– +(3.80,2.24) +(1.90,3.62) – +(3.80,2.24) +(1.90,3.62) – +(1.90,2.62) +(0.00,2.24) – +(0.95,1.93) +(1.31,0.81) – +(0.73,0.00) +(2.49,0.81) – +(3.08,0.00) +(2.85,1.93) – +(3.80,2.24) +(1.90,2.62) – +(2.49,0.81) +(2.49,0.81) – +(0.95,1.93) +(0.95,1.93) – +(2.85,1.93) +(2.85,1.93) – +(1.31,0.81) +(1.31,0.81) – +(1.90,2.62) +(0.95,1.93) node +(0.00,2.24) node +(1.90,2.62) node +(1.90,3.62) node +(2.85,1.93) node +(2.49,0.81) node +(1.31,0.81) node +(0.73,0.00) node +(3.08,0.00) node +(3.80,2.24) node ]{};

-0.6 cm

An *out-regular set*, abbreviated OR-set, of a graph $G$ that is not the empty graph is a set $Q$ of vertices such that $|N(u) \cap (
V \setminus Q )| = |N(v) \cap ( V \setminus Q)| > 0$ for every two vertices $u$ and $v$ in $Q$. We remark that if $G \ne {{\overline{K}}}_n$, then $G$ contains a vertex $v$ that is not isolated in $G$ and the set $\{v\}$ is an OR-set in $G$. Hence every graph that is not empty has an OR-set. The *out-regular number* of a non-empty graph $G$, denoted by ${\xi_{\rm or}}(G)$, is the maximum cardinality of an OR-set. By convention, if $G = {{\overline{K}}}_n$, we define ${\xi_{\rm or}}(G) = 0$. An OR-set of $G$ of cardinality ${\xi_{\rm or}}(G)$ is called an ${\xi_{\rm or}}(G)$-set. Since every graph that is not empty has an OR-set, we have 