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Let G be a connected graph. A set S ⊆ V (G) is convex 2-dominating if S is both convex and 2-dominating. The minimum cardinality among all convex 2-dominating sets in G, denoted by γ2con(G), is called the convex 2-domination number of G. In this paper, we initiate the study of convex 2- domination in graphs. We show that any two positive integers a and b with 6 ≤ a ≤ b are, respectively, realizable as the convex domination number and convex 2-domination number of some connected graph. Furthermore, we characterize the convex 2-dominating sets in the join, corona, lexicographic product, and Cartesian product of two graphs and determine the corresponding convex 2-domination number of each of these graphs. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A subset S of vertices of G is a geodetic hop dominating set if it is both a geodetic set and a hop dominating set. The geodetic hop domination number of G is the minimum cardinality among all geodetic hop dominating sets in G. In this paper, we characterize the geodetic hop dominating sets in the join of two graphs. These characterizations which use the concept of pointwise non-dominating 2-path closure absorbing set are, in turn, used to determine the geodetic hop domination number of the join of graphs. Moreover, a realization result involving the hop domination number and geodetic hop domination number is also obtained. [ABSTRACT FROM AUTHOR]

A dominating set D ⊆ V (G) is called a connected co-independent dominating set of G , where G is a finite simple undirected graph, if D is a connected dominating set of G and V (G) ∖ D is an independent set. The minimum cardinality of such a set, denoted by γ c , coi (G) , is called the connected co-independent domination number of G. In this paper, we study this concept and the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the connected co-independent dominating sets in these types of graphs and determine the exact values of their connected co-independent domination numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and do an initial investigation of a variant of Grundy hop domination in a graph called the connected Grundy hop domination. We show that the connected Grundy hop domination number lies between the connected hop domination and Grundy hop domination number of a graph. In particular, we give realization results involving connected hop domination, connected Grundy hop domination, and Grundy hop domination numbers. Moreover, we determine the connected Grundy hop domination numbers of some graph. [ABSTRACT FROM AUTHOR]

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called weakly convex hop dominating if for every two vertices x, y ∈ C, there exists an x-y geodesic P(x, y) such that V (P(x, y)) ⊆ C and for every v ∈ V (G)\C, there exists w ∈ C such that dG(v,w) = 2. The minimum cardinality of a weakly convex hop dominating set of G, denoted by γwconh(G), is called the weakly convex hop domination number of G. In this paper, we introduce and initially investigate the concept of weakly convex hop domination. We show that every two positive integers a and b with 3 ≤ a ≤ b are realizable as the weakly convex hop domination number and convex hop domination number of some connected graph. Furthermore, we characterize the weakly convex hop dominating sets in some graphs under some binary operations. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the concepts of Grundy domination and Grundy hop domination in graphs and give some realization results involving these parameters. We show that the Grundy domination number and Grundy hop domination number of a graph G are generally incomparable (one is not always less than or equal the other). It is shown that their absolute difference can be made arbitrarily large. Moreover, the Grundy domination numbers of some graphs are determined. [ABSTRACT FROM AUTHOR]

A subset S of V (G), where G is a simple undirected graph, is hop dominating if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2 and it is hop differentiating if N²G[u] ∩ S ̸= N²G[v] ∩ S for any two distinct vertices u, v ∈ V (G). A set S ⊆ V (G) is hop differentiating hop dominating if it is both hop differentiating and hop dominating in G. The minimum cardinality of a hop differentiating hop dominating set in G, denoted by γdh(G), is called the hop differentiating hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the hop differentiating hop dominating sets in graphs under some binary operations. [ABSTRACT FROM AUTHOR]

