Let G be a graph with edge set E containing no isolated vertices. A 2-rainbow edge dominating function of G is a function f from E to the family of all subsets of {1, 2} such that for each edge e ∈ E with f (e) = ᶲ, where . The minimum value of a 2-rainbow edge dominating function f of G is called the 2-rainbow edge domination number of G, denoted by . A Roman {2}-edge dominating function of G is 0, 1, 2} such that for each edge e ∈ E with g(e) = 0. The minimum value of a Roman {2}-edge dominating function g of G is called the Roman {2}-edge domination number of G, denoted by . An edge dominating set of G is a set F ⊆ E such that each edge not in F is adjacent to an edge in F . The edge domination number of G is the minimum cardinality among all edge dominating sets of G. In this paper, we first prove that for any tree T . Secondly, for any tree T , we present a lower bound on (T ) in terms of and characterize the trees T for which . Finally, it is known that for any graph G, and we characterize the trees T for which .

For a connected graph G=(V,E), the dominating set in graph G is a subset of vertices F⊂V such that every vertex of V−F is adjacent to at least one vertex of F. The minimum cardinality of a dominating set of G, denoted by γ(G), is the domination number of G. The edge dominating set in graph G is a subset of edges S⊂E such that every edge of E−S is adjacent to at least one edge of S. The minimum cardinality of an edge dominating set of G, denoted by γ′(G), is the edge domination number of G. In this paper, we characterize all trees and claw-free cubic graphs with equal domination and edge domination numbers, respectively.

For a graph G(V,E) with size n any edge f∈E, a set S^'⊆E is said to be power edge dominating set of graphs G if each edge e∈E-S^' is dominated in S^' by the following rules if : (i) an edge f in G is in power edge dominating set (in short PEDS), then it dominates itself and dominates all the adjacent edges of f, (ii) an observed edge g in G has m >1 adjacent edges and if m-1 of these edges are observed earlier, then the remaining non-observed edge is also observed by g∈G. The minimum cardinality of a power edge domination number of G denoted by γ_ped^' (G). In this paper we introduce a new notion called power edge domination number and discuss the power edge domination number of corona product of certain graphs.

The secure edge dominating set of a graph G is an edge dominating set F with the property that for each edge e ∈ E − F , there exists f ∈ F adjacent to e such that ( F − { f } ) ∪ { e } is an edge dominating set. In this paper, we obtained upper bounds for edge domination and secure edge domination number for Mycielski of a tree.

The Lilavati Prize stands as a hallmark of excellence in the field of mathematics, recognizing outstanding contributions to the discipline. This study delves into the domination parameters exhibited by past Lilavati Prize winners, aiming to uncover the underlying factors contributing to their exceptional achievements. Through a meticulous analysis of their mathematical prowess, academic backgrounds, research methodologies, and societal impact, this research seeks to delineate the common traits and strategies employed by these laureates. Utilizing a combination of quantitative data analysis and qualitative assessments, we aim to provide insights into the intricate interplay between individual brilliance, collaborative endeavors, and institutional support in shaping mathematical excellence. By identifying key domination parameters among Lilavati Prize winners, this study not only offers valuable insights into the dynamics of mathematical achievement but also provides guidance for fostering future talent in the field. In this present article, the collaboration graph of Lilavathi prize winners (2010-2022) was obtained with 14 nodes and 78 links. It will be quite time consuming to share a piece of information from a given node to any other node even though the whole network is connected. Moreover it demands more resources in terms of time, memory etc. To optimize the use of above resources the idea of computing the split domination, strong split domination, strong nonsplit domination, inverse domination, inverse non split domination, edge, outer connected, total connected, connected edge domination number and just excellent assumes significance. In this paper we aimed to find some particular domination parameters namely Split, Strong, Strong split, Inverse, connected, edge connected, outer connected , just excellent; domination & total domination number for the collaboration graph of Lilavathi prize winners.

Graph theory is one of the most advanced branches of discrete mathematics with variety of applications to different branches of Science and Technology. For a positive integer ???? > 1, the involutory addition Cayley graph ????+(????????, ????????), is the graph whose vertex set is ???????? ={0,1,2,3, … , ???? − 1} and edge set ????(????????) = {???????? /????, ???? ???? ????????, ???? + ???? ???? ????????}, where ???????? = {???? ???? ???????? ∶ ????2 ≡ 1(???????????? ????)} is the set of involutory elements of ????????.By taking Involutory Addition Cayley Graphs ????+(????????, ????????) the author’s evaluated graph related Connected edge domination numbers and Entire domination numbers. In this paper, Connected edge domination number, Entire domination number of the Involutory addition cayley graphs ????+(????????, ????????) were discussed.

