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By using SPQR trees to encode the possible embeddings of the planar part of an apex graph, it is possible to compute a drawing of the graph in the plane in which the only crossings involve the apex vertex, minimizing the total number of crossings, in polynomial time.
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K, every vertex is an apex.
The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex.
Any other graph G is an apex graph if and only if none of the forbidden minors is a minor of G.
These forbidden minors include the seven graphs of the Petersen family, three disconnected graphs formed from the disjoint unions of two of K Despite the complete set of forbidden minors remaining unknown, it is possible to test whether a given graph is an apex graph, and if so, to find an apex for the graph, in linear time.
And if v itself is removed, any other vertex may be chosen as the apex.
By the Robertson–Seymour theorem, because they form a minor-closed family of graphs, the apex graphs have a forbidden graph characterization.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.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.To address this hypothesis, (1) I reviewed published information on the relationship between morphology and movement in fishes and (2) I investigated the relationship between movement performance (dispersion, rate of movement, activity space, and home range) of individual tiger sharks against their morphology (pre-caudal length, caudal fin area, aspect ratio, and body condition).Data generated as a result of this thesis provides the first quantitative assessment of the potential relationship between intraspecific variation in movement performance and aspects of morphology for a large migratory fish in the wild.For, if G is an apex graph with apex v, then any contraction or removal that does not involve v preserves the planarity of the remaining graph, as does any edge removal of an edge incident to v.If an edge incident to v is contracted, the effect on the remaining graph is equivalent to the removal of the other endpoint of the edge.