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

Abstract

Let $G^{\to }$ be a simple connected directed graph on $n\ge 2$ vertices and let $V^{\ast }$ be a non-empty subset of $V(G^{\to })$ and denote the undirected subgraph induced by $V^{\ast }$ by, $⟨V^{\ast }⟩.$ We show that the \emph{competition graph} of the Jaco graph $J_n(1),n\in \mathbb{N},n\ge 5,$ denoted by $C(J_n(1))$ is given by:\\ \\ $C(J_n(1))=⟨V^{\ast }{⟩}_{V^{\ast }=\{v_i|3\le i\le n-1\}}-\{v_i{v}_{m_i}|m_i=i+d_{J_n(1)}^{+}(v_i),3\le i\le n-2\}\cup \{v_1,v_2,v_n\}.$\\ \\ Further to the above, the concept of the \emph{grog number} $g(G^{\to })$ of a simple connected directed graph $G^{\to }$ on $n\ge 2$ 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.

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