Robertson, Seymour & Thomas (1993a) used this fact in their proof of the case k = 6 of the Hadwiger conjecture, the statement that every 6-chromatic graph has the complete graph K A graph family F has bounded local treewidth if the graphs in F obey a functional relationship between diameter and treewidth: there exists a function ƒ such that the treewidth of a diameter-d graph in F is at most ƒ(d).
The apex graphs do not have bounded local treewidth: the apex graphs formed by connecting an apex vertex to every vertex of an n × n grid graph have treewidth n and diameter 2, so the treewidth is not bounded by a function of diameter for these graphs.
The null graph is also counted as an apex graph even though it has no vertex to remove.
Apex graphs are closed under the operation of taking minors and play a role in several other aspects of graph minor theory: linkless embedding, Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph.
With this terminology, the connection between apex graphs and local treewidth can be restated as the fact that apex-minor-free graph families are the same as minor-closed graph families with bounded local treewidth.
The concept of bounded local treewidth forms the basis of the theory of bidimensionality, and allows for many algorithmic problems on apex-minor-free graphs to be solved exactly by a polynomial-time algorithm or a fixed-parameter tractable algorithm, or approximated using a polynomial-time approximation scheme.
Here, I investigated relationships between variation in individual movement performance of a marine apex predator, the tiger shark (Galeocerdo cuvier), and individual differences in morphometric aspects of body and fin shape.
My null hypothesis is the scale and complexity of individual shark movement is not related to individual variation in body and fin shape.
If G is an apex graph with apex v, and τ is the minimum number of faces needed to cover all the neighbors of v in a planar embedding of G\, then G may be embedded onto a two-dimensional surface of genus τ − 1: simply add that number of bridges to the planar embedding, connecting together all the faces into which v must be connected.
For instance, adding a single vertex to an outerplanar graph (a graph with τ = 1) produces a planar graph.