A Static Index Number and Out Degree (In Degree) Domination Number of Total Directed Graphs

The primary objective of this research is to introduce and analyse the concept of the static index number in the context of directed and total directed graphs. This study focuses on exploring the structural properties of two newly defined components, the engender edge set and the sturdy set, which play a crucial role in characterizing connectivity within directed graphs. A directed graph is said to be a minimally connected sub digraph, and if we remove any one edge from the graph, then the graph will be disconnected. The minimum cardinality of a set <img src=image/13443264_01.gif> is said to be the Engender edge set of <img src=image/13443264_05.gif> if (i) <img src=image/13443264_02.gif> (ii) <img src=image/13443264_03.gif> is a minimally connected subdigraph. The set of edges <img src=image/13443264_04.gif> is called a sturdy set of <img src=image/13443264_05.gif> and the minimum cardinality of a sturdy set is called a static index number of a directed graph <img src=image/13443264_05.gif> and it is denoted by <img src=image/13443264_06.gif>. This parameter serves as a novel measure of structural stability and minimal connectivity in directed graphs. In this work, we establish several theoretical results and derive bounds for the static index number of both directed and total directed graphs. Furthermore, we examine its relationship with other graph-theoretic parameters, particularly the out-degree and in-degree domination numbers. The findings provide deeper insight into the interplay between domination and minimal connectivity in directed networks and contribute to the broader understanding of graph invariants in digraph theory.