Enrico C Enriquez

University of San Carlos (Philippines) | USC · Department of Computer Science and Mathematics

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ () ∖ , there exists ∈ such that , ∈ (). A dominating set S is called a fair dominating set if for each distinct vertices , ∈ () ∖ , | () ∩ | = | () ∩ |. Further, if is a minimum fair dominating set of , then a fair dominating set ⊆ () ∖ is called an inv...

Let G=(V(G),E(G)) be a connected simple graph. A subset S of V(G) is a dominating set of G if for every u∈V(G\S), there exists v∈S such that uv∈E(G). A dominating set D is called a restrained dominating set if for each u∈V(G)\D there exist v∈V(D) and z∈V(G)\D(z≠u) such that u is adjacent to v and z. Further, if D is a minimum restrained dominating...

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...

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ ()\ , there exists ∈ such that ∈ (). A dominating set is called a super dominating set if for very vertex ∈ ()\ , there exists ∈ such that () ∩ (()\) = { }. A super dominating set is called a secure super dominating set if for every vertex ∈ ()\ , there...

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ () ∖ , there exists ∈ such that ∈ (). A dominating set S is called a fair dominating set if for each distinct vertices. ∈ () ∖ , | () ∩ | = | () ∩ |. Further, if is a minimum fair dominating set of , then a fair dominating set ⊆ () ∖ is called an invers...

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ () ∖ , there exists ∈ such that ∈ (). A dominating set is called a secure dominating set if for each ∈ () ∖ there exists ∈ such that is adjacent to and (∖ { }) ∪ { } is a dominating set. A secure dominating set is called a perfect secure dominating set...

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ () ∖ , there exists ∈ such that ∈ (). A dominating set is called a restrained dominating set if for each ∈ () ∖ there exist ∈ and ∈ () ∖ (≠) such that is adjacent to and. A restrained dominating set is called a perfect restrained dominating set of if ea...

Let = ((), ()) be a connected simple graph. A subset of () is a dominating set of if for every ∈ () ∖ , there exists ∈ ℎ ℎ ∈ (). A dominating set is an inverse dominating set with respect to a minimum dominating set of if ⊆ () ∖. An inverse dominating set is called a super inverse dominating set of if for every vertex ∈ () , there exists ∈ such tha...

In this paper, we characterize the fair restrained dominating set in the Cartesian product of two graphs and give some important properties of the lexicographic product of two graphs.

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] in...

In this paper, we give the characterization of a fair restrained dominating set in the corona of two nontrivial connected graphs and give some important results.

In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results.

A fair dominating set ⊆ is a super fair dominating set (or-set) if for every ∈ ∖ , there exists ∈ such that ∩ (∖) = { }. The minimum cardinality of an-set, denoted by , is called the super fair domination number of. In this paper, we characterize the super fair dominating set of the corona and lexicographic product of two graphs.

Let G be a connected simple graph. A dominating set is a fair dominating set in if for every two distinct vertices and from , that is, every two distinct vertices not in have the same number of neighbors from A fair dominating set is a fair secure dominating set if for each there exists such that and the set is a dominating set of The minimum cardi...

Let be a connected simple graph. A dominating set is a fair dominating set in if every two distinct vertices not in have the same number of neighbors from , that is, for every two distinct vertices and from , A fair dominating set is a fair restrained dominating set if every vertex not in is adjacent to a vertex in and to a vertex in Alternately, a...

Let í µí±® = (í µí±½(í µí±®), í µí±¬(í µí±®)) be a simple graph. A a weakly convex dominating set í µí±ºofí µí±® is called super weakly convex dominating set of í µí±®if for every vertex í µí²– ∈ í µí±½ í µí±® ∖ í µí±ºthere exists a vertexí µí²— ∈ í µí±º ∩ í µí±µ í µí±® (í µí²–) such thatí µí±µ í µí±® í µí²— ∩ í µí±½ í µí±® ∖ í µí±º = {í µí²–}. The...

Let í µí°º be a connected simple graph. A set í µí±† ⊆ í µí±‰(í µí°º) is a doubly connected dominating set if it is dominating and both 〈 í µí±†〉 and 〈 í µí±‰ í µí°º ∖ í µí±†〉 are connected. The doubly connected domination number of í µí°º, denoted by í µí»¾ í µí±í µí± (í µí°º), is the smallest cardinality of a doubly connected dominating set í µ...