Intuitionistic Fuzzy Graphs (InFGs) serve as a sophisticated framework for modeling complex and uncertain phenomena across diverse domains, such as decision-making, economics, medicine, computer science, and engineering. In this research, we develop and analyse the properties of jump graphs in the context of InFGs. The vertex set of the jump graph J(G) of a graph G is defined as the edge set of G, with adjacency between vertices in J(G) established if and only if the corresponding edges in G are non-incident. We systematically construct sequences of jump graphs for InFGs through iterative processes and investigate the structural characteristics of these sequences. Moreover, we introduce the concept of an effective edge dominating set for jump graphs of InFGs and rigorously determine the effective edge domination number for certain classes of graphs. These contributions enhance the theoretical foundation of InFGs and extendtheir applicability to solving real-world problems characterized by uncertainty and complexity

Applications using domination in graphs can be found across multiple domains.When there is a set number of resources (such as fire departments and healthcare facilities) and the goal is to reduce the distance that someone must travel in order to reach the most nearby facility, domination emerges in facility positioning problems.Domination notions can also be found in land mapping problems (e.g., limiting the quantity of places where an assessor has to visit in order to obtain measurements of elevation for an entire region), tracking telecommunications or electrical infrastructure, and tasks involving spotting squads of senators. A comparable issue arises when efforts are made to minimize the quantity of facilities needed to serve every individual and the ideal distance to service is established. Considering the graph G = [ {V}, {E}]. Let the set I V {G} is a secure - vertex - edge dominating set of G, suppose every edge, y E [G], then there exists a vertex V I so that V stands up for y . i.e., The vertex in I defends the edges incident on that vertex and the edges which lie next to the incident edges. A secure - vertex - edge dominating set I of a graph G has the characteristic of being a dominant set where every vertex z V – I either follows a vertex or a vertex adjacent to the incident edges of z, x I such that (I- {x}) {z} is a dominating set. The secure - vertex - edge domination number in G is the least cardinality of secure - vertex - edge domination and is depicted by . We have commenced researching this new parameter and have found the secure - vertex - edge dominance number of several standard graphs and the middle graphs of some standard graphs. In the current analysis, the secure - vertex - edge dominance number of a few designated specific graphs such as Bull Graph, Durer Graph, Heawood Graph, Moser Spindle Graph and etc., was discovered.

A matching M in a graph G is an induced matching if the largest degree of the subgraph of G induced by M is equal to one. A dominating induced matching (DIM) of G is an induced matching that dominates every edge of G. It is well known that, if they exist, all dominating induced matchings of G are of the same size. The dominating induced matching number of G, denoted by dim ( G ) , is the size of any dominating induced matching of G. In this paper, we continue the study of dominating induced matchings. We prove that, if G has a DIM, then the induced matching number of G is equal to the independence number of its line graph L(G) and to the edge domination number of G. It is also shown that dim ( G ) ≤ 2 dim ( L ( G ) ) , provided that both G and L(G) have a DIM. We also present some bounds on dim ( G ) . In particular, for a tree T with a DIM we show that ⌈ n − l + 1 3 ⌉ ≤ dim ( T ) ≤ ⌊ n − 1 + l 3 ⌋ , where l is the number of leaves. Moreover, for a regular graph G we establish some Nordhaus-Gaddum type bounds.

Let G=V;E be a simple graph with vertex set V and edge set E. In a graph G, a subset of edges denoted by M is referred to as an edge-dominating set of G if every edge that is not in M is incident to at least one member of M. A set M⊆E is the locating edge-dominating set if for every two edges e1,e2∈E−M, the sets Ne1∩M and Ne2∩M are nonempty and different. The edge domination number γLG of G is the minimum cardinality of all edge-dominating sets of G. The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.

