Let D=(V, A) be a digraph with minimum indegree at least one and girth at least k, where k≥2 is an integer. In this paper, the following results were proved. A digraph D has a (k,l)-kernel if and only if its line digraph L(D) does, and the number of (k,l)-kernels in D is equal to the number of (k,l)-kernels in L(D), where 1≤l
给一张图 G，如果 C 是在包括和 | C 下面最大的 G 的一张完全的潜水艇图，潜水艇图 C 被称为 G 的一个派系|⩾
2。G 的派系横过的集合 S 是 G 的一套顶点以便 S 遇见 G 的所有派系。派系横过的数字，作为 τ
表示了 C (G), 是在 G 的一个派系横过的集合的最小的集的势。G 的派系图，作为 K 表示了(G) ，因为如果并且仅当在 G 的相应派系有非空的交叉，顶点，和二个顶点是邻近的，图被拿 G 的派系获得。让 F 是图 G 的一个班以便 F ={ G |K (G) 是一棵树 } 。在这篇论文，在有独立派系横过的集合的 F 的图被显示出并且这样 τ
C (G)/|G |⩽
为所有 G ∈
的 1/2F。
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.
Er Fang SHANMoo Young SOHNXu Dong YUANMichael A. HENNING
Let G = (V,E) be a simple graph without isolated vertices. For positive integer k, a 3-valued function f : V → {-1,0,1} is said to be a minus total k-subdominating function (MTkSF) if sum from (u∈N(v)) to f(u)≥1 for at least k vertices v in G, where N(v) is the open neighborhood of v. The minus total k-subdomination number γkt(G) equals the minimum weight of an MTkSF on G. In this paper, the values on the minus total k-subdomination number of some special graphs are investigated. Several lower bounds on γkt of general graphs and trees are obtained.
A function f:V(G)→｛-1,1｝ defined on the vertices of a graph G is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. An STDF f is minimal if there does not exist a STDF g: V(G)→ {-1,1}, f≠g, for which g(v)≤f(v) for every v∈V(G). The weight of a STDF is the sum of its function values over all vertices. The signed total domination number of G is the minimum weight of a STDF of G, while the upper signed domination number of G is the maximum weight of a minimal STDF of G. In this paper, we present sharp upper bounds on the upper signed total domination number of a nearly regular graph.