In this paper, we initiate the study of super connected dominating set of a graph í µí°º by giving the super connected domination number of some special graphs. Further, we shows that given positive integers í µí±˜, í µí±š and í µí±› such that í µí±› ≥ 2 and 1 ≤ í µí±˜ ≤ í µí±š ≤ í µí±› − 1, there exists a connected graph í µí°º with |í µí±‰(í µí°º...

The perfect dominating set í µí±† is called a perfect doubly connected dominating set if it is doubly connected set in í µí°º. The minimum cardinality of a perfect doubly connected dominating set is called a perfect doubly connected domination number of í µí°º and is denoted by í µí»¾ í µí±í µí±‘í µí± (í µí°º). In this paper we investigate the co...

Let í µí°º = (í µí±‰(í µí°º), í µí°¸(í µí°º)) be a simple graph. A set í µí±† ⊆ í µí±‰(í µí°º) is called a convex dominating set of a graph í µí°º if for every vertex í µí±¢ ∈ í µí±‰ í µí°º ∖ í µí±†, there exists í µí±£ ∈ í µí±† ∩ í µí± í µí°º (í µí±¢) and í µí°¼ í µí°º [í µí±†] = í µí±†. It is a super convex dominating set if í µí± í µí°º í µí±£...

Let be a connected simple graph. A convex dominating set of is a convex secure dominating set, if for each element there exists an element such that and is a dominating set. The convex secure domination number of denoted by is the minimum cardinality of a convex secure dominating set of A convex secure dominating set of cardinality will be called a...

Let G be a connected simple graph. A set S of vertices of a graph G is an outer-convex dominating set if every vertex not in S is adjacent to some vertex in S and V (G) \ S is a convex set. In this paper we characterize the outer-convex dominating sets in the composition and Cartesian product of two connected graphs. It is shown that the outer-conv...

In this paper, we initiate the study of super fair dominating set of a graph í µí°º by giving the super fair domination number of some special graphs. Further, we shows that given positive integers í µí±˜, í µí±š and í µí±› such that í µí±› ≥ 2 and 1 ≤ í µí±˜ ≤ í µí±š ≤ í µí±› − 1, there exists a connected graph í µí°º with |í µí±‰(í µí°º)| = í µí±...

Let í µí±® be a connected simple graph. A subset í µí±º of a vertex set í µí±½(í µí±®) is a dominating set of í µí±® if for every vertex í µí²— ∈ í µí±½(í µí±®)\ í µí±º, there exists a vertex í µí²™ ∈ í µí±º such that í µí²™í µí²— is an edge of í µí±®. Let í µí±« be a minimum dominating set in í µí±®. The dominating set í µí±º ⊆ í µí±½ í µí±® ∖ í µ...

Let í µí°º be a connected simple graph. A weakly convex dominating set í µí±† of í µí°º is a weakly convex doubly connected dominating set if í µí±† is a doubly connected dominating set of í µí°º. The weakly convex doubly connected domination number of í µí°º, denoted by í µí»¾ í µí±í µí±í µí± í µí±¤ (í µí°º), is the smallest cardinality of a co...

A subset í µí±† of í µí±‰(í µí°º) is a dominating set of í µí°º if for every í µí±£ ∈ í µí±‰(í µí°º)\í µí±†, there exists í µí±¥ ∈ í µí±† such that í µí±¥í µí±£ ∈ í µí°¸(í µí°º). An identifying code of a graph í µí°º is a dominating set í µí° ¶ ⊆ í µí±‰(í µí°º) such that for every í µí±£ ∈ í µí±‰(í µí°º), í µí± í µí°º [í µí±£] ∩ í µí° ¶ is distinc...

Let í µí°º be a simple graph. A set í µí±† of vertices of a graph í µí°º is an outer-clique dominating set if every vertex not in í µí±† is adjacent to some vertex in í µí±† and the subgraph induced by í µí±‰ í µí°º ∖ í µí±† is clique. In this paper, we give the characterization of the outer-clique dominating sets resulting from the corona and Cart...

Let í µí°º be a simple graph. A set í µí±† of vertices of a graph í µí°º is an outer-clique dominating set if every vertex not in í µí±† is adjacent to some vertex in í µí±† and the subgraph induced by í µí±‰ í µí°º ∖ í µí±† is clique. In this paper, we will show that given positive integers í µí±˜, í µí±š and í µí±› such that 1 ≤ í µí±˜ ≤ í µí±š ≤...

