Acyclic

Acyclic describes structures or systems that contain no cycles. In mathematics and computer science, a cycle is a closed path that begins and ends at the same element without repeating edges or vertices (except the starting/ending vertex). A graph is called acyclic when it has no such cycles. In undirected graphs this implies that every connected component is a tree, while in directed graphs the absence of directed cycles means a topological ordering exists.

In undirected graphs, an acyclic connected graph is a tree, characterized by having exactly one less edge than its number of vertices (|E| = |V| − 1). A graph with multiple acyclic components is a forest. These structures are foundational in data organization, network design, and many combinatorial algorithms.

In directed graphs, a structure with no directed cycles is a directed acyclic graph (DAG). DAGs admit a topological ordering, which sequences vertices so that every directed edge goes from an earlier to a later vertex. This property makes DAGs central to scheduling, dependency resolution, data flow analysis, and spreadsheet recalculation models.

In chemistry, acyclic (open-chain) compounds lack rings. Acyclic hydrocarbons such as alkanes and alkenes contrast with cyclic compounds that feature ring structures. The acyclic or cyclic nature of a molecule influences its stability, reactivity, and naming conventions.

The term is also used in broader contexts to indicate freedom from circular dependencies or feedback loops. Cycle detection algorithms, such as depth-first search, are commonly employed to verify acyclicity in graphs and dependency networks.