Related papers: Improved Outerplanarity Bounds for Planar Graphs
A graph is beyond-planar if it can be drawn in the plane with a specific restriction on crossings. Several types of beyond-planar graphs have been investigated, such as k-planar if every edge is crossed at most k times and RAC if edges can…
A graph is NIC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share at most one common end vertex. NIC-planarity generalizes IC-planarity, which allows a vertex to be…
We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most…
A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining…
We strengthen the connections between electrical transformations and homotopy from the planar setting---observed and studied since Steinitz---to arbitrary surfaces with punctures. As a result, we improve our earlier lower bound on the…
A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a…
In this paper, we give new, tight subexponential lower bounds for a number of graph embedding problems. We introduce two related combinatorial problems, which we call String Crafting and Orthogonal Vector crafting, and show that these…
A $k$-inner planar graph is a planar graph that has a plane drawing with at most $k$ {internal vertices}, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a $0$-inner planar graph. In…
In the minimum planarization problem, given some $n$-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a…
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph…
Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…
A plane graph is rectilinear planar if it admits an embedding-preserving straight-line drawing where each edge is either horizontal or vertical. We prove that rectilinear planarity testing can be solved in optimal $O(n)$ time for any plane…
We define the \emph{visual complexity} of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to…
A matching of a graph is a set of edges without common end vertex. A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. Recently, Biedl and Wittnebel proved that every 1-planar graph…
The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have…
The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that…
The clique number of an undirected graph $G$ is the maximum order of a complete subgraph of $G$ and is a well-known lower bound for the chromatic number of $G$. Every proper $k$-coloring of $G$ may be viewed as a homomorphism (an…
A planar map is outerplanar if all its vertices belong to the same face. We show that random uniform outerplanar maps with $n$ vertices suitably rescaled by a factor $1/ \sqrt{n}$ converge in the Gromov-Hausdorff sense to…
We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge…
Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we…