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We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows.…
Graph-based methods are known to be successful in many machine learning and pattern classification tasks. These methods consider semi-structured data as graphs where nodes correspond to primitives (parts, interest points, segments, etc.)…
A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is…
We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result. More precisely, it is shown that the algebraic connectivity of a surface S, defined as the supremum of a(G) over all…
We work with combinatorial maps to represent graph embeddings into surfaces up to isotopy. The surface in which the graph is embedded is left implicit in this approach. The constructions herein are proof-relevant and stated with a subset of…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…
A signed graph (SG) is a graph where edges carry sign information attached to it. The sign of a network can be positive, negative, or neutral. A signed network is ubiquitous in a real-world network like social networks, citation networks,…
Problems in scientific computing, such as distributing large sparse matrix operations, have analogous formulations as hypergraph partitioning problems. A hypergraph is a generalization of a traditional graph wherein "hyperedges" may connect…
We have developed a new embedding method for solving scalar hyperbolic conservation laws on surfaces. The approach represents the interface implicitly by a signed distance function following the typical level set method and some embedding…
Graph embedding techniques have attracted growing interest since they convert the graph data into continuous and low-dimensional space. Effective graph analytic provides users a deeper understanding of what is behind the data and thus can…
We propose the Graph Space Embedding (GSE), a technique that maps the input into a space where interactions are implicitly encoded, with little computations required. We provide theoretical results on an optimal regime for the GSE, namely a…
In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some…
Graph embedding aims at learning a vector-based representation of vertices that incorporates the structure of the graph. This representation then enables inference of graph properties. Existing graph embedding techniques, however, do not…
It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$.…
We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural…
An embedding is a mapping from a set of nodes of a network into a real vector space. Embeddings can have various aims like capturing the underlying graph topology and structure, node-to-node relationship, or other relevant information about…
Graph is an important data representation which appears in a wide diversity of real-world scenarios. Effective graph analytics provides users a deeper understanding of what is behind the data, and thus can benefit a lot of useful…
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…
Graph embedding provides an efficient solution for graph analysis by converting the graph into a low-dimensional space which preserves the structure information. In contrast to the graph structure data, the i.i.d. node embedding can be…