Lower Bounds on the Majority Domination Number of Graphs

Let G=(V,E) be a simple graph. For any real valued function f∶V→R and SV, let f(S)=∑ u∈S?f(u). A majority dominating function is a function f∶V→{-1,1} such that f(N)≥1 for at least half the vertices v∈V. Then majority domination number of a graph G is γ maj(G)=min{f(V)|f is a majority dominating function on G}. We obtain lower bounds on this parameter and generalize some results of Henning.

On Minus Paired-Domination in Graphs

The study of minus paired domination of a graph G=(V,E) is initiated. Let SV be any paired dominating set of G , a minus paired dominating function is a function of the form f∶V→{-1,0,1} such that f(v)= 1 for v∈S, f(v)≤0 for v∈V-S , and f(N)≥1 for all v∈V . The weight of a minus paired dominating function f is w(f)=∑f(v) , over all vertices v∈V . The minus paired domination number of a graph G is γ - p( G )=min{ w(f)|f is a minus paired dominating function of G }. On the basis of the minus paired domination number of a graph G defined, some of its properties are discussed.

Signed Total Domination in Graphs

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.

Roman Domination Number and Domination Number of a Tree

A Roman dominating function on a graph G = （V, E） is a function f ： V→{0, 1, 2} satisfying the condition that every vertex u for which f（u） = 0 is adjacent to at least one vertex v for which f（v） = 2. The weight of a Roman dominating function is the value f（V） = Σu∈Vf（u）. The minimum weight of a Roman dominating function on a graph G, denoted by γR（G）, is called the Roman dominating number of G. In this paper, we will characterize a tree T with γR（T） = γ（T） ＋ 3.

Signed total domatic number of a graph

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.

On Signed Domination of Grid Graph

Let G(V, E) be a finite connected simple graph with vertex set V(G). A function is a signed dominating function f : V(G)→{−1,1} if for every vertex v ∈ V(G), the sum of closed neighborhood weights of v is greater or equal to 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate the signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 6, 7 and arbitrary n.

The Generalization of Signed Domination Number of Two Classes of Graphs

Let be a graph. A function is said to be a Signed Dominating Function (SDF) if holds for all . The signed domination number . In this paper, we determine the exact value of the Signed Domination Number of graphs and for , which is generalized the known results, respectively, where and are denotes the k-th power graphs of cycle and path .

On Minus Domination and Signed Domination in Graphs

In this paper we obtain some lower bounds for minus and signed domination numbers. We also prove and generalize a conjecture on the minus domination number for bipartite graph of order n, which was proposed by Jean Dunbar et al [1].

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V (G) and an edge set E(G). A total signed domination function of G is a function f : V (G) ∪ E(G) → {?1, 1}. The weight of f is w(f) = Σ x∈V(G)∪E(G) f(x). For an element x ∈ V (G) ∪ E(G), we define $f[x] = \sum\nolimits_{y \in N_T [x]} {f(y)} $ . A total signed domination function of G is a function f : V (G) ∪ E(G) → {?1, 1} such that f[x] ? 1 for all x ∈ V (G) ∪ E(G). The total signed domination number γ s * (G) of G is the minimum weight of a total signed domination function on G.In this paper, we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values of γ s * (G) when G is C n and P n .

The Spatio-temporal Pattern of Regional Land Use Change and Eco-environmental Responses in Jiangsu, China

Land use change and its eco-environmental responses are foci in geographical research. As a region with uneven economic development, land use change and eco-environmental responses across Jiangsu Province are relevant to China＇s overall development pattern. The external function of regional land use changes during different stages of economic development. In this study, we proposed a novel classification system based on the dominant function of land use according to ＂production-ecology-life＂, and then analyzed land use change and regional eco-environmental responses from a functional perspective of regional development. The results showed that from 1985 to 2008, land use change features in Jiangsu were that productive land area decreased and eco- logical and living land areas increased. Land use changes in southern Jiangsu were the most dramatic. In southern and central parts of Jiangsu the agricultural production function weakened and urban life service function strengthened; in northern Jiangsu, the mining production function＇s comparative advantage highlighted that the rural life service function was weakening. Ecological environmental quality decreased slightly in Jiangsu and its three regions. The maximum contribution rate to ecological environmental change occurred in southern Jiangsu and the minimum rate was located in the north. Eco-environmental quality deteriorated in southern and central Jiangsu, related to expanding construction land in urban and rural areas. Ecological environmental quality deterioration in northern Jiangsu is probably due to land development and consolidation. The main reason for improvements in regional ecological environments is that agricultural production land was converted to water ecological land across Jiangsu.

On Signed Edge Total Domination Numbers of Graphs

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）.

