Given a (not necessarily proper) coloring of a digraph \( c:V(D)\rightarrow\mathds{N}\), let \( OC(v)\) denote the set of colors assigned to the out-neighbors of \(v\). Similarly, let \( IC(v)\) denote the set of colors assigned to the in-neighbors of \(v\). Then \(c\) is a set coloring of \(D\) provided \((u,v) \in A(D)\) implies \( OC(u) \neq OC(v)\). Analogous to the set chromatic number of a graph given by Chartrand, \(et\) \(al.\) \([3]\), we define \( \chi_s(D) \) as the minimum number of colors required to produce a set coloring of \(D\). We find bounds for \(\chi_s(D)\) where \(D\) is a digraph and where \(D\) is a tournament. In addition we consider a second set coloring, where \((u,v) \in A(D)\) implies \( OC(u) \neq IC(v)\).

Let \(\mathcal{C}\) be a finite family of distinct boxes in \(\mathbb{R}^d\), with \(G\) the intersection graph of \(\mathcal{C}\), and let \(S = \cup\{C : C \in \mathcal{C}\}\). For each block of \(G\), assume that the corresponding members of \(\mathcal{C}\) have a staircase convex union. Then when \(S\) is staircase starshaped, its staircase kernel will be a staircase convex set. Moreover, this result (and others) will hold for more general families \(\mathcal{C}\) as well.

The domination chain \(\iota_r(G) \leq \gamma(G) \leq \iota(G) \leq \beta_o(G) \leq \Gamma(G) \leq IR(G)\), which holds for any graph \(G\), is the subject of much research. In this paper, we consider the maximum number of edges in a graph having one of these domination chain parameters equal to \(2\) through a unique realization. We show that a specialization of the domination chain still holds in this setting.