Through researching into the independent and dominating set in graph theory, this paper has put forward a concept of independent-dominating set and has expounded and proved the internal relationship among the following three concepts: independent-dominating set, maximal independent set and minimal dominating set. On the basis of arranging the vertices set of a given graph in a reasonable sequence，this paper has given out some optimun solving algorithms with a low complexity O(n³)，which has settled satisfactorily and efficiently the solving problem of maximum independent (-dominating) set and minimum (independent-) dominating set in graph theory. The method and the algorithms provided in this paper can certainly be used for reference in studying the problems of graph coloring, covering and matching and so on.