Sebastian Wiederrecht

Institute for Basic Science | IBS · DIMAG

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, oth...

\noindent By a seminal result of Valiant, computing the permanent of $(0,1)$-matrices is, in general, $\#\mathsf{P}$-hard. In 1913 P\'olya asked for which $(0,1)$-matrices $A$ it is possible to change some signs such that the permanent of $A$ equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the bia...

A colouring of a digraph as defined by Neumann-Lara in 1982 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural di...

We introduce a new kernelization tool, called rainbow matching technique, that is appropriate for the design of polynomial kernels for packing problems. Our technique capitalizes on the powerful combinatorial results of [Graf, Harris, Haxell, SODA 2021]. We apply the rainbow matching technique on two (di)graph packing problems, namely the Triangle-...

The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every $t\in\mathbb{N},$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most $c_{t}$ vertices, to graphs that can be seen as the union of some grap...

Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, oth...

The map of relations between the different directed width measures in general still has some blank spots. In this work we fill in many of these open relations for the restricted class of semicomplete digraphs. To do this we show the parametrical equivalence between directed path-width, directed tree-width, DAG-width, and Kelly-width. Moreover, we s...

A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a...

A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B and a cycle C in B is called conformal if B-V(C) has a perfect matching. We show that there do not exist two disjoint alternating paths that form a cross over a conformal cycle C in a brace B if and only if...

An orientation of a graph G is Pfaffian if every even cycle C such that G-V(C) has a perfect matching has an odd number of edges oriented in either direction of traversal. Graphs that admit a Pfaffian orientation permit counting the number of their perfect matchings in polynomial time. We consider a strengthening of Pfaffian orientations. An orient...

Matching minors are a specialised version of minors fit for the study of graphs with perfect matchings. The first major appearance of matching minors was in a result by Little who showed that a bipartite graph is Pfaffian if and only if it does not contain K3,3 as a matching minor. Later it was shown, that K3,3-matching minor free bipartite graphs...

Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can becomputed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise bas...

We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors and in terms of the structure of the cycle hypergraph to a digraph D, i.e. the hypergraph on vertex set V(D) whose hyperedges correspond to the vertex sets of directed cycles in D.

In this paper we generalise the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. We define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of the even dicycle problem. Then we...

Many well-known -hard algorithmic problems on directed graphs resist efficient parameterizations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investiga...

We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors. Moreover, we show that there is a linear relation between the hypertree-width of the dual of the cycle hypergraph of D, i. e. the hypergraph with vertices V (D) where every hyperedge corresponds to a directed cycle in D, and the directed treewidth of D. Based...

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 o...

Many well-known NP-hard algorithmic problems on directed graphs resist efficient parametrisations with most known width measures for directed graphs, such as directed treewidth, DAG-width, Kelly-width and many others. While these focus on measuring how close a digraph is to an oriented tree resp. a directed acyclic graph, in this paper, we investig...

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a...

A graph G is called matching covered if it is connected and every edge is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition that can be considered a generalisation of directed treewidth. We show that the perfect matching width of every bipartite matching covered...

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 o...

A strong edge colouring is a proper colouring of the edges of a graph such that no two edges that are incident with a common edge receive the same colour. The square of a graph G is obtained from G by adding edges between vertices of distance exactly 2. Therefore the strong edge colouring problem can be transformed to the problem of finding a prope...

The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a...

This paper studies the fundamental problem of how to reroute $k$ unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner and \emph{fast}. This scheduling problem has applications in traffic engineering in communication networks and has recently received much...

A strong edge colouring is a proper colouring of the edges of a graph such that no two edges that are incident with a common edge receive the same colour. The square of a graph G is obtained from G by adding edges between vertices of distance exactly 2. Therefore the strong edge colouring problem can be transformed to the problem of finding a prope...

A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a more general coloring concept can be defined: The strong coloring of a graph. So a k-strong coloring is a colo...

We introduce the concept of matching connectivity as a notion of connectivity in graphs admitting perfect matchings which heavily relies on the structural properties of those matchings. We generalise a result of Robertson, Seymour and Thomas for bipartite graphs with perfect matchings in order to obtain a concept of alternating paths that turns o...

In this work we investigate the chordality of squares and line graph squares of graphs. We prove a sufficient condition for the chordality of squares of graphs not containing induced cycles of length at least five. Moreover, we characterize the chordality of graph squares by forbidden subgraphs. Transferring that result to line graphs allows us to...

We initiate the theoretical study of a fundamental practical problem: how to schedule the congestion-free rerouting of $k$ flows? The input to our problem are $k$ path pairs: for each of the $k$ unsplittable flows (of a certain demand), there is an old and a new path along which the flow should be routed. As different flows can interfere on the phy...