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G = （V, E） be a simple graph. A function f ： E → {＋1,-1} is called a signed cycle domination function （SCDF） of G if ∑e∈E（C） f（e） ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc（G） = min{∑e∈E f（e）｜ f is an SCDF of G}. This paper will characterize all maxima] planar graphs G with order n ≥ 6 and γ′sc（G） =n.

Delimiting Ecological Space and Simulating Spatial-temporal Changes in Its Ecosystem Service Functions based on a Dynamic Perspective: A Case Study on Qionglai City of Sichuan Province, China

Delimiting ecological space scientifically and making reasonable predictions of the spatial-temporal trend of changes in the dominant ecosystem service functions(ESFs) are the basis of constructing an ecological protection pattern of territorial space, which has important theoretical significance and application value. At present, most research on the identification, functional partitioning and pattern reconstruction of ecological space refers to the current ESFs and their structural information, which ignores the spatial-temporal dynamic nature of the comprehensive and dominant ESFs, and does not seriously consider the change simulation in the dominant ESFs of the future ecological space. This affects the rationality of constructing an ecological space protection pattern to some extent. In this study, we propose an ecological space delimitation method based on the dynamic change characteristics of the ESFs, realize the identification of the ecological space range in Qionglai City and solve the problem of ignoring the spatial-temporal changes of ESFs in current research. On this basis, we also apply the Markov-CA model to integrate the spatial-temporal change characteristics of the dominant ESFs, successfully realize the simulation of the spatial-temporal changes in the dominant ESFs in Qionglai City’s ecological space in 2025, find a suitable method for simulating ecological spatial-temporal changes and also provide a basis for constructing a reasonable ecological space protection pattern. This study finds that the comprehensive quantity of ESF and its annual rate of change in Qionglai City show obvious dynamics, which confirms the necessity of considering the dynamic characteristics of ESFs when identifying ecological space. The areas of ecological space in Qionglai city represent 98307 ha by using the ecological space identification method proposed in this study, which is consistent with the ecological spatial distribution in the local ecological civilization construction plan. This confirms the reliability of the ecological space identification method based on the dynamic characteristics of the ESFs. The results also show that the dominant ESFs in Qionglai City represented strong non-stationary characteristics during 2003–2019,which showed that we should fully consider the influence of the dynamics in the dominant ESFs on the future ESF pattern during the process of constructing the ecological spatial protection pattern. The Markov-CA model realized the simulation of spatial-temporal changes in the dominant ESFs with a high precision Kappa coefficient of above 0.95, which illustrated the feasibility of using this model to simulate the future dominant ESF spatial pattern. The simulation results showed that the dominant ESFs in Qionglai will still undergo mutual conversions during 2019–2025 due to the effect of the their non-stationary nature. The ecological space will still maintain the three dominant ESFs of primary product production, climate regulation and hydrological regulation in 2025, but their areas will change to 32793 ha, 52490 ha and 13024 ha, respectively. This study can serve as a scientific reference for the delimitation of the ecological conservation redline, ecological function regionalization and the construction of an ecological spatial protection pattern.

EXISTENCE OF 0-1 UNIVERSAL MINIMAL TOTAL DOMINATING FUNCTIONS

In this paper, we study the existence of 0-1 universal minimal total dominating functions in a graph. We establish a formulation of linear inequalities to characterize universal minimal total dominating functions and show that for a kind of graphs whose adjacent matrices are balanced, the existence of universal minimal total dominating functions coincides with that of 0-1 ones. It is also proved that for general graphs, the problem of testing the existence of 0-1 universal minimal total dominating functions is NP-hard.

On the Mixed Minus Domination in Graphs

Let G=(V,E)be a graph,for an element x∈V∪E,the open total neighborhood of x is denoted by N_(t)(x)={y|y is adjacent to x or y is incident with x,y∈V∪E},and Nt[x]=Nt(x)∪{x}is the closed one.A function f:V(G)∪E(G)→{−1,0,1}is said to be a mixed minus domination function(TMDF)of G if∑_(y∈Nt[x])f(y)≥1 holds for all x∈V(G)∪E(G).The mixed minus domination numberγ′_(tm)(G)of G is defined as γ′_(tm)(G)=min{∑x∈V∪E f(x)|f is a TMDF of G.In this paper,we obtain some lower bounds of the mixed minus domination number of G and give the exact values ofγ′_(tm)(G)when G is a cycle or a path.

SOME RESULTS ON UNIVERSAL MINIMAL TOTAL DOMINATING FUNCTIONS

In this paper, we introduce the concepts of redundant constraint and exceptional vertex which play an important role in the characterization of universal minimal total dominating functions (universal MTDFs), and establish some further results on universal MTDFs in general graphs. By extending these results to trees, we get a necessary and sufficient condition for universal MTDFs and show that there is a good algorithm for deciding whether a given tree has a universal MTDF.