Competition Graphs of Jaco Graphs and the Introduction of the Grog Number of a Simple Connected Graph

Abstract: Let $G^\rightarrow$ be a simple connected directed graph on $n \geq 2$ vertices and let $V^*$ be a non-empty subset of $V(G^\rightarrow)$ and denote the undirected subgraph induced by $V^*$ by, $\langle V^* \rangle.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1), n \in \Bbb N, n \geq 5,$ denoted by $C(J_n(1))$ is given by:\ \ $C(J_n(1)) = \langle V^* \rangle_{V^* = {v_i|3 \leq i \leq n-1}} - {v_iv_{m_i}| m_i = i + d^+_{J_n(1)}(v_i), 3 \leq i \leq n-2} \cup {v_1, v_2, v_n}.$\ \ Further to the above, the concept of the \emph{grog number} $g(G^\rightarrow)$ of a simple connected directed graph $G^\rightarrow$ on $n \geq 2$ vertices as well as the general \emph{grog number} of the underlying graph $G$, will be introduced. The \emph{grog number} measures the efficiency of an \emph{optimal predator-prey strategy} if the simple directed graph models an ecological predator-prey web.\ \ We also pose four open problems for exploratory research.