Related papers: Shortest path embeddings of graphs on surfaces
An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled…
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…
A connected graph $G$ with at least $2m+2n+2$ vertices is said to have property $E(m,n)$ if, for any two disjoint matchings $M$ and $N$ of size $m$ and $n$ respectively, $G$ has a perfect matching $F$ such that $M\subseteq F$ and $N\cap…
We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path…
We prove that if $f_g: (\Sigma,g) \rightarrow (\mb{S}^{2+p},\tg)$ is a smooth minimal isometric embedding of a Riemannian surface $(\Sigma,g)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area preserving conformal deformations of $g$ on…
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…
An isometric embedding of a graph into a metric space is an embedding of the vertices such that the smallest number of edges connecting any two vertices equals to the distance in the metric space between the images. In this paper, we study…
We study embeddings of a graph $G$ in a surface $S$ by considering representatives of different classes of $H_1(S)$ and their intersections. We construct a matrix invariant that can be used to detect homological invariance of elements of…
We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric $X$. Computing such an embedding (exactly or approximately) is a non-trivial task even when $X$ is the…
Let $\varphi:F_1\to F_2$ be an injective morphism of free groups. If $\varphi$ is geometric (i.e. induced by an inclusion of oriented compact connected surfaces with nonempty boundary), then we show that $\varphi$ is an isometric embedding…
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 prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some…
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…
We prove that, for every $k=1,2,...,$ every shortest-path metric on a graph of pathwidth $k$ embeds into a distribution over random trees with distortion at most $c$ for some $c=c(k)$. A well-known conjecture of Gupta, Newman, Rabinovich,…
Graph embeddings deal with injective maps from a given simple, undirected graph $G=(V,E)$ into a metric space, such as $\mathbb{R}^n$ with the Euclidean metric. This concept is widely studied in computer science, see \cite{ge1}, but also…
We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…
How much cutting is needed to simplify the topology of a surface? We provide bounds for several instances of this question, for the minimum length of topologically non-trivial closed curves, pants decompositions, and cut graphs with a given…
The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap…
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut…
A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph $G$ with no minor isomorphic to a fixed graph $H$ has a certain structure. The structure can then be exploited to deduce far-reaching…