Let 𝐺 be a connected simple graph. A nonempty subset 𝑆 of the vertex set 𝑉(𝐺) is a clique in 𝐺 if the graph 〈𝑆 〉 induced by 𝑆 is complete. A clique 𝑆 in 𝐺 is a clique dominating set if it is a dominating set. A clique dominating set 𝑆 is a clique secure dominating set in 𝐺 if for every vertex 𝑢 ∈ 𝑉 𝐺 ∖ 𝑆 , there exists a vertex 𝑣 ∈ 𝑆 ∩ 𝑁𝐺(𝑢), such...

Let í µí°º be a connected simple graph. A weakly convex dominating set í µí±† of í µí°º is a weakly convex doubly connected dominating set if í µí±† is a doubly connected dominating set of í µí°º. The weakly convex doubly connected domination number of í µí°º, denoted by í µí»¾ í µí±í µí±í µí± í µí±¤ (í µí°º), is the smallest cardinality of a co...

Let G = (V(G), E(G)) be a 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 V G ∖ S. It is a super restrained dominating set if for every vertex u ∈ V G ∖ S, there exists v ∈ S such that N G v ∩ (V G ∖ S) = {u}. The minimum cardinality of a super restrained dominatin...

Let í µí°º be a connected simple graph. A restrained dominating set í µí±† of a graph í µí°º, is a restrained secure dominating set of í µí°º if for each í µí±¢ ∈ í µí±‰ í µí°º ∖ í µí±†, there exists í µí±£ ∈ í µí±† such that í µí±¢í µí±£ ∈ í µí°¸(í µí°º) and the set í µí±† ∖ í µí±£ ∪ {í µí±¢ } is a dominating set of í µí°º. The minimum cardinality...

Let í µí°º = (í µí±‰(í µí°º), í µí°¸(í µí°º)) be a simple graph. A set í µí±† ⊆ í µí±‰(í µí°º) is called a secure dominating set of a graph í µí°º if for every vertex í µí±¢ ∈ í µí±‰(í µí°º) ∖ í µí±†, there exists í µí±£ ∈ í µí±† ∩ í µí± í µí°º (í µí±¢) such that (í µí±† ∖ {í µí±£}) ∪ {í µí±¢} is dominating. It is a super secure dominating set if...

Abstract: Let 𝐺 be a connected simple graph. A nonempty subset 𝑆 of the vertex set 𝑉(𝐺) is a clique in 𝐺 if the graph 〈 𝑆 〉 induced by 𝑆 is complete. A clique 𝑆 in 𝐺 is a clique dominating set if it is a dominating set. A clique dominating set 𝑆 of 𝑉(𝐺) is a restrained clique dominating set if for each 𝑢 ∈ 𝑉 𝐺 ∖ 𝑆, there exists 𝑧 ∈ 𝑉 𝐺 ∖ 𝑆 such th...

Let G be a connected simple graph. A restrained convex dominating set S in a connected graph G is a secure restrained convex dominating set, if for each element u in V(G)\S there exists an element v in S such that uv E(G) and (S\v u is a restrained convex dominating set. The secure restrained convex domination number of G, denoted by g src (G), is...

Let be a connected simple graph. A set ⊆ () is a doubly connected dominating set if it is dominating and both 〈 〉 and 〈 ()\ 〉 are connected. A convex dominating set of is a convex doubly connected dominating set if is a doubly connected dominating set of. The convex doubly connected domination number of , denoted by (), is the smallest cardinality...

Let G be a connected simple graph. A nonempty subset S of the vertex set V (G) is a clique in G if the graph hSi induced by S is complete. A clique S in G is a clique dominating set if it is a dominating set. Let C be a minimum clique dominating set in G. The clique dominating setS ⊆ V (G)\ C is called an inverse clique dominating set with respect...

Let G = (V (G), E(G)) be a simple connected graph. A dominating set S in G is called a secure dominating set in G if for every u∈V (G) \S, there exists v∈S ∩ NG(u) such that (S \ {v}) ∪ {u} is a dominating set. The minimum cardinality of secure dominating set is called the securedomination number of G and is denoted by γs(G). A secure dominatingset...

