Perfect Domination Research Articles

Sub-Restrained Perfect Domination In Graphs

Sub-Restrained Perfect Domination In Graphs

Disjoint Perfect Domination in the Cartesian Product of Two Graphs

Let G be a connected simple graph. A dominating set S ⊆ V (G) is called a perfect dominating set of G if every u ∈ V (G)\S is dominated by exactly one element of S. Let D be a minimum perfect dominating set of G. A perfect dominating set S ⊂ (V (G) \ D) is called an inverse perfect dominating set of G with respect to D. A disjoint perfect dominating set of G is the set C = D ∪ S ⊆ V (G). Furthermore, the disjoint perfect domination number, denoted by γpγp(G), is the minimum cardinality of a disjoint perfect dominating set of G. A disjoint perfect dominating set of cardinality γpγp(G) is called γpγp-set. In this paper, we give some property of the disjoint perfect dominating set in the Cartesian products of two graphs.

Certified Perfect Domination in Graphs

Let $ G = (V(G),E(G)) $ be a simple connected graph. A set $ S \subseteq V(G) $ is called a certified perfect dominating set of $ G $ if every vertex $ v \in V(G)\setminus S $ is dominated by exactly one element $ u \in S $, such that $ u $ has either zero or at least two neighbors in $ V(G)\setminus S $. The minimum cardinality of a certified perfect dominating set of $ G $ is called the \textit{certified perfect domination number} of $ G $ and denoted by $ \gamma_{cerp}(G) $. A certified perfect dominating set $ S $ of $ G $ with $ \lvert S \rvert = \gamma_{cerp}(G) $ is called a $ \gamma_{cerp} $-set. In this paper, the author focuses on several key aspects: a characterization of the certified perfect dominating set, determining the exact values of the certified perfect domination number for specific graphs, and investigating the certified perfect domination number of graphs resulting from the join of two graphs. Furthermore, some relationships between the certified dominating set, the perfect dominating set, and the certified perfect dominating set of a graph $ G $ are established.

Generalization of restrained triple connected outer perfect domination number for square of grid graphs

Generalization of restrained triple connected outer perfect domination number for square of grid graphs

Total Perfect Roman Domination

A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑v∈Vf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)≤3γp(G) for every graph G, and we characterized trees T of order n≥3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties.

Extended formulations for perfect domination problems and their algorithmic implications

Given an undirected graph G=(V,E), a subset D⊆V is called a vertex dominating set (VDS) if every vertex of V either belongs to D or is adjacent to a vertex of D. Additionally, a VDS D is called perfect if every vertex of V∖D is adjacent to a single vertex of D. Finally, the Perfect Vertex Domination Problem (PVDP) asks for a perfect VDS D with the smallest cardinality possible. Domination extends very naturally to the edges of G=(V,E) and the Perfect Edge Domination Problem (PEDP) asks for a perfect edge dominating set with as few edges as possible. We propose new formulations for PVDP and PEDP. They rely on structural properties of perfect dominating sets and are computationally compared with their counterparts from the literature. For the new PEDP formulation, in particular, running times for standard state-of-the-art mixed integer programming codes are shown to frequently lead to speed-ups of orders of magnitude, over their corresponding performances for the remaining PEDP formulations.

Results on Coregular Perfect Domination of Line Graph and Relation with Different Dominations of Graph

Results on Coregular Perfect Domination of Line Graph and Relation with Different Dominations of Graph

Perfect Domination Ratios of Archimedean Lattices

An Archimedean lattice is an infinite graph constructed from a vertex-transitive tiling of the plane by regular polygons. A dominating set of vertices is a perfect dominating set if every vertex that is not in the set is dominated exactly once. The perfect domination ratio is the minimum proportion of vertices in a perfect dominating set. Seven of the eleven Archimedean lattices can be efficiently dominated, which easily determines their perfect domination ratios. The perfect domination ratios are determined for the four Archimedean lattices that can not be efficiently dominated.

A New Approach of Perfect Domination in Product of Interval-Valued Fuzzy Incidence Graphs With Application

Fuzzy graphs, also known as fuzzy incidence graphs, are a useful and well-organized tool for encapsulating and resolving a variety of real-world situations involving ambiguous data and information. In this investigation article, we introduced the chance of interval-valued fuzzy incidence graphs (IVFIGs) alongside their specific properties. The operations of Cartesian product (CP), Tensor product (TP) in IVFIGs are additionally examined. The technique to compute the degree (DG) of IVFIGs acquired by CP and TP is examined. Some significant hypotheses to figure the DG of the vertices of IVFIGs gained by CP and TP are explained. An innovative idea of perfect domination in CP of two IVFIGs and TP of two IVFIGs utilizing incidence pair are presented and gotten the connection between them. Eventually, genuine utilization of perfect domination number (PDN) to discover which countries (country) have the best education policies among various countries is inspected.

