Paths

Paths are terms used across disciplines to denote routes or sequences of points connected by relationships. In everyday language it describes a navigable route between two places; in mathematics and computer science it carries precise technical meanings that can differ by context.

In graph theory, a path is a sequence of vertices with consecutive vertices joined by edges. A simple path has no repeated vertices; a cycle starts and ends at the same vertex. Path concepts include shortest paths, Hamiltonian paths (visiting each vertex exactly once), and Eulerian paths (traversing each edge once). Algorithms such as Dijkstra’s, Bellman-Ford, and A* are used to find paths in various graphs.

In topology and geometry, a path is a continuous map from the unit interval into a space, providing a notion of connectivity. Path-connectedness means any two points can be joined by a path. This notion underpins many theoretical results about space structure and continuity.

In computing, a path identifies a file or directory within a file system. Path forms include absolute paths, which specify a location from the root, and relative paths, which depend on a current directory. Different operating systems use different separators, such as / on Unix-like systems and \ on Windows.

In physical planning and recreation, a path or trail is a designed ground route for pedestrians or animals. Designers consider width, surface, gradient, drainage, signage, and accessibility to accommodate users with varying mobility.