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The $t$-onion star is the digraph obtained from a star with $2t$ leaves by replacing every edge by a triple of arcs, where in $t$ triples we orient two arcs away from the center, and in the remaining $t$ triples we orient two arcs towards the center. Note that the $t$-onion star contains, as an immersion, every digraph on $t$ vertices where each vertex has outdegree at most $2$ and indegree at most $1$, or vice versa. We investigate the structure in digraphs that exclude a fixed onion star as an immersion. The main discovery is that in such digraphs, for some duality statements true in the undirected setting we can prove their directed analogues. More specifically, we show the next two statements. There is a function $f\colon \mathbb{N}\to \mathbb{N}$ satisfying the following: If a digraph $D$ contains a set $X$ of $2t+1$ vertices such that for any $x,y\in X$ there are $f(t)$ arc-disjoint paths from $x$ to $y$, then $D$ contains the $t$-onion star as an immersion. There is a function $g\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ satisfying the following: If $x$ and $y$ is a pair of vertices in a digraph $D$ such that there are at least $g(t,k)$ arc-disjoint paths from $x$ to $y$ and there are at least $g(t,k)$ arc-disjoint paths from $y$ to $x$, then either $D$ contains the $t$-onion star as an immersion, or there is a family of $2k$ pairwise arc-disjoint paths with $k$ paths from $x$ to $y$ and $k$ paths from $y$ to $x$., Comment: 14 pages, 5 figures

For a fixed simple digraph $H$ without isolated vertices, we consider the problem of deleting arcs from a given tournament to get a digraph which does not contain $H$ as an immersion. We prove that for every $H$, this problem admits a polynomial kernel when parameterized by the number of deleted arcs. The degree of the bound on the kernel size depends on $H$., Comment: Full version of a paper accepted to ESA 2022

Tuza famously conjectured in 1981 that in a graph without k+1 edge-disjoint triangles, it suffices to delete at most 2k edges to obtain a triangle-free graph. The conjecture holds for graphs with small treewidth or small maximum average degree, including planar graphs. However, for dense graphs that are neither cliques nor 4-colorable, only asymptotic results are known. Here, we confirm the conjecture for threshold graphs, i.e. graphs that are both split graphs and cographs, and for co-chain graphs with both sides of the same size divisible by 4., Comment: 14 pages, 11 figures, Accepted to European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB) 2021

Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width., Comment: 23 pages, 4 figures

We consider the Erd\H{o}s-P\'osa property for immersions and topological minors in tournaments. We prove that for every simple digraph $H$, $k\in \mathbb{N}$, and tournament $T$, the following statements hold: (i) If in $T$ one cannot find $k$ arc-disjoint immersion copies of $H$, then there exists a set of $\mathcal{O}_H(k^3)$ arcs that intersects all immersion copies of $H$ in $T$. (ii) If in $T$ one cannot find $k$ vertex-disjoint topological minor copies of $H$, then there exists a set of $\mathcal{O}_H(k\log k)$ vertices that intersects all topological minor copies of $H$ in $T$. This improves the results of Raymond [DMTCS '18], who proved similar statements under the assumption that $H$ is strongly connected., Comment: 15 pages, 1 figure

We consider the problem of determining the inducibility (maximum possible asymptotic density of induced copies) of oriented graphs on four vertices. We provide exact values for more than half of the graphs, and very close lower and upper bounds for all the remaining ones. It occurs that, for some graphs, the structure of extremal constructions maximizing density of its induced copies is very sophisticated and complex., Comment: Added appendices on usage of updated codes and verification of the certificates

A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $O(9^k\cdot n^9)$, and also give a polynomial-time 9-approximation algorithm., Comment: Extended abstract accepted to International Symposium on Parameterized and Exact Computation (IPEC'20)