设G=(V(G),E(G))是一个图,k是一个正整数.称一个顶点子集S为G的k-控制集,若V(G)S中的每个顶点在S中至少有k个邻点,我们用rk(G)表示k-控制集的最小阶数.令d1≤d2≤…≤dn为图G的度序列.当n为偶数时,度序列中位数m(G)=dn/2+1,当n为奇数时,度序列中位数m(G)=dn+1/2.一个仍未解决的Graffiti.pc猜想说:对任一n个顶点的连通图G,r2(G)≤n-m(G)+1.首先我们证明了此猜想的一个弱形式:r2(G)≤n-d1+1.此外,通过拓展此猜想在二部图上的结果,我们证明了对最小度不小于2的无三角形图G,r2(G)≤n-△(G),其中△(G)为图G的最大度.众所周知,每一个其边数不少于顶点数的图都包含一个圈.我们将此结论推广到超图上.进而得到上述猜想对所有分裂图都成立.“,”Let G =(V(G),E(G)) be a graph and k ≥ 1 integer.A subset S C V(G) is called a k-dominating set of G if each vertex of V(G) S is adjacent to at least k vertices of S in G.The k-domination number of G,denoted by γk(G),is the cardinality of a minimum k-dominating set of G.Let d1 ≤ d2 ≤ … ≤ dn be the degree sequence of G.The upper median m(G) of the degree sequence is defined as m(G) =dnn+1/2 if n is odd and m(G) =dn/2+1 ifn is even.An attractive conjecture of Graffiti.pc still remains open:for any connected graph G of order n,γ2(G) ≤ n--m(G) + 1.First we show a weaker version of the conjecture:γ2(G) ≤ n-δ(G) + 1.By extending a known result on bipartite graphs,we show that for any triangle-free graph G with δ(G) ≥ 2,γ2(G) ≤ n-△(G),where △(G) is the maximum degree of G.It is a folklore that any graph G of size m and order n with m ≥ n contains a cycle.By extending it to hypergraphs,we are able to verify the validity of the Graffiti.pc conjecture for all split graphs.