Let G=(V, E) be a simple graph without an isolate. A subset T of V is a total dominating set of G if for any there exists at least one vertex such that .The total domination number γ1(G) of G is the minimum order of a total dominating set of G. This paper proves that if G is a connected graph with n≥3 vertices and minimum degree at least two.
Each vertex of a graph G = （V, E） is said to dominate every vertex in its closed neighborhood. A set S C V is a double dominating set for G if each vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double dominating number dd（G）. In this paper, new relationships between dd（G） and other domination parameters are explored and some results of [1] are extended. Furthermore, we give the Nordhaus-Gaddum-type results for double dominating number.
With positive integers r,t and n,where n≥rt and t≥2,the maximum number of edges of a simple graph of order n is estimated,which does not contain r disjoint copies of K_r for r=2 and 3.
Aim To research new characterization and circuit property of binary matroid. Methods Constract the modular pairs of hyperplanes of a a matroid. Results and Conclusion It is proved that a matroid M on finite set S is binary if and only if for any two distinct hyper-planes H1 and H2, if H1H2S ,and H1 and H2 are modular pair, then S-(H1H2) is a hyperplande .And a necessary and sufficient condition for a binary matroid to have a k-circuit is obtained.
