Some New Results on Domination and Independent Dominating Set of Some Graphs

One area of graph theory that has been studied in great detail is dominance in graphs. Applications for dominating sets are numerous. In wireless networking, dominant sets are used to find effective paths inside ad hoc mobile networks. They have also been used in the creation of document summaries and safe electrical grid systems. A set <I>S</I>⊆<I>V</I> is said to be dominating set of <I>G</I> if for every <i>v </i>є <I>V</I>-<I>S</I> there exists a vertex <i>u</i> є <I>S</I> such that <i>uv</i> є <I>E</I>. The dominance number of <I>G</I>, represented by <i>γ</i>(<I>G</I>), is the lowest cardinality of vertices among the dominating set of <I>G</I>. A classic NP-complete decision problem in computational complexity theory determines whether, given a graph <I>G</I> and input <I>K</I>, <i>γ</i>(<I>G</I>) ≤ <I>K</I>. This is known as the dominating set issue. Consequently, it is thought that calculating <i>γ</i>(<I>G</I>) for each given graph <I>G</I> may not be possible to do with a feasible algorithm. In addition to efficient approximation tactics, there exist efficient exact techniques for various graph classes. If there are no neighboring vertices in a subset <I>S</I>, then <I>S</I>⊆<I>V</I> is an independent set. Additionally, the empty set and the subset with just one vertex are independent. An independent dominating set of <I>G</I> is a set <I>S</I> of vertices in a graph <I>G</I> that is both an independent and a dominating set of <I>G</I>. This paper's primary goal is to investigate the dominance and independent dominating set of many graphs, including the line graph, the alternate triangular belt graph, the bistar graph, the triangular snake graph, and others.