Related papers: Excluded minors for the Klein Bottle II. Cascades
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing…
The structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus is studied. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the…
We determine the excluded minors characterising the class of countable graphs that embed into some compact surface.
We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $\gamma$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an…
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of…
We find the minimum number $k=\mu'(\Sigma)$ for any surface $\Sigma$, such that every $\Sigma$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that…
In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of a large class of k-dimensional grid graphs and effective two-sided bounds for the genus of any 3-dimensional grid graph, both in terms of…
Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He…
We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface $\Sigma$ and integers $\Delta$ and $k$, determine the maximum order $N(\Delta,k,\Sigma)$ of a graph embeddable in $\Sigma$ with maximum degree…
Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular…
We show that, for every n and every surface $\Sigma$, there is a graph U embeddable on $\Sigma$ with at most cn^2 vertices that contains as minor every graph embeddable on $\Sigma$ with n vertices. The constant c depends polynomially on the…
We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal…
A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…
Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T,…
It is well known that every sufficiently large connected graph has, as an induced subgraph, $K_n$, $K_{1,n}$, or an $n$-vertex path. A 2023 paper of Allred, Ding, and Oporowski identified a set of unavoidable induced subgraphs of…
The Graph Minors Structure Theorem (GMST) of Robertson and Seymour states that for every graph $H,$ any $H$-minor-free graph $G$ has a tree-decomposition of bounded adhesion such that the torso of every bag embeds in a surface $\Sigma$…
Kuratowski proved that a finite graph embeds in the plane if it does not contain a subdivision of either K_5 or K_{3,3}, called Kuratowski subgraphs. A conjectured generalization of this result to all nonorientable surfaces says that a…
In this paper we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional…
A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly…
A general position map $f:K\to M$ of a $k$-dimensional simplicial complex to a $2k$-dimensional manifold (for $k=1$, of a graph to a surface) is a $\mathbb Z_2$-embedding if $|f\sigma \cap f\tau|$ is even for any non-adjacent $k$-faces…