Locating Edge Domination Number of Some Classes of Claw-Free Cubic Graphs

For a graph G = (V,E), a set S ⊆ V(S ⊆ E) is a restrained dominating (restrained edge dominating) set if every vertex (edge) not in S is adjacent (incident) to a vertex (edge) in S and to a vertex (edge) in V - S(E-S). The minimum cardinality of a restrained dominating (restrained edge dominating) set of G is called restrained domination (restrained edge domination) number of G, denoted by γr (G) (γre(G). The restrained edge domination number of some standard graphs are already investigated while in this paper the restrained edge domination number like degree splitting, switching, square and middle graph obtained from path.

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 $$G = (V, E)$$ , 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).

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.

Let G=(V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v in V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v in V, there is exactly one d in D dominating v. An edge set M is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph. We give a unified framework for investigating the complexity of these problems on various classes of graphs. In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs. We extend the two problems to hypergraphs and show that ED remains NP-complete on alpha-acyclic hypergraphs, and is solvable in polynomial time on hypertrees, while EED is polynomial on alpha-acyclic hypergraphs and NP-complete on hypertrees.

In this paper, the new kind of parameter Regular total semi - µ strong (weak) edge domination number in an intuitionistic fuzzy graph is defined and established the parametric conditions. Another new kind of parameter an equitable regular total semi - µ strong (weak) edge domination number is defined and established the parametric conditions. The properties of Regular total semi - µ strong (weak) edge domination number and an equitable regular total semi - µ strong (weak) edge domination number domination number are discussed.

Let (G) and ' (G) be the domination number and edge dom- ination number, respectively. The lexicographical Product G1 • G2 of graph of G1 and G2 has vertex set V (G1 • G2) = V (G1) × V (G2) and edge set E(G1 •G2) = {(u1v1)(u2v2)| (u1u2 ∈ E(G1))∪(u1 = u2 and v1v2 ∈ E(G2))}. In this paper, we determined generalization of domination and edge domination number on lexicographical product of complete graphs and any simple graph.

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.

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 $$S $$ (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.

The total graph of a graph $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\cup E(G)$ with $c_1,c_2 \in V(G)\cup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to (or incident on) each other in $G$. The total chromatic number $\chi''(G)$ of a graph $G$ is defined to be the chromatic number of its total graph. The well-known Total Coloring Conjecture or TCC states that for every simple finite graph $G$ having maximum degree $\Delta(G)$, $\chi''(G)\leq \Delta(G) + 2$. In this paper, we consider two ways to weaken TCC: (1) Weak TCC: This conjecture states that for a simple finite graph $G$, $\chi''(G) = \chi(T(G)) \leq\Delta(G) + 3$. While weak TCC is known to be true for 4-colorable graphs, it has remained open for 5-colorable graphs. In this paper, we settle this long pending case. (2) Hadwiger's Conjecture for total graphs: We can restate TCC as a conjecture that proposes the existence of a strong $\chi$-bounding function for the class of total graphs in the following way: If $H$ is the total graph of a simple finite graph, then $\chi(H) \leq\omega(H) + 1$, where $\omega(H)$ is the clique number of $H$. A natural way to relax this question is to replace $\omega(H)$ by the Hadwiger number $\eta(H)$, the number of vertices in the largest clique minor of $H$. This leads to the Hadwiger's Conjecture (HC) for total graphs: if $H$ is a total graph then $\chi(H) \leq \eta(H)$. We prove that this is true if $H$ is the total graph of a graph with sufficiently large connectivity. A second motivation for studying Hadwiger's conjecture for total graphs is the following: Consider the class of split graphs whose vertex set is partitioned into an independent set $A$ and a clique $B$, with the following additional constraints: (1) Each vertex in $B$ has exactly 2 neighbours in $A$; (2) No two vertices in $B$ have the same neighbourhood in $A$. It is known that (European Journal of Combinatorics, 76, 159-174,2019) if Hadwiger's conjecture is proved for the squares of this special class of split graphs, then it holds also for the general case. Of course, proving the conjecture for this specialzed-looking case is indeed difficult since it is only a reformulation of the general case, and therefore it is natural to consider the difficulty level of Hadwiger's conjecture for the squares of graph classes defined by slighly modifying the above class of graphs. A natural structural modification is to assume that both $A$ and $B$ are independent sets, keeping everything else same. It turns out that the squares of this modified class of graphs is exactly the class of total graphs. From this perspective, it is not really surprising that HC on Total Graphs is also challenging. On the other hand, we show that weak TCC implies HC on total graphs. This perhaps suggests that the latter is an easier problem than the former.

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.

Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various fields. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artificial intelligence, decision-making theory, etc. Many research studies today are trying to find other applications for domination in their field of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufficient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body’s defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance.

Approximation algorithms for partially covering with edges

For any graph G, let α′(G) and α′min(G) be the maximum cardinality and minimum cardinality among all maximal matchings in G, respectively, and let γ′(G) and γt ′(G) be the edge domination number and edge total domination number of G, respectively. In this paper, we first show some properties of maximal matchings and further determine the exact values of α′(G) and α′min(G) for a complete multipartite graph G. Then, we disclose relationships between maximal matchings and minimal edge dominating sets, and thus obtain the exact values of γ′(G) and γt′(G) for a complete multipartite graph G.