A topological drawing of a graph is fan-planar if for each edge e the edges crossing e have a common endpoint on the same side of e, and a fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by two forbidden patterns, one of which is the configuration where e is crossed by two independent edges and the other where e is crossed by incident edges with the co...

The notion of a planar st graph (also known as e-bipolar planar graph) is essentially equivalent to that of a progressive plane graph, which was introduced by Joyal and Street in the theory of graphical calculus for tensor categories. Fraysseix and Mendez have shown a bijection between equivalence classes of planar st embeddings of a directed graph G and the conjugate orders of the edge poset o...

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding...

A graph G is called “planar” if there is a way to draw it in the plane (e.g. on a piece of paper) such that there are no crossings among edges, except of course of the endpoints of edges which may coinside upon common vertices (shared by more than one edge). Given in another way: a graph G is planar if a way exists to draw it in the plane such that any of the plane's points are at most occupied...

Planar graphs originated with the studies of polytopes and of maps. The skeleton (edges) of a threedimensional polytope provide a planar graph. We obtain a planar graph from a map by representing countries by vertices, and placing edges between countries that touch each other. Assuming each country is contiguous, this gives a planar graph. While planar graphs were introduced for practical reaso...

Read’s algorithm for constructing a planar layout of a graph G produces a straight-line embedding of G, by using a sequence of triangulations. Let F denote any face of G. In this paper, Read’s algorithm is modified. A straight-line embedding is constructed in which F forms the outer face, such that its vertices lie on a convex regular polygon. It is proved that the method always works. Usually ...

We consider the capacitated domination problem, which models a service-requirement assigning scenario and which is also a generalization of the dominating set problem. In this problem, we are given a graph with three parameters defined on the vertex set, which are cost, capacity, and demand. The objective of this problem is to compute a demand assignment of least cost, such that the demand of e...

We consider the capacitated domination problem, which models a service-requirement assigning scenario and which is also a generalization of the dominating set problem. In this problem, we are given a graph with three parameters defined on the vertex set, which are cost, capacity, and demand. The objective of this problem is to compute a demand assignment of least cost, such that the demand of e...

Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is ...

In this paper, we present algorithms to produce orthogonal drawings of arbitrary graphs. As opposed to most known algorithms, we do not restrict ourselves to graphs with maximum degree 4. The best previous result gave an (m ? 1) (m 2 + 1)-grid for graphs with n nodes and m edges. We present algorithms for two scenarios. In the static scenario, the graph is given completely in advance. We produc...