Dominations in Intutionistic Fuzzy Directed Graphs with Applications towards Influential Graphs

Objectives: To introduce a contemporary aspect notorious as upper g-eccentric point set, upper g-eccentric number, upper g-eccentric dominating set, upper g-eccentric domination number in fuzzy graphs and related concepts. Methods: Strong arc in fuzzy graphs concepts are used to find upper g-eccentric number and upper g-eccentric domination number. Findings: The new idea upper g-eccentric domination in fuzzy graphs will leads to numerous application such as cell phone tower formation in remote areas. Novelty: Using the concepts g-distance, geodesics and strong arc in fuzzy graphs, the upper g-eccentric dominating set and upper g-eccentric domination number in fuzzy graphs are obtained. Some points are observed in this concept and some theorem related to the upper g-eccentric dominant set, its numbers are proved. Keywords: g-Eccentric Dominating Set; g-Eccentric Domination Number; Upper g-Eccentric Domination Number

Strong arcs in fuzzy graphs

The concept of domination is introduced within the context of signed fuzzy graphs (signed-FGs), along with examples, as a novel metric to evaluate graph stability under varying conditions. This metric particularly focuses on dominant sets and integrity measures, providing a well-rounded approach to assessing the structural stability of signed- FGs. The necessity of fulfilling the domination integrity condition in evaluating the performance of signed-FGs is highlighted through a discussion on its formulation and an analysis of its upper and lower bounds. An algorithm for identifying strong arcs and classifying them is presented, along with an algorithm for identifying signed fuzzy trees. Furthermore, the role of symmetry in signed-FGs is explored, revealing that symmetrical structures often correspond to higher domination integrity, thus contributing to the improved stability and predictability of the graphs. The paper also establishes important connections with classical graph varieties, such as complete graphs and their variations, demonstrating that changes in domination integrity increase with certain parameters. Additionally, real-life scenarios where these concepts are applicable serve to complement the theoretical results. The case study findings illustrate the significance of domination integrity in practical contexts by emphasizing various instances where it can be determined and utilized. Such instances include identifying independent dominant sets in path and cycle diagrams, as well as estimating the lower bounds of domination integrity in these structures. The estimation of domination integrity using block graph methods is underscored as crucial for enhancing the efficiency of signed-FG applications.

Types of arcs in a fuzzy graph

The concept of domination in graphs is an inevitable topic in the field of graph theory. The sources of domination can be seen in a wide range of real-world situations, such as radio broadcasting, computer communication networks, school bus routing, electrical power networks, influence in social networks, surveying, resource allocation, and even transporting hazardous materials. A dominating set (DS) of a finite, simple, connected, and undirected graph <img src=image/13441473_01.gif>, or simply <img src=image/13441473_02.gif> with a non-empty node set <img src=image/13441473_03.gif> and line set <img src=image/13441473_04.gif>, is a set <img src=image/13441473_05.gif> such that every node, <img src=image/13441473_06.gif> is adjacent to a node, <img src=image/13441473_07.gif>. The domination number (DN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_08.gif>, is the cardinality of a minimum DS. Similarly, set <img src=image/13441473_09.gif> of nodes is called an independent set (IS) if no two nodes in <img src=image/13441473_10.gif> are adjacent to each other. An independent dominating set (IDS) of <img src=image/13441473_02.gif> is a set <img src=image/13441473_11.gif> that is both dominating and independent in <img src=image/13441473_02.gif>. The independent domination number (IDN) of <img src=image/13441473_02.gif>, represented by <img src=image/13441473_12.gif>, is the cardinality of a minimum IDS. Further, let <img src=image/13441473_02.gif> be with node set <img src=image/13441473_13.gif> where each <img src=image/13441473_14.gif> is a set of nodes having at least two nodes of the same degree and <img src=image/13441473_15.gif>. The degree splitting graph (DSG) of <img src=image/13441473_02.gif>, denoted by <img src=image/13441473_16.gif>, is formed from <img src=image/13441473_02.gif> by adding nodes <img src=image/13441473_17.gif> and joining to each node of <img src=image/13441473_14.gif> for <img src=image/13441473_18.gif>. In this paper, we derive the IDN of certain graphs such as lotus graph, line graph of sunlet graph, butterfly graph, shipping graph, ladder graph, fan graph, double fan graph, and their degree splitting graphs.

Let G be a fuzzy graph. Let γ(G), γp(G) denote respectively the domination number, the point set domination number of a fuzzy graph. A dominating set D of a fuzzy graph is said to be a point set dominating set of a fuzzy graph if for every S⊆V-D there exists a node d∈D such that 〈S ∪ {d}〉 is a connected fuzzy graph. The minimum cardinality taken over all minimal point set dominating set is called a point set domination number of a fuzzy graph G and it is denoted by γp(G). In this paper we concentrate on the point set domination number of a fuzzy graph and obtain some bounds using the neighbourhood degree of fuzzy graphs.