Let G′ be a simple, connected, and undirected (UD) graph with the vertex set M(G′) and an edge set N(G′). In this article, we define a function f:M∪N⟶0,1 as a fractional mixed dominating function (FMXDF) if it satisfies fRmx=∑yϵRmxfy≥1 for all x∈MG′∪NG′, where Rmx indicates the closed mixed neighbourhood of x, that is the set of all y∈MG′∪NG′ such that y is adjacent to x and y is incident with x and also x itself. Here, pf=∑x∈M∪Nfx is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by γfm∗G′ and is designated as the lowest poundage among all FMXDFs of G′. We compute the FMXDN of some common graphs such as paths, cycles, and star graphs, the middle graph of paths and cycles, and shadow graphs. Furthermore, we compute upper bounds for the sum of the two fractional dominating parameters, resulting in the inequality γf1′Τ+γfm∗Τ≤r+p−radΤ−α, where γf1′ and γfm∗ are the fractional edge domination number and FMXDN, respectively. Finally, we compare γfm∗ to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.

Abstract The notion of Plithogenic vertex domination and Plithogenic edge domination in Plithogenic product fuzzy graphs (PPFGs) has been newly introduced and discussed based on P-weights of P-vertices and P-edges respectively with properties, results and examples. Plithogenic vertex domination number and Plithogenic edge domination number for a Plithogenic product fuzzy graph (PPFG), and the bounds and algorithms for and the maximum and minimum values and algorithms for the same are also investigatedthe same are also investigated. Perfect Plithogenic vertex domination and perfect Plithogenic edge domination are presented as special cases of Plithogenic vertex domination and Plithogenic edge domination respectively. It is observed that the P-vertex (P-edge) with the maximum P-weight exists in every Plithogenic vertex (edge) dominating set of a Plithogenic product fuzzy graph. The novelty in these ideas of domination is that a better perception of reality can be obtained through the analysis of both, Plithogenic vertex domination and Plithogenic edge domination, in any Plithogenic product fuzzy graphical structure.

Abstract A set of edges of a graph is an edge dominating set if every edge of intersects at least one edge of , and the edge domination number is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study for graphs which are the incidence graph of some incidence structure , with an emphasis on the case when is a symmetric design. In particular, we show in this latter case that determining is equivalent to determining the largest size of certain incidence‐free sets of . Throughout, we employ a variety of combinatorial, probabilistic and geometric techniques, supplemented with tools from spectral graph theory.

Let G = (V, E) be a graph. The total onto minus edge dominating function is a function f: E → {−1, 0, 1} such that f is onto and f (N(e)) ≥ 1 for all e ∈ E(G). The total onto minus edge domination number of a graph G is a minimum weight of a set of all total onto minus edge dominating functions of G and it is denoted by .
 In this paper we discuss about the total onto minus edge domination number for some graphs : Star graph, Bistar graph, Friendship graph and Flower graph.

Domination versus edge domination on claw-free graphs

In this paper, Some bounds, theorems and results on whole edge domination number in bipolar fuzzy graph are established with support of some examples. The concepts of perfect, complete perfect and semi-perfect whole edge domination in bipolar fuzzy graph are discussed and investigated with some of their properties and also results on perfect contributed via the support of some examples.

An edge dominating set of a graph is said to be an odd (even) sum degree edge dominating set (osded (esded) - set) of G if the sum of the degree of all edges in X is an odd (even) number. The odd (even) sum degree edge domination number is the minimum cardinality taken over all odd (even) sum degree edge dominating sets of G and is defined as zero if no such odd (even) sum degree edge dominating set exists in G. In this paper, the odd (even) sum degree domination concept is extended on the co-dominating set E-T of a graph G, where T is an edge dominating set of G. The corresponding parameters co-odd (even) sum degree edge dominating set, co-odd (even) sum degree edge domination number and co-odd (even) sum degree edge domination value is defined. Further, the exact values of the above said parameters are found for some standard classes of graphs. The bounds of the co-odd (even) sum degree edge domination number are obtained in terms of basic graph terminologies. The co-odd (even) sum degree edge dominating sets are characterized. The relationships with other edge domination parameters are also studied.

The concepts of Neutrosophic secure edge domination number and neutrosophic total secure edge domination number in single valued neutrosophic graphs (SVNG) with strong arcs are introduced and analysed in this paper, and some of their properties are studied. The relationship between the neutrosophic secure edge dominance number and its inverse is presented. The concepts inverse neutrosophic total edge domination set and inverse neutrosophic total edge domination number are also defined. Some of these concepts' properties are investigated.

Let {\ \ \gamma}^e(G) be the edge domination number of a graph. A “web graph” W(s,t) is obtained from the Cartesian product of cycle graph of order s\ and path graph of order\ t. In this paper, edge domination number of the web graph is determined. Mathematical subject classification: 05C69

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.