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The concept of vulnerability in a communication network plays an important role when there is a disruption in the network. There exist several graph parameters that measure the vulnerability of a communication network. Domination integrity is one of the vulnerability parameters that measure the performance of a communication network. In this paper, we introduce the concept of paired domination integrity of a graph as a new measure of graph vulnerability. Let G = (V,E) be a simple, connected graph. A set of vertices in a graph G, say S, is a paired dominating set if the following two conditions are satisfied: (i) every vertex of G has a neighbor in S and (ii) the subgraph induced by S contains a perfect matching. The paired domination integrity of G, denoted by PDI(G), is defined as PDI(G) = min{|S| + m(G − S) : S is a paired dominating set of G}, where m(G − S) is the order of the largest component in the induced subgraph of G − S. In this paper, we determine few bounds relating paired domination integrity with other graph parameters and the paired domination integrity of some classes of graphs. [ABSTRACT FROM AUTHOR]

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center. [ABSTRACT FROM AUTHOR]

Vagueness and uncertainty are two distinct models are represented by Fuzzy sets and Soft sets. The combination of Soft sets and simple graphs produces soft graphs which is also an interesting concept to deal with uncertainty problems. Any communication network can be modeled as a graph whose nodes are the processors (stations) and a communication link as an edge between corresponding nodes. The stability of a communication network is a very important factor for the network designers to reconstruct the it after the failure of certain stations or communication links. Two essential quantities in an analysis of the vulnerability of a communication network are (1) the number of nodes that are not functioning and (2) the size of a maximum order of a remaining sub network within which mutual communications can still occur. C. A. Barefoot, et. al. [13] introduced the concept of integrity. The extension of such a vulnerability parameter is studied in fuzzy graphs. Since neutrosophic soft graphs are the most generalized network structure where we can define and study the importance of the vulnerability parameters is made in this manuscript. Also, we introduce the domination integrity of neutrosophic soft graphs and explain with suitable examples. Few bounds are obtained.

Let H = (V, E) be a graph and xy ∈ E (H). Then x strongly dominates y if deg(x) ⩾ deg(y). A subset S of V is said to be a strong dominating set if every node y ∈ V – S is strongly dominated by some node x in H and is denoted by sd-set. The strong domination number γs (H) is the minimum cardinality of a strong dominating set. In this paper, we introduce a new vulnerability parameter called strong domination integrity in graphs. Strong domination integrity of some families of graphs are determined and its bounds are also obtained. The proposed parameter is applied in water distribution network system to identify the influential group of nodes within the network. Fuzzy graphs can be used to model uncertain networks. By using membership values of strong arcs, strong domination integrity is extended to fuzzy graphs as a new vulnerability parameter. In this study, we investigate the strong domination integrity for complete bipartite fuzzy graphs, complete fuzzy graphs and bounds are also derived. Some basic results and theorems are obtained. This vulnerability parameter is also applied in the transportation network systems. [ABSTRACT FROM AUTHOR]

The stability of a communication network has a great importance in network design. There are several vulnerability measures used to determine the resistance of network to the disruption in this sense. Domination theory provides a model to measure the vulnerability of a graph network. A new vulnerability measure of domination integrity was introduced by Sundareswaran in his Ph.D. thesis (Parameters of vulnerability in graphs (2010)) and defined as DI(G) = min{|S| + m(G − S):S ∈ V(G)} where m(G − S) denotes the order of a largest component of graph G − S and S is a dominating set of G. The domination integrity of an undirected connected graph is such a measure that works on the whole graph and also the remaining components of graph after any break down. Here we determine the domination integrity of wheel graph W1,n, Ladder graph Ln, Sm,n, Friendship graph Fn, Thorn graph of Pn and Cn which are commonly used graph models in network design. [ABSTRACT FROM AUTHOR]

The domination integrity of graph G is defined by D I (G) = min { | S | + m (G − S) : S   is a dominating set of   G } where m (G − S) denotes the order of the largest component in G − S. This parameter is a measures of vulnerability of a graph. In this paper, we determine the domination integrity of middle graph of graph B n , graph E p t and graph H n 2 . We also obtain some results for cartesian product and corona. [ABSTRACT FROM AUTHOR]

In this paper, we propose the application of the concept of power domination integrity to an electric power network. A phasor measurement unit (PMU) is used to analyze and control the power system by measuring voltage phase in electrical nodes and transmission lines. Due to the high cost of PMUs, it is necessary to minimize the number of PMUs such that the depth of observability is ensured. Placing PMUs in a network can be formulated as a graph theoretic problem of finding the minimum number of nodes (PMUs) in a graph that has a maximum number of links with other nodes. To achieve this, the concept of domination in graph theory is applied to power networks by redefining "adjacency" of a vertex as an "observed" vertex. The power domination number identifies the number of PMUs to be placed. The proposed concept of power domination integrity gives not only the minimum number of PMUs but also identifies the optimal locations for PMU placement in an electric power network. [ABSTRACT FROM AUTHOR]

The domination integrity of a connected graph G = (V (G);E(G)) is denoted as DI(G) and defined by DI(G) = min{|S| + m(G - S)}, where S is a dominating set and m(G - S) is the order of a maximum component of G-S. This paper discusses domination integrity of line splitting graph and central graph of some graphs. [ABSTRACT FROM AUTHOR]

C.A. Barefoot, et. al. [4] introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. I(G) = min{|S| + m(G - S) : S ⊂ V (G)}, where m(G - S) denotes the order of the largest component in G - S. Unlike the connectivity measures, integrity shows not only the difficulty to break down the network but also the damage that has been caused. A subset S of V (G) is said to be an I-set if I(G) = |S| + m(G - S). We introduced a new vulnerability parameter in [4], namely domination integrity of a graph G. It is a defined as DI(G) = min{|S| + m(G - S)}, where S is a dominating set of G and m(G - S) denotes the order of the largest component in G - S. K.S. Bagga, et. al. [2] gave a formula for I(K2 x Cn). In this paper, we give a correct formula for I (K2 x Cn). We find some results on the integrity and domination integrity of gear graphs. [ABSTRACT FROM AUTHOR]

In a graph G, those dominating sets S which give minimum value for |S| + m(G--S), where m(G--S) denotes the maximum order of a component of G--S, are called dominating integrity sets of G (briefly called DI-sets of G). This concept combines two important aspects namely domination and integrity in graphs. In this paper, we show that the decision problem domination integrity is NP-complete even when restricted to planar or chordal graphs. [ABSTRACT FROM AUTHOR]

The stability of a communication network is one of the important parameters for network designers and users. A communication network can be considered to be highly vulnerable if the destruction of a few elements cause large damage and only few members are able to communicate. In a communication network several vulnerability measures like binding number, toughness, scattering number, integrity, tenacity, edge tenacity and rupture degree are used to determine the resistance of network to the disruption after the failure of certain nodes (vertices) or communication links (edges). Domination theory also provides a model to measure the vulnerability of a graph network. The domination integrity of a simple connected graph is one such measure. Here we determine the domination integrity of square graph of path as well as the graphs obtained by composition (lexicographic product) of two paths. [ABSTRACT FROM AUTHOR]