If you’re interested in presenting an open problem in “Graphs and Matroids” at the event on December 14th, please email me at [rose.mccarty @ uwaterloo.ca] by the end of the day on December 7th (Anywhere on Earth). After that we need to finalize the schedule.
Tangles are a way to capture clusters or highly cohesive regions in graphs: a tangle in a graph declares for each low-order separation one of the two sides of that separation as ‘big’, and does so in such a way that no three ‘small’ sides together cover the graph. Most types of clusters in graphs — for instance clique- or grid-minors — give rise to such a tangle by declaring as ‘big’ for each separation the side which contains the majority of the vertices of that cluster.
An open question in the theory of tangles is whether a converse to this holds: given a tangle in a graph, can we find a set of vertices such that the ‘big’ side of each separation is precisely the side which contains the majority of the set’s vertices? A positive answer to this would validate the intuition that the sides declared as ‘big’ by a tangle are in fact big in some sense.
In joint work with Christian Elbracht and Maximilian Teegen we show that a fractional version of this is true: given a tangle we construct a weight function on the vertices such the ‘big’ side of each separation is exactly the side with higher total weight. We also show that our result extends to tangles of hypergraphs as well as to edge tangles, which are based on cuts rather than vertex separations. Curiously though one cannot achieve both extensions simultaneously: there are edge tangles of hypergraphs that are not defined by such a weight function on the vertices.