Let G=(V(G),E(G)) be a nontrivial connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)∖S, there exists x∈S such that xv ∈E(G). A set S⊆V(G) is said to be an outer-connected dominating set in G if S is dominating and either S=V(G) or ⟨V(G)∖S⟩ is connected. The outer-connected domination number of G is the minimum cardinality of an outer-connected dominating set of G, denoted by γ ̃_c (G). A fair dominating set in graph G is a dominating set S such that all vertices in V(G)∖S are dominated by the equal number of vertices in S. The fair domination number of G is the minimum cardinality of a fair dominating set of G, denoted by γ_fd (G). A nonempty subset S⊆V (G) is an outer-connected fair dominating set of G, if S is a fair dominating set of G and the subgraph ⟨V(G)∖S⟩ induced by V(G)∖S is connected. The outer-connected fair domination number of G is the minimum cardinality of an outer-connected fair dominating set of G, denoted by γ ̃_cfd (G). In this paper, we initiate the study of the concept and we show the existence of a connected graph G with |V(G)| = n and γ ̃_cfd (G) = k for all positive integer k. Further, give the outer-connected fair domination number of some special graphs.

Let be a connected simple graph. A subset of is a dominating set of if for every , there exists such that. A set is said to be an outer-connected dominating set in if is dominating and either or is connected. Let be a minimum dominating set of . A nonempty subset is an outer-connected inverse dominating set of , if is an inverse dominating set with respect to and the subgraph induced by is connected. The outer-connected inverse domination number of , is denoted by , that is, the minimum cardinality of an outer-connected inverse dominating set of . In this paper, we initiate the study of the concept and give the outer-connected inverse domination number of some special graphs. Further, we give the characterization of the outer-connected inverse dominating set in the join of two nontrivial connected 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.

Let G be a nontrivial connected graph. A dominating set D⊆V(G) is called a doubly connected dominating set of G if both 〈D〉 and 〈V(G)\D〉 are connected. Let D be a minimum connected dominating set of G. If S⊆V(G)\D is a connected dominating set of G, then S is called an inverse doubly connected dominating set of G with respect to D. Furthermore, the inverse doubly connected domination number, denoted by γ_cc^(-1) (G) is the minimum cardinality of an inverse doubly connected dominating set of G. An inverse doubly connected dominating set of cardinalities γ_cc^(-1) (G) is called γ_cc^(-1)-set. In this paper, we characterized the inverse doubly connected domination in the lexicographic product of two graphs and give some important results.

Let G be a connected simple graph. A set S⊆V(G) is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in (G)∖S . A set S of vertices of a graph G is an outer-restrained dominating set if every vertex not in S is adjacent to some vertex in S and V(G)∖S is a restrained set. The outer-restrained domination number of G, denoted by (γ_r ) ̃(G) is the minimum cardinality of an outer-restrained dominating set of G. An outer-restrained set of cardinality (γ_r ) ̃(G) will be called a (γ_r ) ̃(G) -set. This study is an extension of an existing research on outer-restrained domination in graphs. In this paper, we characterized the outer-restrained domination in graphs under the lexicographic product of two graphs.

Let G be a nontrivial connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈ V(G)\S, there exists x∈S such that xv∈E(G). Let D be a minimum dominating set of G. If V(G)\D contains a dominating set say S of G, then S is called an inverse dominating set with respect to D. A fair dominating set in a graph G (or FD-set) is a dominating set S such that all vertices not in S are dominated by the same number of vertices from S ; that is, every two vertices not in S has the same number of neighbors in S. An inverse dominating subset S of a vertex set V(G) is said to be fair inverse dominating set if for every vertex v∈ V(G)\S is dominated by the same number of the vertex in S. A fair inverse domination number is the minimum cardinality of a fair inverse dominating set S in G. In this paper, we initiate the study of the concept and give the fair inverse domination number of some special graphs. Further, we give the characterization of the fair inverse dominating set in the join of two nontrivial connected graphs.

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v ∈ V(G)∖S, there exists x∈S such that xv ∈ E(G). An identifying code S of a graph G is a dominating set S⊆V(G) such that for every v ∈ V(G), N_G [v]∩S is distinct. An identifying code of a graph G is an identifying restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V(G)∖S. Alternately, an identifying code of a graph S⊆V(G) is an identifying restrained dominating set if N[S]=V(G) and 〈V(G)∖S〉 is a subgraph without isolated vertices. The minimum cardinality of an identifying restrained dominating set of G, denoted by γ_r^ID (G), is called the identifying restrained domination number of G. In this paper, we initiate the study of the concept and give the domination number of some special graphs. Further, we show the characterization of the identifying restrained dominating set in the join of two nontrivial connected graphs.

