Glueloop

Glueloop is a theoretical construct used in graph theory and topology to describe a closed path formed by gluing together multiple loops at shared vertices or edges. The central operation, gluing, identifies selected elements of different loops according to a compatibility rule, yielding a single composite loop that can reuse a vertex multiple times.

Formal construction: Given a labeled directed graph G, a glueloop is an equivalence class of finite sequences of closed walks w1, w2, ..., wk in G such that wi shares a vertex with wi+1 and the joining is performed by a gluing function that matches the relevant edge labels and incidences. The resulting object is considered up to reparameterization along the sequence.

Properties: A glueloop may be reducible if it decomposes into two nontrivial glueloops; otherwise it is irreducible. Its length equals the sum of the lengths of the constituent walks, counting multiplicities. The arrangement of glue points can create self-intersections, affecting the loop's topology.

Applications and examples: In algebraic topology, glueloops can model elements of loop spaces with constrained concatenation. In network models, they represent cyclic routing that must visit a prescribed sequence of hubs. A simple example is gluing two unit loops at a common vertex to form a figure-eight; more complex glueloops glue longer sequences to produce multi-bridge cycles.

History and usage: The term glueloop appears in contemporary theoretical discussions as a descriptive label for loop-gluing constructions. Definitions vary by context, and the concept is not universally standardized across subfields.