Pathsdetermined

Pathsdetermined is a term sometimes used in graph theory and algorithm design to describe a framework in which the set of meaningful or representative paths between vertices is fixed by a rule rather than chosen ad hoc. In this view, a graph G=(V,E) is paired with a path-determination rule R that assigns to each pair of vertices (s,t) a predefined subset P_R(s,t) of simple s-t paths. The rule is intended to depend only on available data such as endpoints, edge weights, or a fixed tie-breaking order, and to yield a well-defined collection of paths for every pair.

Common interpretations of path determination include: shortest-path determinism, where P_R(s,t) is the set of all shortest s-t paths under a given metric; lexicographic determinism, where paths are ordered by a fixed vertex-label sequence and a unique path is chosen; and canonical path rules, which select a single representative path according to a fixed policy. The resulting path sets are used to study properties of redundancy, reliability, and routing.

Computational considerations vary with the rule. Enumerating all paths can be exponential in size, but many practical R functions yield polynomially bounded or single-path outputs, enabling efficient queries for connectivity, alternative routing options, or path diversity.

Applications span network routing with predictable path selection, circuit design, and theoretical investigations into determinism in combinatorial structures. Note that there is no universally standardized formal definition for pathsdetermined, and the term may be used with different, context-specific meanings in different works.