Some Studies on Dominating Sets and Restrained Dominating Sets

In general an undirected graph is an interval graph (IG), if the vertex set V can be put into one-to-one correspondence with a set of intervals on the real line , such that two vertices are adjacent in G if and only if their corresponding intervals have non-empty intersection. The set I is called an interval representation of G and G is referred to as the intersection graph Let be any interval family where, each is an interval on the real line and , for . Here is called the left end point labeling and is the right end point labeling of . Circular - arc graphs are introduced as generalization of Interval graphs. If we bend the real line into a circle, then any family of intervals of the real line is transformed into a family of arcs of the circle. Therefore, every interval graph is a circular - arc graph. But, the converse need not be true. However both these classes of graphs have received considerable attention in the literature in recent years and have been studied extensively. A circular - arc graph is the intersection graph of a set of arcs on the circle.