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 ...

A graph is 1-planar, if it admits a 1-planar embedding, where each edge has at most one crossing. Unfortunately, testing 1-planarity of a graph is known as NP-complete. This paper considers the problem of testing 2-planarity of a graph, in particular,testing full outer-2planarity of a graph. A graph is fully-outer-2-planar, if it admits a fully-outer-2-planar embedding such that every vertex is...

∗ Corresponding author. Abstract In this paper we introduce an application of genetic algorithms (GAs) with the problem of drawing of level planar graph or hierarchical planar graph, and explore the potential use of GAs to solve this particular problem. Given a l evel planar graph, we want to find a geometric position of every vertex (layout) in a straight-line grid drawing without any edge-int...

A graph G has a planar cover if there exists a planar graph H , and a homomorphism φ : H → G that maps the neighbours of each vertex bijectively. Each graph that has an embedding in the projective plane also has a finite planar cover. Negami conjectured the converse in 1988. This conjecture holds as long as no minor-minimal non-projective graph has a finite planar cover. From the list there rem...

Generalizing Jones’s notion of a planar algebra, we have previously introduced an A2-planar algebra capturing the structure contained in the double complex pertaining to the subfactor for a finite SU(3) ADE graph with a flat cell system. We now introduce the notion of modules over an A2-planar algebra, and describe certain irreducible Hilbert A2-TL-modules. We construct an A2-graph planar algeb...

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighte...

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no K3,3 minor, and consequently for planar graphs. Carsten Thomassen made the following conjecture [1, 7]: Conjecture 1 (Thomassen) Every longest circuit of a 3-connected graph has a chord. That conjecture has been proved for planar graphs with minimum degree at...

We propose a fixed-parameter tractable algorithm for the Max-Cut problem on embedded 1-planar graphs parametrized by the crossing number k of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithm recursively reduces a 1-planar graph to at most 3 planar graphs, using edge removal and node contraction. The Max-Cut problem...

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a represen...

Steven Finch August 25, 2004 A graph of order  consists of a set of  vertices (points) together with a set of edges (unordered pairs of distinct points). Note that loops and multiple parallel edges are automatically disallowed. Two vertices joined by an edge are called adjacent. Two graphs  and  are isomorphic if there is a one-to-one map from the vertices of  to the vertices of  that pre...