相关论文: Excluding a full grid minor
The pinched sphere is the pseudo-surface $\mathbb{S}^{\circ}_0$ obtained by identifying two distinct points of the sphere. We provide a structural characterization of graphs excluding an $\mathbb{S}^{\circ}_0$-embeddable graph as a minor.…
In this paper, we classify all the finite groups $G$ such that the commuting graph $\Gamma_C(G)$, order-sum graph $\Gamma_{OS}(G)$ and non-inverse graph $\Gamma_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite…
In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the…
The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no…
A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs…
We study the structure of generalized Baumslag-Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of…
The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about…
The Kuratowski-Wagner Theorem asserts that a graph is planar if and only if it does not have either $K_{3,3}$ or $K_5$ as a minor. Using this Wagner obtained a precise description of all graphs with no $K_{3,3}$ minor and all graphs with no…
Removing all connections between two vertices s and z in a graph by removing a minimum number of vertices is a fundamental problem in algorithmic graph theory. This (s,z)-separation problem is well-known to be polynomial solvable and serves…
A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of…
For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_\alpha(G) \) matrix, proposed by Nikiforov, is…
We generalise structure tree theory, which is based on removing finitely many edges, to removing finitely many vertices. This gives a significant generalization of Tutte's tree decomposition of 2-connected graphs into 3-connected blocks.…
We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.
Halin conjectured that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. This has recently been proven by the second author. In this paper, we strengthen this result by establishing the…
We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
Given a graph $H$, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to $H$, if and only if $H$ is a forest.
We identify all minimal chordal graphs that are not circular-arc graphs, thereby resolving one of ``the main open problems'' concerning the structures of circular-arc graphs as posed by Dur{\'{a}}n, Grippo, and Safe in 2011. The problem had…
Halin conjectured 20 years ago that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. We prove Halin's conjecture. This implies a forbidden minor characterisation for the property of having…
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a…