Edge Domination Research Articles

A conjecture on the lower bound of the signed edge domination number of 2-connected graphs

In this short note, we construct an infinite family of counterexamples to a conjecture on the lower bound of the signed edge domination number of 2-connected graphs. We propose two problems in order to revise the original conjecture.

Domination Edge Integrity of Corona Products of C_n with P_m, C_m, K_(1,m)

Vulnerability is the most important concept in analysis of communication networks to disruption. Any network can be modelled by graphs so measures defined on graphs gives an idea in design. Integrity is one of the well-known vulnerability measures interested in remaining structure of a graph after any failure. Domination is also an another popular concept in network design. Nowadays new vulnerability measures take a great role in any failure not only on nodes also on links which have special properties. A new measure domination edge integrity of a connected and undirected graph was defined such as 𝐷𝐼′(𝐺)=𝑚𝑖𝑛{ |𝑆|+𝑚(𝐺−𝑆):𝑆 ⊆ 𝐸(𝐺)} where 𝑚(𝐺−𝑆) is the order of a maximum component of 𝐺−𝑆 and 𝑆 is an edge dominating set. In this paper some results concerning this parameter on corona products of graph structures 𝐶𝑛⊙𝐶𝑚, 𝐶𝑛⊙𝑃𝑚, 𝐶𝑛⊙𝐾1,𝑚 are presented.

The edge-to-edge geodetic domination number of a graph

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

On the resolving strong domination number of graphs: a new notion

The study of metric dimension of graph G has widely given some results and contribution of graph research of interest, including the domination set theory. The dominating set theory has been quickly growing and there are a lot of natural extension of this study, such as vertex domination, edge domination, total domination, power domination as well as the strong domination. In this study, we initiate to combine the two above concepts, namely metric dimension and strong domination set. Thus we have a resolving strong domination set. We have obtained the resolving strong domination number, denoted by γrst(G), of some graphs.

Edge domination and 2-independence in trees

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.

Domination, Edge Domination and Roman Domination in Human Chain Graph

In this paper, we determine domination number, edge domination number and roman domination number of a human chain graph.

Restrained Edge Domination Number of Some Path Related 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.

Vertex-Edge Roman Domination

A vertex-edge Roman dominating function (or just ve-RDF) of a graph G = (V,E) is a function f : V (G) →{0, 1, 2} such that for each edge e = uv either max{f(u),f(v)}≠0 or there exists a vertex w such that either wu ∈ E or wv ∈ E and f(w) = 2. The weight of a ve-RDF is the sum of its function values over all vertices. The vertex-edge Roman domination number of a graph G, denoted by γveR(G), is the minimum weight of a ve-RDF G. In this paper, we initiate a study of vertex-edge Roman dominaton. We first show that determining the number γveR(G) is NP-complete even for bipartite graphs. Then we show that if T is a tree different from a star with order n, l leaves and s support vertices, then γveR(T) ≥ (n − l − s + 3)∕2, and we characterize the trees attaining this lower bound. Finally, we provide a characterization of all trees with γveR(T) = 2γ′(T), where γ′(T) is the edge domination number of T.

Connected vertex–Edge dominating sets and connected vertex–Edge domination polynomials of triangular ladder

Connected vertex–Edge dominating sets and connected vertex–Edge domination polynomials of triangular ladder

Bounding and approximating minimum maximal matchings in regular graphs

The edge domination number γe(G) of a graph G is the minimum size of a maximal matching in G. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on γe(G) for regular and non-regular graphs G in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

An Efficient Energy Aware Semigraph-Based Total Edge Domination Routing Algorithm in Wireless Sensor Networks

Wireless sensor networks (WSNs) are group of spatially distributed atomic sensors which are deployed widely in unattended environments. Each sensor node is capable of monitoring environmental parameters like temperature, pressure, humidity, vibration etc. And WSNs has wide range of applications starting from home automation to military surveillance. However, one of the noticeable issues in WSNs is its efficient energy usage because sensor nodes are battery powered, which cannot be replaced or recharged once deployed in target area. Since communication module consumes most of the node power, energy efficient topology construction and routing algorithm design plays the vital role for energy conservation. For energy efficient optimized topology construction selected nodes are being chosen to construct a virtual backbone. One of the principle practices followed for virtual backbone construction for optimized topology is Dominating Set (DS) of graph theory. However, generating a virtual backbone by DS or Connected Dominating Set (CDS) is NP-hard problem because of larger network size. To overcome this issue, in this paper an energy aware Total Edge Domination based semigraph model (TEDS) is proposed. The performance ratio of proposed TEDS algorithm is measured as $$\left( {3 + {{ln\Delta^{\prime}}} + \frac{1}{{{{\Delta^{\prime}}}}}} \right)\left| {{{opt}}} \right|$$ , where |opt| represents size of the network constructed using proposed model. The time complexity of the proposed model is measured as O(m) and message complexity $${\text{O}}\left( {mn + \frac{m}{{\Delta^{\prime}}}} \right)$$ , where $$\Delta^{\prime}$$ is the maximum edge degree of the network. In addition, the performance of the network is simulated using the NS-2 simulator, and the performance metrics were studied.

