Related papers: Graph Skeletons and Diminishing Minors

Interpreting a node-link graph is enhanced if similar subgraphs (or motifs) are depicted in a similar manner; that is, they have the same visual form. Small motifs within graphs may be perceived to be identical when they are structurally…

In this paper, we develop a coarse analogue of treewidth. We prove that a graph $G$ admits a tree-decomposition in which each bag is contained in the union of a bounded number of balls of bounded radius, if and only if $G$ admits a…

We show that the fundamental groups of any two closed irreducible non-geometric graph-manifolds are quasi-isometric. This answers a question of Kapovich and Leeb. We also classify the quasi-isometry types of fundamental groups of…

Given two graphs $G$ and $H$, we say that $G$ contains $H$ as an induced minor if a graph isomorphic to $H$ can be obtained from $G$ by a sequence of vertex deletions and edge contractions. We study the complexity of Graph Isomorphism on…

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local…

We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs…

The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known…

We give an algorithm that, given graphs $G$ and $H$, tests whether $H$ is a minor of $G$ in time ${\cal O}_H(n^{1+o(1)})$; here, $n$ is the number of vertices of $G$ and the ${\cal O}_H(\cdot)$-notation hides factors that depend on $H$ and…

Given a graph where vertices are partitioned into $k$ terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The…

Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We…

In this paper, we characterise graphs that are quasi-isometric to graphs with bounded treewidth. Specifically, we prove that a graph is quasi-isometric to a graph with bounded treewidth if and only if it has a tree-decomposition where each…

A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We investigate the differences in complexity between Minimum Matching…

Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating C_H(G). Previous results cover only a few specific instances of this general…

A mapping $\alpha : V(G) \to V(H)$ from the vertex set of one graph $G$ to another graph $H$ is an isometric embedding if the shortest path distance between any two vertices in $G$ equals the distance between their images in $H$. Here, we…

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than…

The graph minor structure theorem by Robertson and Seymour shows that every graph that excludes a fixed minor can be constructed by a combination of four ingredients: graphs embedded in a surface of bounded genus, a bounded number of…

We present Graph Random Neural Features (GRNF), a novel embedding method from graph-structured data to real vectors based on a family of graph neural networks. The embedding naturally deals with graph isomorphism and preserves the metric…

Understanding the structure of minor-free metrics, namely shortest path metrics obtained over a weighted graph excluding a fixed minor, has been an important research direction since the fundamental work of Robertson and Seymour. A…

Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge idealof $G$. In our previous work we showed that L. Lov\'asz's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph…

The present paper is the first one in the sequence of papers about a simple class of {\em framed $4$-graphs}; the goal of the present paper is to collect some well-known results on planarity and to reformulate them in the language of {\em…