Embedding a Graph-Like Continuum in Some Surface

Embedding a Graph-Like Continuum in Some Surface

We show that a graph-like continuum embeds in some surface if and only if it does not contain one of: a generalized thumbtack; or infinitely many K3,3’s or K5’s that are either pairwise disjoint or all have just a single point in common.

Graph Embedding

Some of the parameters used to analyze the efficiency of an embedding are dilation, expansion, edge congestion and wirelength. If e = (u, v) ∈E (G), then the length of Pf (e) in H is called the dilation of the edge e. The maximal dilation over all edges of G is called the dilation of the embedding f. The dilation of embedding G into H is the minimum dilation taken over all embeddings f of G int...

Isometric Embedding and Continuum ISOMAP

Recently, the Isomap algorithm has been proposed for learning a nonlinear manifold from a set of unorganized high-dimensional data points. It is based on extending the classical multidimensional scaling method for dimension reduction. In this paper, we present a continuous version of Isomap which we call continuum isomap and show that manifold learning in the continuous framework is reduced to ...

Embedding a Forest in a Graph

For p ≥ 1, we prove that every forest with p trees whose sizes are a1, . . . , ap can be embedded in any graph containing at least ∑p i=1(ai + 1) vertices and having a minimum degree at least ∑p i=1 ai.

Isometri Embedding and Continuum ISOMAP