Introduction to the De-Jaenisch Number of a Graph

This paper introduces a new novel graph parameter, called the \( \textit{De-Jaenisch number} \) of a graph. The parameter is inspired by the classical Queens problem of Carl Ferdinand de Jaenisch which in itself is inherently linked to domination theory in graphs. The De-Jaenisch number quantifies the stability of minimal dominating sets under vertex insertions along edges. We formally define the lower and upper De-Jaenisch numbers, \( \psi^-(G) \) and \( \psi^+(G) \) and establish foundational properties of these parameters. Exact values are determined for families of graphs. The results presented here lay the groundwork for further theoretical developments and applications of this parameter in domination theory and related areas.