The notion of 1-plane and 1-planar graphs was first introduced by Ringel in connection with the problem of simultaneous coloring of the vertices and faces of plane graphs. In particular, in contrast to planarity testing, testing 1-planarity of a given graph is an NP-complete problem. Even ...
1-平面图的边上界 1-平面图:图G可以画在平面上满足任意条边至多交叉一次的图。是平面图的一种推广。 定理:LetGbe a simple1-planar graph with|V(G)|≥3. Then, we have|E(G)|≤4|V(G)|−8. 证明:令G是V(G)|≥3的简单1-平面图. 我们球面S2上给G加边,得到新的1-平面图,允许重边,但是不...
1-Planar GraphCombinatorial ProblemThe bondage number of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph a domination number greater than the domination number of G. In this paper, we prove that for a 1-planar graph G....
A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved that every planar graph admits an odd 9-coloring; they also conjecture...
A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. However, there are 1-planar graphs that do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a twocoloring of the edges ...
A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is...
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges,...
A graph is outer 1-planar (o1p) if it can be drawn in the plane such that all vertices are in the outer face and each edge is crossed at most once. o1p graphs generalize outerplanar graphs, which can be recognized in linear time, and specialize 1-planar graphs, whose recognition is...
1、Planar Graphs,2,2020/9/4,College of Computer Science & Technology, BUPT,Planar Graphs 平面图,A graph is called planar if it can be drawn in the plane in such a way that no two edges cross. Example of a planar graph: The clique on 4 nodes.,3,2020/9/4,College of Computer ...
[Graph Drawing, 2023] investigated the problem of cop-number for the class of $1$-planar graphs, which are graphs that can be embedded in the plane such that each edge is crossed at most once. They showed that unlike planar graphs which require just three cops, 1-planar graphs have an...