Upper bounds on the signed edge domination number of a graph

A signed edge domination function (or SEDF) of a simple graph G=(V,E) is a function f:E→{1,−1} such that ∑e′∈N[e]f(e′)≥1 holds for each edge e∈E, where N[e] is the set of edges in G that share at least one endpoint with e. Let γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved for the two cases vodd(G)=0 and veven(G)=0, where vodd(G) (resp. veven(G)) is the number of odd (resp. even) vertices in G. This article proves Xu’s conjecture for veven(G)∈{1,2}. We also show that for any simple graph G of order n, γs′(G)≤n+vodd(G)∕2 and γs′(G)≤n−2+veven(G) when veven(G)>0, and thus γs′(G)≤(4n−2)∕3. Our result improves the best current upper bound of γs′(G)≤⌈3n∕2⌉.

Bounds on total edge domination number of a tree

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is the total edge dominating set if every edge in [Formula: see text] is adjacent to at least one edge in [Formula: see text]. The minimum cardinality of a total edge dominated set, denoted by [Formula: see text], is called the total edge domination number of a graph [Formula: see text]. We prove that for every tree [Formula: see text] of diameter at least two with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize the trees attaining each of the bounds.

DOMINATION AND EDGE DOMINATION IN TREES

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.

Complexity and characterization aspects of edge-related domination for graphs

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).

Domination versus edge domination

We propose the conjecture that the domination number γ(G) of a Δ-regular graph G with Δ≥1 is always at most its edge domination number γe(G), which coincides with the domination number of its line graph. We prove that γ(G)≤1+2(Δ−1)Δ2Δγe(G) for general Δ≥1, and γ(G)≤76−1204γe(G) for Δ=3. Furthermore, we verify our conjecture for cubic claw-free graphs.

The complexity of total edge domination and some related results on trees

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 \(E\backslash F\) (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 \(\gamma ^{\prime }(G)\) (resp. \(\gamma _t^{\prime }(G)\)). 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 \(\gamma ^{\prime }(G)\leqslant \gamma ^{\prime }_{t}(G)\leqslant 2\gamma ^{\prime }(G)\) and characterize the trees T which obtain the upper or lower bounds in the inequality.

Edge superconductivity in multilayer WTe2 Josephson junction.

WTe2, as a type-II Weyl semimetal, has 2D Fermi arcs on the (001) surface in the bulk and 1D helical edge states in its monolayer. These features have recently attracted wide attention in condensed matter physics. However, in the intermediate regime between the bulk and monolayer, the edge states have not been resolved owing to its closed band gap which makes the bulk states dominant. Here, we report the signatures of the edge superconductivity by superconducting quantum interference measurements in multilayer WTe2 Josephson junctions and we directly map the localized supercurrent. In thick WTe2 (n}{}sim 60{rm{ nm}}), the supercurrent is uniformly distributed by bulk states with symmetric Josephson effect (n}{}| {I_c^ + ( B )} | {=} | {I_c^ - ( B )} | ). In thin WTe2 (10 nm), however, the supercurrent becomes confined to the edge and its width reaches up to n}{}1.4{rm{ mu m }}and exhibits non-symmetric behavior n}{}| {I_c^ + ( B )} | ne | {I_c^ - ( B )} |. The ability to tune the edge domination by changing thickness and the edge superconductivity establishes WTe2 as a promising topological system with exotic quantum phases and a rich physics.

RETRACTED ARTICLE: An investigation of unicyclic graphs in which the isolate bondage number is equal to three in graph network theory

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 outer-connected vertex edge domination number in Cartesian product graphs

Let G be a simple, finite and connected graph. An outer-connected vertex edge dominating set, abbreviated OCVEDS of G is a subset D of vertices of G such that D is a vertex edge dominating set and the graph G\\D is connected. The outer-connected vertex edge domination number of G is a minimum cardinality of a OCVEDS of G and is denoted by . This paper focuses on the OCVEDS in Cartesian product of graphs.