Related papers: SEFE with No Mapping via Large Induced Outerplane …

In a book embedding of a graph G, the vertices of G are placed in order along a straight-line called spine of the book, and the edges of G are drawn on a set of half-planes, called the pages of the book, such that two edges drawn on a page…

Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their…

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon)…

We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are…

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine…

Low-dimensional representations, or embeddings, of a graph's nodes facilitate several practical data science and data engineering tasks. As such embeddings rely, explicitly or implicitly, on a similarity measure among nodes, they require…

A directed graph $G$ is upward planar if it admits a planar embedding such that each edge is $y$-monotone. Unlike planarity testing, upward planarity testing is NP-hard except in restricted cases, such as when the graph has the…

A matching is a set of edges without common endpoint. It was recently shown that every 1-planar graph (i.e., a graph that can be drawn in the plane with at most one crossing per edge) that has minimum degree 3 has a matching of size at…

We prove that every connected triangle-free graph on $n$ vertices contains an induced tree on $\exp(c\sqrt{\log n})$ vertices, where $c$ is a positive constant. The best known upper bound is $(2+o(1))\sqrt n$. This partially answers…

Let P be a set of n > 2 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles uvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n-3, then…

We show that for all $n \equiv 0 \pmod{6}$, $n \geq 18$, there is an orientable triangular embedding of the octahedral graph on $n$ vertices that can be augmented with handles to produce a genus embedding of the complete graph of the same…

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which…

A classical result of Koml\'os, S\'ark\"ozy and Szemer\'edi states that every $n$-vertex graph with minimum degree at least $(1/2+ o(1))n$ contains every $n$-vertex tree with maximum degree $O(n/\log{n})$ as a subgraph, and the bounds on…

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves…

Geometric embedding of graphs in a point set in the plane is a well known problem. In this paper, the complexity of a variant of this problem, where the point set is bounded by a simple polygon, is considered. Given a point set in the plane…

Graph embedding has been proven to be efficient and effective in facilitating graph analysis. In this paper, we present a novel spectral framework called NOn-Backtracking Embedding (NOBE), which offers a new perspective that organizes graph…

Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into…

A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently,…

A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges…

This paper studies optimal-area visibility representations of $n$-vertex outer-1-plane graphs, i.e. graphs with a given embedding where all vertices are on the boundary of the outer face and each edge is crossed at most once. We show that…