Let G be an undirected connected graph with vertex and edge sets V (G) and E(G), respectively. A set C ⊆ V (G) is called convex hop dominating if for every two vertices x, y ∈ C, the vertex set of every x-y geodesic is contained in C and for every v ∈ V (G) \ C, there exists w ∈ C such that dG(v, w) = 2. The minimum cardinality of convex hop dominating set of G, denoted by γconh(G), is called the convex hop domination number of G. In this paper, we show that every two positive integers a and b, where 2 ≤ a ≤ b, are realizable as the connected hop domination number and convex hop domination number, respectively, of a connected graph. We also characterize the convex hop dominating sets in some graphs and determine their convex hop domination numbers. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of an undirected graph G is a clique if every two distinct vertices in S are adjacent. A clique is a superclique if for every pair of distinct vertices v, w ∈ S, there exists u ∈ V (G) \ S such that u ∈ NG(v) \ NG(w) or u ∈ NG(w) \ NG(v). The maximum cardinality of a clique (resp. superclique) in G is called the clique (resp. superclique) number of G. In this paper, we determine the clique and superclique numbers of some graphs. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A subset S of vertices of G is a geodetic hop dominating set if it is both a geodetic and a hop dominating set. The geodetic hop domination number of G, γhg(G), is the minimum cardinality among all geodetic hop dominating sets in G. Geodetic hop dominating sets in a graph resulting from some binary operations have been characterized. These characterizations have been used to determine some tight bounds for the geodetic hop domination number of each of the graphs considered. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A hop dominating set S ⊆ V (G) is called an outer-connected hop dominating set if S = V (G) or the subgraph ⟨V (G) \ S⟩ induced by V (G) \ S is connected. The minimum size of an outer-connected hop dominating set is the outer-connected hop domination number γfch(G). A dominating set of size γfch(G) of G is called a γfch-set. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the outer-connected hop dominating sets in the join, corona and lexicographic products of graphs, and determine bounds of the outer-connected hop domination number of each of these graphs. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is called a hop independent hop dominating set of G if S is both hop independent and hop dominating set of G. The minimum cardinality of hop independent hop dominating set of G, denoted by γhih(G), is called the hop independent hop domination number of G. In this paper, we show that the hop independent hop domination number of a graph G lies between the hop domination number and the hop independence number of graph G. We characterize these types of sets in the shadow graph, join, corona, and lexicographic product of two graphs. Moreover, either exact values or bounds of the hop independent hop domination numbers of these graphs are given. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. Let S = (v1, v2, · · ·, vk) be a sequence of distinct vertices of G and let Ŝ = {v1, v2, . . ., vk}. Then S is a legal closed hop neighborhood sequence of G if N² G[vi ]\∪i−1 j=1N ² G[vj ] ̸= ∅ for each i ∈ {2, · · ·, k}. If, in addition, Ŝ is a hop dominating set of G, then S is called a Grundy hop dominating sequence. The maximum length of a Grundy hop dominating sequence in a graph G, denoted by γ h gr(G), is called the Grundy hop domination number of G. In this paper, we determine some (extreme) values for the Grundy hop domination number. It is pointed out that the Grundy hop domination number is at least equal to the hop domination. Bounds for the Grundy hop domination numbers of some graphs resulting from some binary operations of two graphs are also obtained. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of a (simple) undirected graph G is a superclique in G if it is a clique and for every pair of distinct vertices v, w ∈ S, there exists u ∈ V (G) \ S such that u ∈ NG(v) \ NG(w) or u ∈ NG(w) \ NG(v). The maximum cardinality among the supercliques in G, denoted by ωs(G), is called the superclique number of G. In this paper, we determine the superclique numbers of some graphs including those resulting from some binary operations of graphs. We will also show that the difference of the clique number and the superclique number can be made arbitrarily large [ABSTRACT FROM AUTHOR]

Let G be a (simple) undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a monophonic eccentric dominating set if every vertex in V (G)\S has a monophonic eccentric vertex in S. The minimum size of a monophonic eccentric dominating set in G is called the monophonic eccentric domination number of G. It shown that the absolute difference of the domination number and monophonic eccentric domination number of a graph can be made arbitrarily large. We characterize the monophonic eccentric dominating sets in graphs resulting from the join, corona, and lexicographic product of two graphs and determine bounds on their monophonic eccentric domination numbers. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop independent set of G if any two distinct vertices in S are not at a distance two from each other, that is, dG(v, w) ≠ 2 for any distinct vertices v, w ∈ S. The maximum cardinality of a hop independent set of G, denoted by αh(G), is called the hop independence number of G. In this paper, we show that the absolute difference of the independence number and the hop independence number of a graph can be made arbitrarily large. Furthermore, we determine the hop independence numbers of some graphs including those resulting from some binary operations of graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G)\S, there exists w ∈ S such that dG(v, w) = 2. A set S ⊆ V (G) is a super hop dominating set if ehpnG(v, V (G)\S) ≠ Ø for each v ∈ V (G) \S, where ehpnG(v, V(G)\S) is the set containing all the external hop private neighbors of v with respect to V(G)\S. The minimum cardinality of a super hop dominating set of G, denoted by γhs(G), is called the super hop domination number of G. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the super hop dominating sets in the join, and lexicographic products of graphs, and determine bounds of the super hop domination number of each of these graphs. [ABSTRACT FROM AUTHOR]

A subset S of V (G), where G is a simple undirected graph, is a hop dominating set if for each v ∈ V (G) nS, there exists w ∈ S such that dG(v;w) = 2 and it is a locating-hop set if NG(v; 2) \ S ≠ NG(v; 2) ∩ S for any two distinct vertices u; v ∈ V (G) n S. A set S ⊆ V (G) is a locating-hop dominating set if it is both a locating-hop and a hop dominating set of G. The minimum cardinality of a locating-hop dominating set of G, denoted by γlh(G), is called the locating-hop domination number of G. In this paper, we investigate some properties of this newly defined parameter. In particular, we characterize the locating-hop dominating sets in graphs under some binary operations. [ABSTRACT FROM AUTHOR]