Let 𝐺 be a connected simple graph. A dominating set 𝑆 ⊆ 𝑉(𝐺) is called a perfect dominating set of 𝐺 if each 𝑢 ∈ 𝑉 𝐺 ∖ 𝑆 is dominated by exactly one element of 𝑆. A set 𝑆 of vertices of a graph 𝐺 is an outer-connected dominating set if every vertex not in 𝑆 is adjacent to some vertex in 𝑆 and the subgraph induced by 𝑉 𝐺 ∖ 𝑆 is connected. A perfect...

Let G be a connected simple graph. A convex dominating set S of V (G) is a secure convex dominating set of G if for each u ∈ V (G) \ S, there exists V ∈ S such that uv ∈ E(G) and the set (S \ {v }) ⋃ {u } is a convex dominating set of G. The minimum cardinality of a secure convex dominating set of G. denoted by γscon(G), is called the secure convex...

In [6], Enriquez and Kiunisala showed that every integers k, m, and n with 1 ≤ k ≤ m < n is realizable as inverse domination number, inverse secure domination number, and order of G respectively and gave the characterization of the inverse secure dominating set with inverse secure domination number of one and two. In this paper, we characterize the...

Let G be a connected simple graph. A restrained dominating set S of the vertex set of G, V (G) is a secure restrained dominating set of G if for each u ∈ V (G)\S, there exists v ∈ S such that uv ∈ E(G) and the set (S\{v}) ∪ {u} is a restrained dominating set of G. The minimum cardinality of a secure restrained dominating set of G, denoted by γsr(G)...

Let G be a connected simple graph. A dominating set S ⊆ V (G) is called a perfect dominating set of G if each u ∈ V (G)\S is dominated by exactly one element of S. The perfect domination number of G, denoted by γp(G), is the minimum cardinality of a perfect dominating set of G. Let D be a minimum perfect dominating set of G. A perfect dominating se...

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 V (G) \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. Let D be a minimum restrained dominating set of G. A restrained...

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 V (G)\S. Let D be a minimum restrained dominating set in G. A restrained dominating set S ⊆ (V (G) \D) is called an inverse restrained dominating set of G with respect to D. The inverse restraine...

Let G be a connected simple graph. A nonempty subset S of the vertex set V(G) is a clique in G if the graph induced by S is complete. A clique S in G is a clique dominating set if it is a dominating set. A clique dominating set S is a clique secure dominating set in G if for every vertex u ∈ V (G) \ S, there exists a vertex v ∈ S ∩ NG(u) such that...

Let G be a connected simple graph. A restrained dominating set S of the vertex set of G, is a secure restrained dominating set of G if for each u ∈V(G)\S , there exists v∈S such that uv∈E(G) and the set (S \ \{v})∪{u} is a restrained dominating set of G. The minimum cardinality of a secure restrained dominating set of G, denoted by γ_ST (G), called...

A dominating set S which is also convex is called a convex dominating set of G. A convex dominating set S of V (G) is a restrained convex dominating set of G if for each u ε V (G) \ S, there exists z ε V (G) \ S such that uz ε E(G). The minimum cardinality of a restrained convex dominating set of G, denoted by γrcon(G), is called the restrained con...

In this paper, we characterize the restrained convex dominating sets in the corona, lexicographic and Cartesian products of two connected graphs and then determine the corresponding restrained convex domination numbers of these graphs.

In this paper, we characterize the secure convex dominating sets in the lexicographic and Cartesian products of two connected graphs and then determine the corresponding secure convex domination numbers of these graphs.

In this paper, we give necessary and sufficient conditions for a subset S of the vertex set of a connected graph G to be a secure convex dominating set. Some realization problems will be given. In particular, we show that given positive integers k and n such that n ≥ 4 and 1 ≤ k ≤ n, there exists a connected graph G with |V (G)| = n and γscon(G) =...

Let G be a connected simple graph. A set S ⊆ V (G) is a doubly connected dominating set if it is dominating and both 〈 S〉 and 〈 V (G)\S 〉 are connected. The doubly connected domination number of G, denoted by γcc(G), is the smallest cardinality of a doubly connected dominating set S of G. A convex dominating set S of G is a convex doubly connected...

In this paper, we characterize the secure convex dominating sets in the corona of two connected graphs and then determine the corresponding secure convex domination numbers of these graphs.