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  2. Cycle (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Cycle_(graph_theory)

    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 ...

  3. Cycle graph - Wikipedia

    en.wikipedia.org/wiki/Cycle_graph

    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.

  4. Hamiltonian path - Wikipedia

    en.wikipedia.org/wiki/Hamiltonian_path

    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 ...

  5. Circuit rank - Wikipedia

    en.wikipedia.org/wiki/Circuit_rank

    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 ...

  6. Cycle rank - Wikipedia

    en.wikipedia.org/wiki/Cycle_rank

    The cycle rank of a directed acyclic graph is 0, while a complete digraph of order n with a self-loop at each vertex has cycle rank n.Apart from these, the cycle rank of a few other digraphs is known: the undirected path of order n, which possesses a symmetric edge relation and no self-loops, has cycle rank ⌊ ⁡ ⌋ (McNaughton 1969).

  7. Cycle basis - Wikipedia

    en.wikipedia.org/wiki/Cycle_basis

    In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any ...

  8. Directed acyclic graph - Wikipedia

    en.wikipedia.org/wiki/Directed_acyclic_graph

    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 ...

  9. Path (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Path_(graph_theory)

    A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).