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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 ...
The 1-vs-2 cycles conjecture or 2-cycle conjecture is an unproven computational hardness assumption asserting that solving the 1-vs-2 cycles problem in the massively parallel communication model requires at least a logarithmic number of rounds of communication, even for a randomized algorithm that succeeds with high probability (having a ...
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, [1] is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley ), and uses a specified set of generators for the group.
A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. In ...
A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph . Similar notions may be defined for directed graphs , where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices ...
A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence; a path forms a cycle if the starting vertex of its first edge equals the ending vertex of its last edge. A directed acyclic graph is a directed graph that ...
The circuit rank controls the number of ears in an ear decomposition of a graph, a partition of the edges of the graph into paths and cycles that is useful in many graph algorithms. In particular, a graph is 2-vertex-connected if and only if it has an open ear decomposition. This is a sequence of subgraphs, where the first subgraph is a simple ...
An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. [4] The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree.