Related papers: A Graph Theoretic Proof of the Tight Cut Lemma
Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model…
The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin…
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic…
A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…
This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing…
Decompositional parameters such as treewidth are commonly used to obtain fixed-parameter algorithms for NP-hard graph problems. For problems that are W[1]-hard parameterized by treewidth, a natural alternative would be to use a suitable…
The `lifting` or `splitting-off` operation on graphs is performed by deleting two edges sv and sw having a common end s and adding a new edge between v and w. Such a lift is considered good if it preserves a certain local edge-connectivity…
We study the set of all decompositions (clusterings) of a graph through its characterization as a set of lifted multicuts. This leads us to practically relevant insights related to the definition of a class of decompositions by must-join…
A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to…
In the Matching Cut problem we ask whether a graph $G$ has a matching cut, that is, a matching which is also an edge cut of $G$. We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there…
We develop a new method for constructing approximate decompositions of dense graphs into sparse graphs and apply it to longstanding decomposition problems. For instance, our results imply the following. Let $G$ be a quasi-random $n$-vertex…
Bidirected graphs (earlier studied by Edmonds, Johnson and, in equivalent terms of skew-symmetric graphs, by Tutte, Goldberg, Karzanov, and others) proved to be a useful unifying language for describing both flow and matching problems. In…
We develop a theory for the existence of perfect matchings in hypergraphs under quite general conditions. Informally speaking, the obstructions to perfect matchings are geometric, and are of two distinct types: 'space barriers' from convex…
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several classes of sparse graphs in the literature, for which only weak regularity lemmas were previously known. These include core-dense graphs, low threshold rank graphs,…
We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr.\ and Real (2024). Our central graph-theoretic tool is the use of tree-cut…
A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge…
This paper is the first from a series of papers that provide a characterization of maximum packings of $T$-cuts in bipartite graphs. Given a connected graph, a set $T$ of an even number of vertices, and a minimum $T$-join, an edge weighting…
In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky…
Structured prediction tasks in machine learning involve the simultaneous prediction of multiple labels. This is typically done by maximizing a score function on the space of labels, which decomposes as a sum of pairwise elements, each…
Let $t$ be a positive real number. A graph is called $t$-tough if the removal of any vertex set $S$ that disconnects the graph leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough.…