The intersection graph of a set of arcs on the circle is called a circular-arc graph. Circular-arc has one vertex for each arc in the set and an edge between every pair of vertices corresponding to arcs that intersect. Let 1 2 { , ,....., } C c c c = n be family of arcs on a circle. In this paper we are taking circular arcs such that if we remove 1 c then there will be a disconnection between left end side intersecting arc of 1 c and right end side intersecting arcs of 1 c .We are writing an algorithm to find an inverse of dominating set with respect to a minimum dominating set of a circular-arc family.