A Hamiltonian cycle (path) of a graph G is a simple cycle (path) which contains all the vertices of G. The Hamiltonian cycle problem asks whether a given graph contains a Hamiltonian cycle. It is NP-complete even for 3-connected planar graphs [3, 61. However, the problem becomes polynomial-time solvable for Cconnected planar graphs: Tutte proved that such a graph necessarily contains a Hamilton...

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computation...

A graph G is L-list colorable if for a given list assignment L= {L(v): v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for any list assignment with |L(v)| k for all v ∈ V , then G is said k-choosable. In [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325–328] and [M. Voigt, A non-3choosable plan...

We prove that the crossing number of an apex graph, i.e. a graph G from which only one vertex v has to be removed to make it planar, can be approximated up to a factor of ∆(G−v)·d(v)/2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally chosen planar embedding of a planar graph. Since the latter problem can be solved in polynomial time, this es...

graph graph and geography the geographical problem Figure 3.11: The Koenigsberg bridge problem and its graphical representation. The problem and its associated graph is shown in Figure 3.11 and this allows us to prove it is impossible to visit all the land masses, crossing each bridge only once. Graphs are a topological construct, and do not have a notion of distance associated with them in gen...

Exploring the faces of a planar graph is a prominent approach to recover from routing failures which may occur during geographic greedy forwarding heuristics. A recently studied variant of planar graph based recovery, termed geographical cluster based routing, performs face exploration along the edges of an overlay graph instead of using the network links directly. For this routing variant it h...

A non-planar graph can only be planarised if it is structurally modified. This work presents a new heuristic algorithm that uses vertices deletion to modify a non-planar graph in order to obtain a planar subgraph. The proposed algorithm aims to delete a minimum number of vertices to achieve its goal. The vertex deletion number of a graph G = (V,E) is the smallest integer k ≥ 0 such that there i...

We use methods from computability theory to answer questions about infinite planar graphs. A graph is computable if there is an algorithm which decides whether given vertices are adjacent. Having a procedure for deciding the edge set might not help compute other properties or features of the graph, however. The goal of this paper is to investigate the extent to which features related to the pla...

A graph is (k1, k2)-colorable if its vertex set can be partitioned into a graph with maximum degree at most k1 and and a graph with maximum degree at most k2. We show that every (C3, C4, C6)-free planar graph is (0, 6)-colorable. We also show that deciding whether a (C3, C4, C6)-free planar graph is (0, 3)-colorable is NP-complete.

We investigate structural properties of planar graphs without triangles or without 4-cycles, and show that every triangle-free planar graph G is edge-( (G) + 1)-choosable and that every planar graph with (G) = 5 and without 4-cycles is also edge-( (G) + 1)-choosable. c © 2003 Elsevier B.V. All rights reserved.