spanningspaths

Spanningspaths is a concept in graph theory describing a collection of paths that collectively cover the vertex set of a graph. The term is used in some literature to address problems of coverage and routing where it is desirable to traverse or reach every node using a small number of continuous traversals.

Formally, let G=(V,E) be a finite undirected graph. A spanningspath cover is a family P={P1,...,Pk} of simple paths in G such that the union of the vertex sets of the Pi equals V. Variants include vertex-disjoint spanningspath covers, and spanning paths (single spanningspath) that visit every vertex at least once; a Hamiltonian path is a spanning path with no repeated vertices.

Optimization problems include finding a spanningspath cover with the minimum number of paths or the minimum total length. These problems are generally computationally challenging and are related to the classic path cover problem and the Hamiltonian path problem. Some formulations admit polynomial-time solutions on special graph classes or under additional constraints.

Relationship to other concepts: spanningspaths generalize the idea of a path cover, and differ from spanning trees since paths may overlap and edges are a higher-level structure. They can be used to model routing, exploration, and maintenance tasks where continuous traversals must reach all locations.

Applications: network design and maintenance routing, autonomous exploration by robots, sensor coverage, and circuit layout planning. Additional research areas include approximation algorithms, exact algorithms for restricted graphs, and heuristics for practical instances.