Related papers: Universal Slope Sets for 1-Bend Planar Drawings
A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote…
We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…
A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…
A subset $M$ of the edges of a graph or hypergraph is hitting if $M$ covers each vertex of $H$ at least once, and $M$ is $t$-shallow if it covers each vertex of $H$ at most $t$ times. We consider the existence of shallow hitting edge sets…
For every $n\in \mathbb{N}$, we present a set $S_n$ of $O(n^{3/2}\log n)$ points in the plane such that every planar 3-tree with $n$ vertices has a straight-line embedding in the plane in which the vertices are mapped to a subset of $S_n$.…
The $d$-independence number of a graph $G$ is the largest possible size of an independent set $I$ in $G$ where each vertex of $I$ has degree at least $d$ in $G$. Upper bounds for the $d$-independence number in planar graphs are well-known…
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.
A graph $G$ is said to be $\mathcal H(n,\Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. It is known that for any $\varepsilon > 0$ and any natural number $\Delta$ there exists $c > 0$ such…
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…
We show that every graph of maximum degree three can be drawn in three dimensions with at most two bends per edge, and with 120-degree angles between any two edge segments meeting at a vertex or a bend. We show that every graph of maximum…
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…
We consider the problem of untangling a given (non-planar) straight-line circular drawing $\delta_G$ of an outerplanar graph $G=(V, E)$ into a planar straight-line circular drawing by shifting a minimum number of vertices to a new position…
A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite $1$-planar graphs with prescribed numbers of vertices in partite sets. Bipartite…
A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to $k$-planar graphs, which can be drawn in the plane with at…
In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a…
The complete bipartite graph $K_{1,3}$ is called a claw. The properties of claw-free graphs have attracted considerable attention, with research on claw-free planar graphs tracing back to Plummer's work in 1989. In this paper, we extend…
A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident…
We study the upward point-set embeddability of digraphs on one-sided convex point sets with at most 1 bend per edge. We provide an algorithm to compute a 1-bend upward point-set embedding of outerplanar $st$-digraphs on arbitrary one-sided…
We give an upper bound on the number of perfect matchings in an undirected simple graph $G$ with an even number of vertices, in terms of the degrees of all the vertices in $G$. This bound is sharp if $G$ is a union of complete bipartite…
A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to turn $\delta$ into a plane…