Simplicial cycles and the computation of simplicial trees
  We generalize the concept of a cycle from graphs to simplicial   complexes. We show that a simplicial cycle is either a sequence of   facets connected in the shape of a circle, or is a cone over such a   structure. We show that a simplicial tree is a connected cycle-free   simplicial complex, and use this characterization to produce an   algorithm that checks in polynomial time whether a simplicial   complex is a tree. We also present an efficient algorithm for   checking whether a simplicial complex is grafted, and therefore   Cohen-Macaulay. 
Massimo Caboara, Sara Faridi, Peter Selinger