Graph Bases and Diagram Commutativity

On 2-skeleta of hypercubes

“…The decomposition of the odd-dimensional hypercube 2-complex into spheres and tori is an example of decomposing an even complex into surfaces, as proposed in [4]. We believe that similar decompositions are possible for even 2-complexes related to complete graphs (i.e., the simplex).…”

Canonical sphere bases for simplicial and cubical complexes

“…Both bases B ′ (n, k) and B(n, k), when k = 1, have recursive properties not possessed by all cycle bases [19]. It is possible that some similar properties hold for k > 1 since the key condition in the recursion, that two cycles being added meet in a common nontrivial path, generalizes -the connected sum of two k-spheres remains a k-sphere.…”

“…The cycle basis B(n, 1) for the hypercube graph Q n has a weaker recursive property, called connected sum [28,19]: Any cycle in the graph can be constructed by iterating the procedure described for robust bases. The first iteration constructs a family of cycles from the original basis.…”

“…A property of cycles is called cooperative [19] if the mod-2 sum of two cycles, which intersect in a nontrivial path, has the property whenever both the summands have it. Any cooperative property which holds for all cycles in a connected-sum basis must hold for all cycles in the graph.…”

“…Cycle bases of graphs have been constructed with special properties such as fundamental (constructable from a spanning tree) [30], minimum (least total number of edges) [26], and robust or connected sum (having rearrangement properties similar to the ordering of facets in a shelling), [27,19].…”