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In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. [1] If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. [2] For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3.
In graph theory, the strength of an undirected graph corresponds to the minimum ratio of edges removed/components created in a decomposition of the graph in question. It is a method to compute partitions of the set of vertices and detect zones of high concentration of edges, and is analogous to graph toughness which is defined similarly for vertex removal.
When the degree is less than or equal to 2 or the diameter is less than or equal to 1, the problem becomes trivial, solved by the cycle graph and complete graph respectively. In graph theory, the degree diameter problem is the problem of finding the largest possible graph G (in terms of the size of its vertex set V) of diameter k such that the ...
In particular, every graph property expressible as a first-order sentence can be tested in linear time for the graphs of bounded expansion. These are the graphs in which all shallow minors are sparse graphs, with a ratio of edges to vertices bounded by a function of the depth of the minor. Even more generally, first-order model checking can be ...
In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions. In a flow network , an s–t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the ...
A bramble of order four in a 3×3 grid graph, consisting of six mutually touching connected subgraphs. In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph.
The graphs can be used together to determine the economic equilibrium (essentially, to solve an equation). Simple graph used for reading values: the bell-shaped normal or Gaussian probability distribution, from which, for example, the probability of a man's height being in a specified range can be derived, given data for the adult male population.
Star graphs with m equal to 1 or 2 need only dimension 1. The dimension of a complete bipartite graph K m , 2 {\displaystyle K_{m,2}} , for m ≥ 3 {\displaystyle m\geq 3} , can be drawn as in the figure to the right, by placing m vertices on a circle whose radius is less than a unit, and the other two vertices one each side of the plane of the ...