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.
Like the apex graphs and the linkless embeddable graphs, the YΔY-reducible graphs are closed under graph minors.
And, like the linkless embeddable graphs, the YΔY-reducible graphs have the seven graphs in the Petersen family as forbidden minors, prompting the question of whether these are the only forbidden minors and whether the YΔY-reducible graphs are the same as the linkless embeddable graphs.
However, a holistic understanding of individual movement patterns is important for maintaining ecosystem diversity and function, as well as understanding evolutionary processes.
This is of particular importance for apex predators, which can impact ecosystems through trophic cascades and often display a high degree of variation in movement patterns.
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.
In fish, different species with varying body shapes differ greatly in locomotor ability.
While many studies have evaluated cross-species differences in the relationship between body morphology and locomotor performance, to date few studies have investigated for individual level differences, especially in natural systems involving large fishes.