Face numbers of down-sets

We compare various viewpoints on down-sets (simplicial complexes), illustrating how the combinatorial inclusion-exclusion principle may serve as an alternative to more advanced methods of studying their face numbers.Comment: 3 pages, accepted to Amer. Math. Monthly, v2: typos fixe

Conical and spherical graphs

We introduce and study the notions of conical and spherical graphs. We show that these mutually exclusive properties, which have a geometric interpretation, provide links between apparently unrelated classical concepts such as dominating sets, independent dominating sets, edge covers, and the homotopy type of an associated simplicial complex. In particular, we solve the problem of characterizing the forests whose dominating sets of minimum cardinality are also independent. To establish these connections, we prove a formula to compute the Euler characteristic of an arbitrary simplicial complex from a set of generators of its Stanley-Reisner ideal