Connected Perfect Domination of Interval-Valued Fuzzy Graphs

Connected Perfect Domination of Interval-Valued Fuzzy Graphs

Perfect Domination, Roman Domination and Perfect Roman Domination in Lexicographic Product Graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.

Perfect Domination Polynomial of Homogeneous Caterpillar Graphs and of Full Binary Trees

Perfect Domination Polynomial of Homogeneous Caterpillar Graphs and of Full Binary Trees

EXPLORING AT MOST TWIN OUTER PERFECT DOMINATION NUMBER FOR GENERAL BINARY TREE AND SOME SPECIAL GRAPHS

A set S⊆ V(G) said to be an at most twin outer perfect dominating set if for every vertex v∈V-S, l⩽|N(v)∩S|⩽2 and < V-S> has at least one perfect matching. The minimum cardinality of at most twin outer perfect dominating set is called the at most twin outer perfect domination number and γ atop (G) denotes this number. This was initiated by G.Mahadevan.et.al., recently. Here we find this number for general Binary-tree. corona product of paths and cycles and lotus inside graph.

Perfect Dominating Set of An Interval-Valued Fuzzy Graphs

Aims/ Objectives: Perfect domination is very much useful in network theory, Electrical stations and several fields of mathematics. In This paper, perfect domination in an intervalvalued fuzzy graphs is defined and studied. Some bounds on perfect domination number γp(G) are provided for several interval-valued fuzzy graphs, such as complete, wheel and star,.. etc. Furthermore, the relationship of γp(G): with some other known parameters in interval-valued fuzzy graphs investigated with some suitable examples.

On the perfect differential of a graph

Let G be a graph of order n(G) and vertex set V(G). Given a set S ⊆ V(G), we define the perfect neighbourhood of S as the set Np (S) of all vertices in V(G)\\S having exactly one neighbour in S. The perfect differential of S is defined to be ∂p (S) = |Np (S)| − |S|. In this paper, we introduce the study of the perfect differential of a graph, which we define as ∂p (G) = max{∂p (S) : S ⊆ V(G)}. Among other results, we obtain general bounds on ∂p (G) and we prove a Gallai-type theorem, which states that ∂p (G) + γp R (G) = n(G), where γp R (G) denotes the perfect Roman domination number of G. As a consequence of the study, we show some classes of graphs satisfying a conjecture stated by Bermudo [Discrete Appl. Math. 232 (2017), 64-72].

Complexity issues of perfect secure domination in graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

Analysing of complementary perfect hop domination numeral of corona products of graphs

Analysing of complementary perfect hop domination numeral of corona products of graphs

Perfect domination number of path graph $P_n$ and its Corona product with another path graph $P_{n-1}$

Perfect domination number of path graph $P_n$ and its Corona product with another path graph $P_{n-1}$

Study on neutrosophic graph with application in wireless network

A neutrosophic network is an extension of an intuitionistic fuzzy network that provides more precision compatibility and flexibility than an intuitionistic fuzzy graph in structuring and modelling many real-life problems. The authors have explored the use of a neutrosophic network for modelling the passive optical network, mobile ad hoc network (MANET), and wireless sensor graph. They have presented the idea of strong arc, weak arc strong domination numbers, and strong perfect domination of neutrosophic network. They have described the method to find the values of strong and strong perfect domination of neutrosophic network. Finally, they use the idea of a strong arc strong domination number in MANET and wireless sensor graphs.

A characterization of perfect Roman trees

A set S of vertices is a perfect dominating set of a graph G if every vertex not in S is adjacent to exactly one vertex of S. The minimum cardinality of a perfect dominating set is the perfect domination number γp(G). A perfect Roman dominating function (PRDF) on a graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex u with f(u)=0 is adjacent to exactly one vertex v for which f(v)=2. The weight of a PRDF is the sum of its function values over all vertices, and the minimum weight of a PRDF of G is the perfect Roman domination number γRp(G). Obviously, for every graph G, γRp(G)≤2γp(G), and those graphs attaining the equality are called perfect Roman graphs. In this paper, we provide a characterization of perfect Roman trees.