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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 $(\delta^*_c(D)+1)$) 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., Comment: 24 pages, 2 figures

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number., Comment: 35 pages, 3 figures

We study the complexity of the problems of finding, given a graph $G$, a largest induced subgraph of $G$ with all degrees odd (called an odd subgraph), and the smallest number of odd subgraphs that partition $V(G)$. We call these parameters ${\sf mos}(G)$ and $��_{\sf odd}(G)$, respectively. We prove that deciding whether $��_{\sf odd}(G) \leq q$ is polynomial-time solvable if $q \leq 2$, and NP-complete otherwise. We provide algorithms in time $2^{O({\sf rw})} \cdot n^{O(1)}$ and $2^{O(q \cdot {\sf rw})} \cdot n^{O(1)}$ to compute ${\sf mos}(G)$ and to decide whether $��_{\sf odd}(G) \leq q$ on $n$-vertex graphs of rank-width at most ${\sf rw}$, respectively, and we prove that the dependency on rank-width is asymptotically optimal under the ETH. Finally, we give some tight bounds for these parameters on restricted graph classes or in relation to other parameters., 24 pages, 8 figures

For a fixed finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem asks, given an $n$-vertex input graph $G,$ for the minimum number of vertices that intersect all minor models in $G$ of the graphs in ${\cal F}$. by Courcelle Theorem, this problem can be solved in time $f_{{\cal F}}(tw)\cdot n^{O(1)},$ where $tw$ is the treewidth of $G$, for some function $f_{{\cal F}}$ depending on ${\cal F}$ In a recent series of articles, we have initiated the programme of optimizing asymptotically the function $f_{{\cal F}}$. Here we provide an algorithm showing that $f_{{\cal F}}(tw) = 2^{O(tw\cdot \log tw)}$ for every collection ${\cal F}$. Prior to this work, the best known function $f_{{\cal F}}$ was double-exponential in $tw$. In particular, our algorithm vastly extends the results of Jansen et al. [SODA 2014] for the particular case ${\cal F}=\{K_5,K_{3,3}\}$ and of Kociumaka and Pilipczuk [Algorithmica 2019] for graphs of bounded genus, and answers an open problem posed by Cygan et al. [Inf Comput 2017]. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. Together with our previous results providing single-exponential algorithms for particular collections ${\cal F}$ [Theor Comput Sci 2020] and general lower bounds [J Comput Syst Sci 2020], our algorithm yields the following complexity dichotomy when ${\cal F} = \{H\}$ contains a single connected graph $H,$ assuming the Exponential Time Hypothesis: $f_H(tw)=2^{\Theta(tw)}$ if $H$ is a contraction of the chair or the banner, and $f_H(tw)=2^{\Theta(tw\cdot \log tw)}$ otherwise., Comment: 51 pages, 17 figures

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$., 15 pages, 1 figure

Given two graphs $G$ and $H$, we define $\textsf{v-cover}_{H}(G)$ (resp. $\textsf{e-cover}_{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also $\textsf{v-pack}_{H}(G)$ (resp. $\textsf{v-pack}_{H}(G)$) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of $G$ that contain a minor isomaorphic to $H$. We denote by $��_r$ the graph with two vertices and $r$ parallel edges between them. When $H=��_r$, the parameters $\textsf{v-cover}_{H}$, $\textsf{e-cover}_{H}$, $\textsf{v-pack}_{H}$, and $\textsf{v-pack}_{H}$ are NP-hard to compute (for sufficiently big values of $r$). Drawing upon combinatorial results in [Minors in graphs of large $��_r$-girth, Chatzidimitriou et al., arXiv:1510.03041], we give an algorithmic proof that if $\textsf{v-pack}_{��_r}(G)\leq k$, then $\textsf{v-cover}_{��_r}(G) = O(k\log k)$, and similarly for $\textsf{v-pack}_{��_r}$ and $\textsf{e-cover}_{��_r}$. In other words, the class of graphs containing ${��_r}$ as a minor has the vertex/edge Erd��s-P��sa property, for every positive integer $r$. Using the algorithmic machinery of our proofs, we introduce a unified approach for the design of an $O(\log {\rm OPT})$-approximation algorithm for $\textsf{v-pack}_{��_r}$, $\textsf{v-cover}_{��_r}$, $\textsf{v-pack}_{��_r}$, and $\textsf{e-cover}_{��_r}$ that runs in $O(n\cdot \log(n)\cdot m)$ steps. Also, we derive several new Erd��s-P��sa-type results from the techniques that we introduce., Some of the results of this paper have been presented in WAOA 2015

For every $r \in \mathbb{N}$, let $��_r$ denote the graph with two vertices and $r$ parallel edges. The $��_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $��_r$. This notion generalizes the usual concept of girth which corresponds to the case $r=2$. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], K��hn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of $��_{r}$-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed $r$, graphs excluding as a minor the disjoint union of $k$ $��_{r}$'s have treewidth $O(k\cdot \log k)$., Some of the results of this paper have been presented in WAOA 2015

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set., Discrete Mathematics & Theoretical Computer Science ; Vol. 20 no. 1 ; 13658050