Edge Domination and Secure Edge Domination in Mycielski Graph of Trees

For a connected graph $$G = (V, E)$$ , a subset F of E is an edge dominating set (resp. a total edge dominating set) if every edge in $$E-F$$ (resp. in E) is adjacent to at least one edge in F, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number) of G, denoted by $$\gamma '(G)$$ (resp. $$\gamma '_t(G)$$ ). In the present paper, we study a parameter, called the semitotal edge domination number, which is squeezed between $$\gamma '(G)$$ and $$\gamma '_t(G)$$ . A semitotal edge dominating set is an edge dominating set S such that, for every edge e in S, there exists such an edge $$e'$$ in S that e either is adjacent to $$e'$$ or shares a common neighbor edge with $$e'$$ . The semitotal edge domination number, denoted by $$\gamma ^{'}_{st}(G)$$ , is the minimum cardinality of a semitotal edge dominating set of G. In this paper, we prove that the problem of deciding whether $$\gamma ^{'}(G)=\gamma ^{'}_{st}(G)$$ or $$\gamma _t^{'}(G)=\gamma ^{'}(G)$$ is NP-hard even when restricted to planar graphs with maximum degree 4. We also characterize trees with equal edge domination and semitotal edge domination numbers (Pan et al. in The complexity of total edge domination and some related results on trees, J Comb Optim, 2020, https://doi.org/10.1007/s10878-020-00596-y , we characterized trees with equal edge domination and total edge domination numbers).

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 set $$S $$ (of vertices) of a graph $$G$$ is termed a dominating set of $$G$$ if each vertex in $$V - S$$ is adjacent to a node in $$ S$$. A dominating set $$ S$$ such as the subgraph induced by $$S$$ has an isolated vertex is termed an isolate dominating set and also the minimum count of an isolate dominating set is termed the isolate domination number of $$G$$ and it is represented by $$\gamma_{is} (G)$$. A subset $$X \subseteq E $$ is said to be an edge dominating set if each edge in $$X - E $$ is adjacent to some edge in $$S$$. The edge domination number is that the count of the smallest edge dominating set of $$G$$ and is employed by $$\gamma^{\prime}$$. A collection of edges $$X$$ of $$E$$ is claimed to be a perfect edge dominating set if each edge not in $$ X$$ is adjacent to precisely one edge in $$X$$. The ideal edge domination number is that the minimum cardinality has taken perfect edge dominating sets of $$G$$ and is denoted by $$\gamma_{p}^{^{\prime}}$$. In this paper, we initiate the survey of bondage related to isolate domination. The isolate bondage number $$b_{is} (G)$$ is outlined to be the minimum cardinality of a collection of edges whose relieved from $$G$$ ends up in a graph $$G^{\prime}$$ fulfilling $$\gamma_{is} (G^{\prime}) > \gamma_{is} (G)$$. We obtain several results for isolate dominating set and identical values of isolate bondage number. Moreover, we investigate some bounds for the isolate bondage number, and this bound is keen and analyze under which conditions the domination parameter and isolate domination parameter are equal. Also, we found some more results for perfect edge domination, and we characterize trees for which $$\gamma^{\prime} = \gamma_{p}^{^{\prime}}$$ and further exciting results.

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 .

A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.

For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.

For a graph G = (V, E) with vertex set V and edge set E, a subset F of E is called an edge dominating set (resp. a total edge dominating set) if every edge in E\\F (resp. in E) is adjacent to at least one edge in F, the minimum cardinality of an edge dominating set (resp. a total edge dominating set) of G is the edge domination number (resp. total edge domination number) of G, denoted by gamma'(G) (resp. gamma'(t)(G)). In the present paper, we first prove that the total edge domination problem is NP-complete for bipartite graphs with maximum degree 3. Then, for a graph G, we give the inequality gamma'(G) <= gamma'(t)(G) <= 2 gamma'(G) and characterize the trees T which obtain the upper or lower bounds in the inequality.

Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with domination number equal to twice edge domination number.

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.

A set $F$ of a graph $G(V, E)$ is an edge dominating set if every edge in $E-F$ is adjacent to some edge in $F$. An edge domination number $\gamma^{\prime}(G)$ of $G$ is the minimum cardinality of an edge dominating set. An edge dominating set $F$ is called a cototal edge dominating set if the induced subgraph $\langle E-F\rangle$ doesnot contain isolated edge. The minimum cardinality of the cototal edge dominating set in $G$ is its domination number and is denoted by $\gamma_{c o t}^{\prime}(G)$. We investigate several properties of cototal edge dominating sets and give some bounds on the cototal edge domination number.

In this paper we further prove more results about edge domination in hypergraphs. In particular we prove necessary & sufficient conditions under which the edge domination number of a hypergraph increases or decreases when a vertex is removed from the hypergraph. We have proved that a subset F of E (G) is an edge dominating set of G iff it is a dominating set in G* (Where G* is dual hypergraph of G) also we have proved that if gE (G-v) > gE (G) & if F is a minimum edge dominating set of G then there is an edge e containing v ' e Î F & Prn [e, F] contains two distinct edges.

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.

Let G = (V,E) be a graph. A function f : E ? [0, 1] is called an edge dominating function if ?x?N[e] f(x)?1 for all e ? E(G), where N[e] is the closed neighbourhood of the edge e. An edge dominating function f is called minimal (MEDF) if for all functions g : E ? [0,1] with g < f, g is not an edge dominating function. The fractional edge domination number ?'f and the upper fractional edge domination number ?'f are defined by ?'f (G) = min{|f| : f is an MEDF of G} and ?'f (G) = max{|f| : f is an MEDF of G}, where |f| = ?e?E f(e). Further we introduce the fractional parameters corresponding to edge irredundance and edge independence, leading to the fractional edge domination chain. We also consider topological properties of the set of all edge dominating functions of G.

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 in a graph [Formula: see text] is a subset [Formula: see text] of [Formula: see text] such that each edge in [Formula: see text] is either in [Formula: see text] or adjacent to at least an edge in [Formula: see text]. The edge domination number [Formula: see text] is the minimum cardinality of an edge dominating set in [Formula: see text]. A subset [Formula: see text] of vertices in [Formula: see text] is [Formula: see text]-independent if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text] The [Formula: see text]-independence number [Formula: see text] is the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text] We show that if [Formula: see text] is a nontrivial tree, then [Formula: see text] and we point out that this inequality is not valid for all bipartite graphs. Moreover, we characterize all trees that achieve equality in this bound.