Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such ...Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such that f(N(v)) ≥1 for every vertex v∈V . The signed total domination number of a graph G equals the minimum weight of a signed total dominating function on G . In this paper, some properties of the signed total domination number of a graph G are discussed.展开更多
For an arbitrary subset P of the reals, a function f ： V →P is defined to be a P-dominating function of a graph G = （V, E） if the sum of its function values over any closed neighbourhood is at least 1. That is, fo...For an arbitrary subset P of the reals, a function f ： V →P is defined to be a P-dominating function of a graph G = （V, E） if the sum of its function values over any closed neighbourhood is at least 1. That is, for every v ∈ V, f（N[v]） ≥ 1. The definition of total P-dominating function is obtained by simply changing ‘closed＇ neighborhood N[v] in the definition of P-dominating function to ‘open＇ neighborhood N（v）. The （total） P-domination number of a graph G is defined to be the infimum of weight w（f） = ∑v ∈ V f（v） taken over all （total） P-dominating function f. Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.展开更多
Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on ...Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi（x） ≤ 1 for each x ∈ V, is called a signed total dominating family （of functions） on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s（G）. The properties of the signed total domatic number dt^s（G） are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.展开更多
Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and are positive integers. A function is said to be a signed -edge cover of if for at least vertices of , wher...Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and are positive integers. A function is said to be a signed -edge cover of if for at least vertices of , where . The value , taking over all signed -edge covers of is called the signed -edge cover number of and denoted by . In this paper we give some bounds on the signed -edge cover number of graphs.展开更多
Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination ...Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.展开更多
文摘Let G=(V,E) be a simple graph. For any real valued function f:V →R, the weight of f is f(V) = ∑f(v) over all vertices v∈V . A signed total dominating function is a function f:V→{-1,1} such that f(N(v)) ≥1 for every vertex v∈V . The signed total domination number of a graph G equals the minimum weight of a signed total dominating function on G . In this paper, some properties of the signed total domination number of a graph G are discussed.
基金Project supported by the National Natural Science Foundation of China （Grant No.10571117）, the Shuguang Plan of Shang- hai Education Devel0pment Foundation （Grant No.06SG42）, and the Natural Science Development Foundation of Shanghai Municipal Commission of Education （Grant No.05AZ04）
文摘For an arbitrary subset P of the reals, a function f ： V →P is defined to be a P-dominating function of a graph G = （V, E） if the sum of its function values over any closed neighbourhood is at least 1. That is, for every v ∈ V, f（N[v]） ≥ 1. The definition of total P-dominating function is obtained by simply changing ‘closed＇ neighborhood N[v] in the definition of P-dominating function to ‘open＇ neighborhood N（v）. The （total） P-domination number of a graph G is defined to be the infimum of weight w（f） = ∑v ∈ V f（v） taken over all （total） P-dominating function f. Similarly, the P-edge and P-star dominating functions can be defined. In this paper we survey some recent progress on the topic of dominating functions in graph theory. Especially, we are interested in P-, P-edge and P-star dominating functions of graphs with integer values.
基金Project supported by the National Natural Science Foundation of China （Grant No.1057117）, and the Science Foundation of Shanghai Municipal Commission of Education （Grant No.05AZ04）.
文摘Let G = （V, E） be a graph, and let f ： V →{-1, 1} be a two-valued function. If ∑x∈N（v） f（x） ≥ 1 for each v ∈ V, where N（v） is the open neighborhood of v, then f is a signed total dominating function on G. A set {fl, f2,… fd} of signed d total dominating functions on G with the property that ∑i=1^d fi（x） ≤ 1 for each x ∈ V, is called a signed total dominating family （of functions） on G. The maximum number of functions in a signed total dominating family on G is the signed total domatic number on G, denoted by dt^s（G）. The properties of the signed total domatic number dt^s（G） are studied in this paper. In particular, we give the sharp bounds of the signed total domatic number of regular graphs, complete bipartite graphs and complete graphs.
文摘Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and are positive integers. A function is said to be a signed -edge cover of if for at least vertices of , where . The value , taking over all signed -edge covers of is called the signed -edge cover number of and denoted by . In this paper we give some bounds on the signed -edge cover number of graphs.
基金Supported by the National Natural Science Foundation of China (Grant No. 11061014)
文摘Let G = （V,E） be a graph.A function f ： E → {-1,1} is said to be a signed edge total dominating function （SETDF） of G if e ∈N（e） f（e ） ≥ 1 holds for every edge e ∈ E（G）.The signed edge total domination number γ st （G） of G is defined as γ st （G） = min{ e∈E（G） f（e）｜f is an SETDF of G}.In this paper we obtain some new lower bounds of γ st （G）.
