so that ${\sf id}_{\mathcal{H}}(G)$ is the minimum $k$ such that $G$ admits a $k$-identification to $\mathcal{H}$, and the problem of \textsc{Identification to $\mathcal{H}$} asks, given a graph $G$ and $k\in\mathbb{N}$, whether ${\sf id}_{\mathcal{H}}(G)\le k$. If we set $\mathcal{H}$ to be the class $\mathcal{F}$ of acyclic graphs, we generate the problem \textsc{Identification to Forest}, which we show to be {\sf NP}-complete. We prove that, when parameterized by the size $k$ of the identification set, it admits a kernel of size $2k+1$. For our kernel we reveal a close relation of \textsc{Identification to Forest} with the \textsc{Vertex Cover} problem. We also study the combinatorics of the \textsf{yes}-instances of \textsc{Identification to $\mathcal{H}$}, i.e., the class $\mathcal{H}^{(k)}:=\{G\mid {\sf id}_{\mathcal{H}}(G)\le k\}$, {which we show to be minor-closed for every $k$} when $\mathcal{H}$ is minor-closed. We prove that the minor-obstructions of $\mathcal{F}^{(k)}$ are of size at most $2k+4$. We also prove that every graph $G$ such that ${\sf id}_{\mathcal{F}}(G)$ is sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$ disjoint triangles, or the \emph{$k$-marguerite} graph, that is the graph obtained by $k$ disjoint triangles by identifying one vertex of each of them into the same vertex., Comment: 18 pages, 5 figures

Given integers $k,c > 0$, we say that a digraph $D$ is $(k,c)$-linked if for every pair of ordered sets $\{s_1, \ldots, s_k\}$ and $\{t_1, \ldots, t_k\}$ of vertices of $D$, there are $P_1, \ldots, P_k$ such that for $i \in [k]$ each $P_i$ is a path from $s_i$ to $t_i$ and every vertex of $D$ appears in at most $c$ of those paths. Thomassen [Combinatorica, 1991] showed that for every fixed $k \geq 2$ there is no integer $p$ such that every $p$-strong digraph is $(k,1)$-linked. Edwards et al. [ESA, 2017] showed that every digraph $D$ with directed treewidth at least some function $f(k)$ contains a large bramble of congestion $2$ and that every $(36k^3 + 2k)$-strong digraph containing a bramble of congestion $2$ and size roughly $188k^3$ is $(k,2)$-linked. Since the directed treewidth of a digraph has to be at least its strong connectivity, this implies that there is a function $L(k)$ such that every $L(k)$-strong digraph is $(k,2)$-linked. This result was improved by Campos et al. [ESA, 2023], who showed that any $k$-strong digraph containing a bramble of size at least $2k(c\cdot k -c + 2) + c(k-1)$ and congestion $c$ is $(k,c)$-linked. Regarding the bramble, although the given bound on $f(k)$ is very large, Masa\v{r}\'{\i}k et al. [SIDMA, 2022] showed that directed treewidth $\mathcal{O}(k^{48}\log^{13} k)$ suffices if the congestion is relaxed to $8$. We first show how to drop the dependence on $c$, for even $c$, on the size of the bramble that is needed in the work of Campos et al. [ESA, 2023]. Then, by making two local changes in the proof of Masa\v{r}\'{\i}k et al. [SIDMA, 2022] we show how to build in polynomial time a bramble of size $k$ and congestion $8$ assuming that a large obstruction to directed treewidth (namely, a path system) is given. Applying these results, we show that there is a polynomial function $g(k)$ such that every $g(k)$-strong digraph is $(k,8)$-linked.

