We prove that if $G$ is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of $G$ is trivial in dimension $d$ for all $d\ge 2$. We also construct a sequence $\{G_d\}$ of graphs such that this homology is nontrivial in dimension $d$ for $d\ge 1$. Finally, we show that the discrete cubical homology induced by certain coverings of $G$ equals the ordinary singular homology of a $2$-dimensional cell complex built from $G$, although in general it differs from the discrete cubical homology of the graph as a whole.

We study the partially ordered set $P(a_1,\ldots, a_n)$ of all multidegrees $(b_1,\dots,b_n)$ of monomials $x_1^{b_1}\cdots x_n^{b_n}$, which properly divide $x_1^{a_1}\cdots x_n^{a_n}$. We prove that the order complex $\Delta(P(a_1,\dots,a_n))$ of $P(a_1,\ldots a_n)$ is (nonpure) shellable by showing that the order dual of $P(a_1,\ldots,a_n)$ is $CL$-shellable. Along the way, we exhibit the poset $P(4,4)$ as a new example of a poset with $CL$-shellable order dual that is not $CL$-shellable itself. For $n = 2$, we provide the rank of all homology groups of the order complex $\Delta ( P(a_1,a_2) )$. Furthermore, we give a succinct formula for the Euler characteristic of $\Delta ( P(a_1,a_2) )$.

Several machine learning methods allow for abstaining from uncertain predictions. While being common for settings like conventional classification, abstention has been studied much less in learning to rank. We address abstention for the label ranking setting, allowing the learner to declare certain pairs of labels as being incomparable and, thus, to predict partial instead of total orders. In our method, such predictions are produced via thresholding the probabilities of pairwise preferences between labels, as induced by a predicted probability distribution on the set of all rankings. We formally analyze this approach for the Mallows and the Plackett-Luce model, showing that it produces proper partial orders as predictions and characterizing the expressiveness of the induced class of partial orders. These theoretical results are complemented by experiments demonstrating the practical usefulness of the approach.

Let P be a monotone property of directed graphs on n vertices, and let $\Delta_n^{P}$ denote the abstract simplicial complex whose simplices are the edge sets of graphs having property P. We prove the following: If "P = acyclic,' then $\Delta_n^{P}$ is homotopy equivalent to the (n-2)-sphere. If "P = not strongly connected,' then $\Delta_n^{P}$ has the homotopy type of a wedge of (n-1)! spheres of dimension 2n-4. The lattice of all posets on {1,2,...,n} plays an important role in the analysis. We also discuss some other properties of directed graphs from this point of view.