Approximating branchwidth on parametric extensions of planarity

The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let $H_{0}$ be a graph embeddedable in the projective plane and $H_{1}$ be a graph embeddedable in the torus. We prove that every $\{H_{0},H_{1}\}$-minor free graph $G$ contains a subgraph $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{0}$ and $H_{1}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For $\{H_{0},H_{1}\}$-minor free graphs, there is a function $f\colon\mathbb{N}\to\mathbb{N}$ and a $(1+\epsilon)$-approximation algorithm for branchwidth, running in time $\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n),$ for every $\epsilon>0$

Excluding Surfaces as Minors in Graphs

We introduce an annotated extension of treewidth that measures the contribution of a vertex set $X$ to the treewidth of a graph $G.$ This notion provides a graph distance measure to some graph property $\mathcal{P}$: A vertex set $X$ is a $k$-treewidth modulator of $G$ to $\mathcal{P}$ if the treewidth of $X$ in $G$ is at most $k$ and its removal gives a graph in $\mathcal{P}.$This notion allows for a version of the Graph Minors Structure Theorem (GMST) that has no need for apices and vortices: $K_k$-minor free graphs are those that admit tree-decompositions whose torsos have $c_{k}$-treewidth modulators to some surface of Euler-genus $c_{k}.$ This reveals that minor-exclusion is essentially tree-decomposability to a ``modulator-target scheme'' where the modulator is measured by its treewidth and the target is surface embeddability. We then fix the target condition by demanding that $\Sigma$ is some particular surface and define a ``surface extension'' of treewidth, where \Sigma\mbox{-}\mathsf{tw}(G) is the minimum $k$ for which $G$ admits a tree-decomposition whose torsos have a $k$-treewidth modulator to being embeddable in $\Sigma.$We identify a finite collection $\mathfrak{D}_{\Sigma}$ of parametric graphs and prove that the minor-exclusion of the graphs in $\mathfrak{D}_{\Sigma}$ precisely determines the asymptotic behavior of {\Sigma}\mbox{-}\mathsf{tw}, for every surface $\Sigma.$ It follows that the collection $\mathfrak{D}_{\Sigma}$ bijectively corresponds to the ``surface obstructions'' for $\Sigma,$ i.e., surfaces that are minimally non-contained in $\Sigma.

Odd-Minors I: Excluding small parity breaks

Given a graph class~$\mathcal{C}$, the $\mathcal{C}$-blind-treewidth of a graph~$G$ is the smallest integer~$k$ such that~$G$ has a tree-decomposition where every bag whose torso does not belong to~$\mathcal{C}$ has size at most~$k$. In this paper we focus on the class~$\mathcal{B}$ of bipartite graphs and the class~$\mathcal{P}$ of planar graphs together with the odd-minor relation. For each of the two parameters, $\mathcal{B}$-blind-treewidth and ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth, we prove an analogue of the celebrated Grid Theorem under the odd-minor relation. As a consequence we obtain FPT-approximation algorithms for both parameters. We then provide FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded $\mathcal{B}$-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded ${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth

Cyclewidth and the Grid Theorem for Perfect Matching Width of Bipartite Graphs

A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching width since its introduction. For the restricted case of bipartite graphs, we show that perfect matching width is equivalent to directed treewidth and thus the Directed Grid Theorem by Kawarabayashi and Kreutzer for directed \treewidth implies Norine's conjecture.Comment: Manuscrip