Independent Domination Number of Certain Graphs and Their Degree Splitting Graphs

The concept of domination in graphs is an inevitable topic in the field of graph theory. The sources of domination can be seen in a wide range of real-world situations, such as radio broadcasting, computer communication networks, school bus routing, electrical power networks, influence in social networks, surveying, resource allocation, and even transporting hazardous materials. A dominating set (DS) of a finite, simple, connected, and undirected graph <img src=image/13441473_01.gif>, or simply <img src=image/13441473_02.gif> with a non-empty node set <img src=image/13441473_03.gif> and line set <img src=image/13441473_04.gif>, is a set <img src=image/13441473_05.gif> such that every node, <img src=image/13441473_06.gif> is adjacent to a node, <img src=image/13441473_07.gif>. The domination number (DN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_08.gif>, is the cardinality of a minimum DS. Similarly, set <img src=image/13441473_09.gif> of nodes is called an independent set (IS) if no two nodes in <img src=image/13441473_10.gif> are adjacent to each other. An independent dominating set (IDS) of <img src=image/13441473_02.gif> is a set <img src=image/13441473_11.gif> that is both dominating and independent in <img src=image/13441473_02.gif>. The independent domination number (IDN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_12.gif>, is the cardinality of a minimum IDS. Further, let <img src=image/13441473_02.gif> be with node set <img src=image/13441473_13.gif> where each <img src=image/13441473_14.gif> is a set of nodes having at least two nodes of the same degree and <img src=image/13441473_15.gif>. The degree splitting graph (DSG) of <img src=image/13441473_02.gif>, denoted by <img src=image/13441473_16.gif>, is formed from <img src=image/13441473_02.gif> by adding nodes <img src=image/13441473_17.gif> and joining to each node of <img src=image/13441473_14.gif> for <img src=image/13441473_18.gif>. In this paper, we derive the IDN of certain graphs such as lotus graph, line graph of sunlet graph, butterfly graph, shipping graph, ladder graph, fan graph, double fan graph, and their degree splitting graphs.