In this paper, we use the monophonic eccentric neighborhoods of a graph to generate a base for some topology on the vertex set of the graph. Using this construction, we describe the subbasic open sets in the join, corona and lexicographic product of two graphs. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph of order n ≥ 4. A subset S ⊆ V (G) is a double hop dominating set (or a double 2 -step dominating set) if | N G [ v , 2 ] ∩ S | ≥ 2 , where N G [ v , 2 ] = { v } ∪ { w ∈ V (G) : d G (v , w) = 2 } , for each v ∈ V (G). The smallest cardinality of a double hop dominating set of G , denoted by γ × 2 h (G) , is the double hop domination number of G. In this paper, we investigate the concept of double hop dominating sets and study it for graphs resulting from some binary operations. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. It is a global hop dominating set of G if it is a hop dominating set of both G and the complement G of G. The minimum cardinality of a hop dominating (global hop dominating) set of G, denoted by γh(G) (resp. γgh(G)), is called the hop domination (resp. global hop domination) number of G. In this paper, we give some realization results involving domination, hop domination, and global hop domination parameters. Also, we give a rectification of a result found in a recent paper of the authors and use this to prove some results in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce AC∗ and ACG∗ -type properties and then, using these conditions along with other concepts, define two Denjoy-type integrals of a function with values in a locally convex topological vector space (LCTVS). We show, among others, that these newly defined integrals are included in the SH integral, a version of the Henstock integral, for LCTVSvalued functions. [ABSTRACT FROM AUTHOR]

In this paper, we present a way of constructing a topology on a vertex set of a graph using monophonic eccentric neighborhoods of the graph G: In this type of construction, we characterize those graphs that induce the indiscrete topology, the discrete topology, and a particular point topology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an operator on a hyper BCI-algebra via application of a left hyper-order. The family consisting of the images of subsets under the operator turns out to be a base for some topology on the hyper BCI-algebra. We investigate some important properties of the induced topology on certain hyper BCI-algebras. In particular, we show that the generated topology on a non-trivial hyper subalgebra of an ordered hyper BCI-algebra coincides with the relative topology on this hyper subalgebra. [ABSTRACT FROM AUTHOR]

A BCH-algebra (H, *, 0) furnished with a topology τ on H (also called a BCH-topology on H) is called a topological BCH-algebra (or TBCH-algebra) if the function *: H × H → H, defined by *((x, y)) = x * y for any x, y ∊ H, is continuous, where the Cartesian product topology on H × H is furnished by τ. In this paper, we give other structural properties of topological BCH-algebras. [ABSTRACT FROM AUTHOR]

Given a hyper BCK-algebra (H, *, 0), we introduce some subsets of H and use them to generate two closure operators on H. In this paper, we show that each of the two closure operators on H can be utilized to form a base for some topology on H. Moreover, we show that each of the induced topologies coincides with a previously known topology on a hyper BCK-algebra. [ABSTRACT FROM AUTHOR]

A set S of vertices of a connected graph G is a semitotal dominating set if every vertex in V (G)nS is adjacent to a vertex in S, and every vertex in S is of distance at most 2 from another vertex in S. A semitotal dominating set S in G is a secure semitotal dominating set if for every v Є V (G) n S, there is a vertex x Є S such that x is adjacent to v and that ... is a semitotal dominating set in G. In this paper, we characterize the semitotal dominating sets and the secure semitotal dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding semitotal domination and secure semitotal domination numbers. [ABSTRACT FROM AUTHOR]

Let G be a (simple) connected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∊ V (G) n S, there exists w ∊ S such that dG(v;w) = 2. The minimum cardinality of a hop dominating set of G, denoted by Ɣh(G), is called the hop domination number of G. In this paper we revisit the concept of hop domination, relate it with other domination concepts, and investigate it in graphs resulting from some binary operations. [ABSTRACT FROM AUTHOR]

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺2(𝑣) = {u ϵ V(G): 𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G2[A] = V(G) \ N_G2 [A] where N_G2 [A] = A ∪ {v ϵ V(G): 𝑁_𝐺2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a graph with degree sequence ⟨d1, d2, · · ·, dn⟩, where d1 ≥ d2 ≥ · · · ≥ dn. The polynomial representation of G is given by fG(x) = Σn i=1 x di = ∆( X G) k=1 akx k, where ak is the number of vertices of G having degree k for each i = 1, 2, · · · n = ∆(G). In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple and undirected graph. A dominating set S ⊆ V (G) is called a differentiating odd dominating set if for every vertex v ∈ V (G), |N[v] ∩ S| ≡ 1(mod 2) and NG[u] ∩ S≠ NG[v] ∩ S for every two distinct vertices u and v in V (G). The minimum cardinality of a differentiating odd dominating set of G, denoted by γo D(G), is called the differentiating odd domination number. In this paper, we discuss differentiating odd dominating sets and give bounds or exact values of the differentiating odd domination numbers of some graphs. We give necessary and sufficient conditions for some graphs to admit a differentiating odd dominating set. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of some graphs and determine the differentiating odd domination numbers of these graphs. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of the zero divisor graph of a hyper BCI-algebra and investigates some of its properties. [ABSTRACT FROM AUTHOR]