Domination Number in Graphs with Minimum Degree Two

摘要

A set D of vertices of a graph G = (V,E) is called a dominating set if every vertex of Vnot in D is adjacent to a vertex of D.In 1996,Reed proved that every graph of order n with minimumdegree at least 3 has a dominating set of cardinality at most 3n/8.In this paper we generalize Reed'sresult.We show that every graph G of order n with minimum degree at least 2 has a dominating set ofcardinality at most (3n+|V_2|)/8,where V_2 denotes the set of vertices of degree 2 in G.As an applicationof the above result,we show that for k > 1,the k-restricted domination number r_k(G,y) < (3n+5k)/8for all graphs of order n with minimum degree at least 3.