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Strangulated graph, a graph in which every peripheral cycle is a triangle; Strongly connected graph, a directed graph in which every edge is part of a cycle; Triangle-free graph, a graph without three-vertex cycles; Even-cycle-free graph, a graph without even cycles; Even-hole-free graph, a graph without even cycles of length larger or equal to 6
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.
Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual 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]
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 ...
In mathematics, a cyclic graph may mean a graph that contains a cycle, or a graph that is a cycle, with varying definitions of cycles. See: Cycle (graph theory), a cycle in a graph; Forest (graph theory), an undirected graph with no cycles; Biconnected graph, an undirected graph in which every edge belongs to a cycle
The cycle space of a planar graph is the cut space of its dual graph, and vice versa. The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis.
More precisely, the definition may allow the graph to have induced cycles of length four, or may also disallow them: the latter is referred to as even-cycle-free graphs. [ 1 ] Addario-Berry et al. (2008) demonstrated that every even-hole-free graph contains a bisimplicial vertex (a vertex whose neighborhood is the union of two cliques ), which ...