Search results
Results from the WOW.Com Content Network
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed ...
A directed cycle graph of length 8. A directed cycle graph is a directed version of a cycle graph, with all the edges being oriented in the same direction. In a directed graph, a set of edges which contains at least one edge (or arc) from each directed cycle is called a feedback arc set.
Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, [3] a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of ...
A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamiltonian-connected if for every pair of vertices there is a Hamiltonian path between the two vertices. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once.
In a planar graph, a cycle basis formed by the set of bounded faces of an embedding necessarily has this property: each edge participates only in the basis cycles for the two faces it separates. Conversely, if a cycle basis has at most two cycles per edge, then its cycles can be used as the set of bounded faces of a planar embedding of its graph.
The cycle graph of a group is not uniquely determined up to graph isomorphism; nor does it uniquely determine the group up to group isomorphism. That is, the graph obtained depends on the set of generators chosen, and two different groups (with chosen sets of generators) can generate the same cycle graph. [2]
For k = 1, the unique forbidden minor is a 3-vertex cycle graph. [5] For k = 2, the unique forbidden minor is the 4-vertex complete graph K 4. [5] For k = 3, there are four forbidden minors: K 5, the graph of the octahedron, the pentagonal prism graph, and the Wagner graph. Of these, the two polyhedral graphs are planar. [8]
If a graph has a cycle double cover, the cycles of the cover can be used to form the 2-cells of a graph embedding onto a two-dimensional cell complex. In the case of a cubic graph, this complex always forms a manifold. The graph is said to be circularly embedded onto the manifold, in that every face of the embedding is a simple cycle in the ...