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K 4 and K 2,3: Graph minor Diestel (2000), [1] p. 107: Outer 1-planar graphs: Six forbidden minors Graph minor Auer et al. (2013) [2] Graphs of fixed genus: A finite obstruction set Graph minor Diestel (2000), [1] p. 275: Apex graphs: A finite obstruction set Graph minor [3] Linklessly embeddable graphs: The Petersen family: Graph minor [4 ...
Graph traversal is a subroutine in most graph algorithms. The goal of a graph traversal algorithm is to visit (and / or process) every node of a graph. Graph traversal algorithms, like breadth-first search and depth-first search, are analyzed using the von Neumann model, which assumes uniform memory access cost. This view neglects the fact ...
Let X be a connected, non-compact Riemann surface.A deep theorem of Heinrich Behnke and Stein (1948) asserts that X is a Stein manifold.. Another result, attributed to Hans Grauert and Helmut Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.
In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is named for Paul Erdős and András Hajnal, who first posed it as an open problem in a paper from 1977. [1]
They showed that, for an n-vertex graph G with m edges, h(G) ≤ 2 n 2 /3 − m + 1 − 1. If every graph has a polynomial Hajós number, this would imply that it is possible to prove non-colorability in nondeterministic polynomial time, and therefore imply that NP = co-NP, a conclusion considered unlikely by complexity theorists. [7]
For example, the two embedded circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus and 2-sphere represent 2-cycles. Cycles form a group under the operation of formal addition, which refers to adding cycles symbolically rather than combining them geometrically.
This two-graph is called the extension of G by x in design theoretic language. [3] In a given switching class of graphs of a regular two-graph, let Γ x be the unique graph having x as an isolated vertex (this always exists, just take any graph in the class and switch the open neighborhood of x) without the vertex x. That is, the two-graph is ...
In graph theory, the graph removal lemma states that when a graph contains few copies of a given subgraph, then all of the copies can be eliminated by removing a small number of edges. [1] The special case in which the subgraph is a triangle is known as the triangle removal lemma .
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