A good edge-labeling (gel for short) of a graph $G$ is a function $\lambda: E(G) \to \mathbb{R}$ such that, for any ordered pair of vertices $(x, y)$ of $G$, there do not exist two distinct increasing paths from $x$ to $y$, where ``increasing'' means that the sequence of labels is non-decreasing. This notion was introduced by Bermond et al. [Theor. Comput. Sci. 2013] motivated by practical applications arising from routing and wavelength assignment problems in optical networks. Prompted by the lack of algorithmic results about the problem of deciding whether an input graph admits a gel, called GEL, we initiate its study from the viewpoint of parameterized complexity. We first introduce the natural version of GEL where one wants to use at most $c$ distinct labels, which we call $c$-GEL, and we prove that it is NP-complete for every $c \geq 2$ on very restricted instances. We then provide several positive results, starting with simple polynomial kernels for GEL and $c$-\GEL parameterized by neighborhood diversity or vertex cover. As one of our main technical contributions, we present an FPT algorithm for GEL parameterized by the size of a modulator to a forest of stars, based on a novel approach via a 2-SAT formulation which we believe to be of independent interest. We also present FPT algorithms based on dynamic programming for $c$-GEL parameterized by treewidth and $c$, and for GEL parameterized by treewidth and the maximum degree. Finally, we answer positively a question of Bermond et al. [Theor. Comput. Sci. 2013] by proving the NP-completeness of a problem strongly related to GEL, namely that of deciding whether an input graph admits a so-called UPP-orientation., Comment: 46 pages, 16 figures

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.

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations., Comment: 58 pages, 7 figures

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that $K_t$-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are $FPT$ parameterized by bipartite treewidth. We provide the following complexity dichotomy when $H$ is a 2-connected graph, for each of $H$-Subgraph-Packing, $H$-Induced-Packing, $H$-Scattered-Packing, and $H$-Odd-Minor-Packing problem: if $H$ is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if $H$ is non-bipartite, then it is solvable in XP-time. We define 1-${\cal H}$-treewidth by replacing the bipartite graph class by any class ${\cal H}$. Most of the technology developed here works for this more general parameter., Comment: Presented in IPEC 2023

Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy) of $D$, which we denote by $\delta^*_c(D)$. It appears to be a nice generalisation of the undirected degeneracy. In this work, using this new definition of cycle-degeneracy, we extend several evidences for Cereceda's conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by $\mathcal{D}_k(D)$, is the undirected graph whose vertices are the $k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they differ on the colour of exactly one vertex. We show that $\mathcal{D}_k(D)$ has diameter at most $O_{\delta^*_c(D)}(n^{\delta^*_c(D) + 1})$ (respectively $O(n^2)$ and $(\delta^*_c(D)+1)n$) when $k$ is at least $\delta^*_c(D)+2$ (respectively $\frac{3}{2}(\delta^*_c(D)+1)$ and $2(\delta^*_c(D)+1)$). This improves known results on digraph redicolouring (Bousquet et al.). Next, we extend a result due to Feghali to digraphs, showing that $\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,\epsilon}(n(\log n)^{d-1})$ when $D$ has maximum average cycle-degree at most $d-\epsilon$. We then show that two proofs of Bonamy and Bousquet for undirected graphs can be extended to digraphs. The first one uses the digrundy number of a digraph and the second one uses the $\mathscr{D}$-width. Finally, we give a general theorem which makes a connection between the recolourability of a digraph $D$ and the recolourability of its underlying graph $UG(D)$. This result directly extends a number of results on planar graph recolouring to planar digraph redicolouring.

We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.

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

The CONTRACTION(vc) problem takes as input a graph $G$ on $n$ vertices and two integers $k$ and $d$, and asks whether one can contract at most $k$ edges to reduce the size of a minimum vertex cover of $G$ by at least $d$. Recently, Lima et al. [JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, CONTRACTION(vc) admits an XP algorithm running in time $f(d) \cdot n^{O(d)}$. They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results: 1. CONTRACTION(vc) is W[1]-hard parameterized by $k + d$. Moreover, unless the ETH fails, the problem does not admit an algorithm running in time $f(k + d) \cdot n^{o(k + d)}$ for any function $f$. In particular, this answers the open question stated in Lima et al. [JCSS 2021] in the negative. 2. It is NP-hard to decide whether an instance $(G, k, d)$ of CONTRACTION(vc) is a yes-instance even when $k = d$, hence enhancing our understanding of the classical complexity of the problem. 3. CONTRACTION(vc) can be solved in time $2^{O(d)} \cdot n^{k - d + O(1)}$. This XP algorithm improves the one of Lima et al. [JCSS 2021], which uses Courcelle's theorem as a subroutine and hence, the $f(d)$-factor in the running time is non-explicit and probably very large. On the other hard, it shows that when $k=d$, the problem is FPT parameterized by $d$ (or by $k$)., Comment: 35 pages, 5 figures

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.