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in the Cartesian product and lexicographic product of two connected graphs.

As secure domination and inverse domination garnered attention from various researchers, the combination of the two also raised a certain amount of curiosity. This paper aimed to investigate the secure inverse domination in graphs which is defined as follows. Let G be a connected simple graph and let D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is an inverse dominating set of G with respect to D. The set S is called a secure inverse dominating set of G if for every u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪{u} is a dominating set of G. The secure inverse domination number of G, denoted by γ_s^((-1) ) (G), is the minimum cardinality of a secure inverse dominating set of G. A secure inverse dominating set of cardinality γ_s^((-1) ) (G) is called γ_s^((-1) )-set. Particularly, the researchers examined and provided the characterization of secure inverse dominating set in the corona and lexicographic product of two graphs in this study. Moreover, the secure inverse domination number of graphs under the binary operations corona and lexicographic product were determined.

Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

This study advances the limitations of current fuzzy multi-objective assignment models by exploring some formulations with fuzzy parameters and fuzzy goals. The first formulation expresses the coefficients of the objective functions and constraints as fuzzy sets. Two crucial fuzzy transformations were adopted, along with the computational process of the epsilon-constrained multi-objective optimisation. On the other hand, the second formulation assumes the fuzzy coefficients of objective functions and constraints and extends such fuzziness by introducing fuzzy constraints. Lastly, the coefficients of the objective functions and the constraints are expressed as crisp sets while allowing permissible constraint violations. The symmetric fuzzy linear programming solution concepts in the domain literature were adopted as part of the computational process in arriving at a model solution. Actual case examples aided these formulations to gain insights into their computational complexity, efficiency, scalability, and flexibility.

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

Let be a connected simple graph. A set is a doubly connected dominating set if it is dominating and both and are connected. The doubly connected domination number of denoted by is the smallest cardinality of a doubly connected dominating set ofA nonempty subsetof the vertex setis a clique in if the graph induced byis complete. A cliquein is a clique dominating set if it is a dominating set. A clique dominating set ofis a clique doubly connected dominating set if is a doubly connected dominating set of The clique doubly connected domination number of, denoted by is the smallest cardinality of a clique doubly connected dominating set of In this paper, we show that every integersand with is realizable as clique doubly connected domination number and order of respectively. Further, we give the characterization of the clique doubly connected dominating set with a clique doubly connected domination numbers of 1 and 2. Finally, we characterize the clique doubly connected dominating sets of the corona and Cartesian product of two graphs.

In this paper, we initiate the study of super connected dominating set of a graph by giving the super connected domination number of some special graphs. Further, we shows that given positive integers and such that and there exists a connected graph with , and . Finally, we characterize the super connected dominating set of the join, corona, and Cartesian product of two graphs.

A subset $S$ of $V(G)$ is a dominating set of $G$ if for every $v \in V(G)\backslash S$, there exists $x \in S$ such that $xv \in E(G)$. An identifying code of a graph $G$ is a dominating set $C\subseteq V(G)$ such that for every $v\in V(G)$, $N_G[v]\cap C$ is distinct. In this paper, we investigate the identifying code of some special graphs and give some important results.

Let be a connected simple graph. A weakly convex dominating set of is a weakly convex doubly connected dominating set if is a doubly connected dominating set of The weakly convex doubly connected domination number of denoted by , is the smallest cardinality of a convex doubly connected dominating set of . In this paper, we show that for each set of integers and with the integers and are realizable as weakly convex doubly connected domination number, convex doubly connected domination number, and order of , respectively. Further, we give the characterization of the weakly convex doubly connected dominating set with weakly convex doubly connected domination numbers of 1 and 2. Finally, we characterize the weakly convex doubly connected dominating sets of the join and corona of two graphs.

Let be a connected simple graph. A subset of a vertex setis a dominating set of if for every vertex there exists a vertexsuch thatis an edge of Let be a minimum dominating set in The dominating set is called an inverse dominating set with respect to A disjoint dominating set of is the setA -dominating set ofis a disjoint set such that The -domination number denoted by z is the minimum cardinality of -dominating set of -dominating set of with cardinality equal to z is called a z -of In this paper, we show that every even integer and integer with is realizable as -domination number and order ofrespectively. Further, we characterize the-dominating sets in the join and corona of two graphs and give some important results.