Search

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]

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]

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 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 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 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]

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 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]

A subset S of V (G), where G is a graph without isolated vertices, is a double dominating set of G if for each x ϵ V (G), ∣NG [x] ∩ S∣ ⩾ 2. This paper, shows that any positive integers a, b and n with 2 ⩽ a < b, b ⩾ 2a and n ⩾ b + 2a - 2, can be realized as domination number, double domination number and order, respectively. It also characterize the double dominating sets in the Cartesian and tensor products of two graphs and determine sharp bounds for the double domination numbers of these graphs. In particular, it show that if G and H are any connected non-trivial graphs of orders n and m respectively, then γ×2(G◻H) ⩽ min {mγ2(G), nγ2(H)}, where γ2, is the 2-domination parameter. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate the concept of global stable locationdomination in graphs. We also characterize the global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs and determine the value of the corresponding global stable locating-domination number. [ABSTRACT FROM AUTHOR]

The concepts of edge Steiner set and edge Steiner number of a graph are investigated in this study. A necessary and sufficient condition for a graph G to satisfy ste(G) = |V (G)|-1, where ste(G) denotes the edge Steiner number of G, is obtained. Edge Steiner sets in the joins of graphs are also studied and the Steiner numbers of these graphs are determined. [ABSTRACT FROM AUTHOR]

Let G be a connected graph with vertex set V (G) and edge set E(G). The open hop neighborhood of vertex v ∈ V (G) is the set NG(v, 2) = {w ∈ V (G) : dG(v, w) = 2}, where dG(v, w) denotes the distance between v and w. A non-empty set S ⊆ V (G) is a locating hop set of G if NG(u, 2) ∩ S ̸= NG(v, 2) ∩ S for every pair of distinct vertices u, v ∈ V (G) \ S. The smallest cardinality of a locating hop set of G, denoted by lhn(G) is called the locating hop number of G. This study focuses mainly on the concept of locating hop set in graphs. Characterizations of locating hop sets in the join and corona of two graphs are given and bounds for the corresponding locating hop numbers of these graphs are determined. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of a (simple) undirected graph G is a locating-dominating set of G if for each v∈V (G) n S, there exists w∈S such tha vw∈E(G) and NG(x) \ S 6= NG(y) \ S for any distinct vertices x and y in V (G) n S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S n fvg is a locating-dominating set of G for each v∈S. The minimum cardinality of a stable locating-dominating set of G, denoted by s l (G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs. [ABSTRACT FROM AUTHOR]

Let G = (V(G), E(G)) be a simple graph. A neighborhood connected k-fair dominating set (nckfd-set) is a dominating set S ⊆ V(G) such that the |N(u) ⋂ S| = k for every u ⲉ V (G)\S and the induced subgraph ⟨N(S)⟩ of S is connected. The neighborhood connected k-fair domination number of G, denoted by γnckfd(G), is the minimum cardinality of an nckfd-set. In this paper, we introduce and investigate the notion of neighborhood connected k-fair domination in graphs. We also characterize such dominating sets in the join, corona, lexicographic and Cartesian products of graphs and determine the exact values or sharp bounds of their corresponding neighborhood connected k-fair domination number. [ABSTRACT FROM AUTHOR]

Let G = (V(G), E(G)) be any undirected graph. Then G induces a topology τG on V(G) with base consisting of sets of the form FG[A] = V(G)\NG[A], where NG[A] = A ∪ { x : xa ∈ E(G) for some a ∈ A } and A ranges over all subsets of V(G). In this paper, we describe the topologies induced by the corona, edge corona, disjunction, symmetric difference, Tensor product, and the strong product of two graphs by determining the subbasic open sets. [ABSTRACT FROM AUTHOR]