In graph theory, a “dominating vertex set” is a subset of vertices in a graph such that every vertex in the graph is either a member of the subset or adjacent to a member of the subset. In other words, the vertices in the dominating set “dominate” the remaining vertices in the graph. Dominating vertex sets are important in graph theory because they can help us understand and analyze the behavior of a graph. For example, in network analysis, a set of dominant vertices may represent key nodes in a network that can influence the behavior of other nodes. Identifying dominant sets in a graph can also help in optimization problems, as it can help us find the minimum set of vertices that can control the entire graph. Now that there are theories about vagueness, it is important to define parallel ideas in vague structures, such as intuitionistic fuzzy graphs. This paper describes a better way to find dominating vertex sets (DVSs) in intuitive fuzzy graphs (IFGs). Even though there is already an algorithm for finding DVSs in IFGs, it has some problems. For example, it does not take into account the vertex volume, which has a direct effect on how DVSs are calculated. To address these limitations, we propose a new algorithm that can handle large-scale IFGs more efficiently. We show how effective and scalable the method is by comparing it to other methods and applying it to water flow. This work’s contributions can be used in many areas, such as social network analysis, transportation planning, and telecommunications.

We study three fundamental geometric optimization problems – independent set, piercing set, and dominating set – on sets of axis-parallel segments in the plane. We consider special cases in which the segments are either unit length or they are anchored on an inclined line (a line with slope \(-1\)). When the segments are anchored on both sides, we prove that all three problems are NP-complete (Throughout, we refer to NP-completeness of problems, as the decision versions of the NP-hard optimization problems we consider are all readily seen to be in NP.); NP-completeness was known for the corresponding problems with axis-parallel rectangles anchored on an inclined line (Correa et al. [4], Mudgal and Pandit [9], Pandit [10]). Further, we prove that the dominating set problem with unit segments in the plane is NP-complete. When the input segments are anchored on one side of the inclined line, there are polynomial-time algorithms for the independent set and piercing set problems.

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.Comment: 17 pages, 4 figures

A vertex subset D of a graph G = (V,E) is said to be a dominating set if every vertex in G is either in D or adjacent to some vertex in D. The minimum cardinality of such a set is the domination number, which is denoted as γ(G). In this paper, we define a sequence associated with the domination concept in graphs and studied the basic properties of the sequence in terms of various parameters of graphs. Using this sequence we order the vertices of a dominating set according its significance and propose Equally Significant Dominating (ESD) graphs. We also introduced domination related topological indices and compute their lower bounds for trees, unicyclic graphs and bicyclic graphs. All the graphs attaining the bounds are characterized.

We present an algorithm for computing the domination number of a planar graph that uses O(c√kn) time, where k is the domination number of the given planar input graph and c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number k is O(√k), and that such a tree decomposition can be found in O(√kn) time. The same technique can be used to show that the DISK DIMENSION problem (find a minimum set of faces that cover all vertices of a given plane graph) can be solved in O(c1√k n) time for c1 = 26√34. Similar results can be obtained for some variants of DOMINATING SET, e.g., INDEPENDENT DOMINATING SET.

We present a fixed-parameter algorithm that constructively solves the $k$-dominating set problem on any class of graphs excluding a single-crossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-minor-free graphs and the $K_{5}$-minor-free graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set, and a collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.

On the maximum number of minimum dominating sets in forests

A clique (convex) dominating set S of G is a 1-movable clique dominating set (resp.1-movable convex dominating set) of G if for every v ∈ S, either S \ {v} is a clique (resp.convex) dominating set or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a clique (resp.convex) dominating set of G.The minimum cardinality of a 1-movable clique (resp.1-movable convex) dominating set of G, denoted by γ 1 mcl (G) (resp.γ 1 mcon (G)), is called the 1-movable clique domination number (resp.1-movable convex domination number) of G.A 1-movable clique dominating set in G with cardinality γ 1 mcl (G) is called a γ 1 mcl -set of G.This paper aims to characterize the 1-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs.The corresponding 1-movable clique domination number of the resulting graph is then determined.Further, it is shown that the concepts of 1-movable clique domination and 1-movable convex domination are equivalent.

A set D of vertices in a graph G is an independent dominating set of G if D is an independent set and every vertex not in D is adjacent to a vertex in D. The independent domination number of G, denoted by i(G), is the minimum cardinality among all independent dominating sets of G. In this paper we show that if T is a nontrivial tree, then i(T) ≥ n(T)+γ(T)−l(T)+2/4, where n(T), γ(T) and l(T) represent the order, the domination number and the number of leaves of T, respectively. In addition, we characterize the trees achieving this new lower bound.