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Artykuły w czasopismach na temat „Resolving connected dominating set”

Autor: Grafiati

Data publikacji: 3 czerwca 2025

Data aktualizacji: 19 lipca 2025

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1

Mahistrado, Angelica Mae, and Helen Rara. "Outer-Connected 2-Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 16, no. 2 (2023): 1180–95. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4771.

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. Let G be a connected graph. A set S ⊆ V (G) is an outer-connected 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or the subgraph ⟨V (G)\S⟩ induced by V (G)\S is connected. The outer-connected 2-resolving hop domination number of G, denoted by γ^c2Rh(G) is the smallest cardinality of an outer-connected 2-resolving hop dominating set of G. This study aims to combine the concept of outer-connected hop domination with the 2-resolving hop dominating sets of graphs. The main results generated in this study include the characterization of outer-conne

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Hausawi, Yasser M., Zaid Alzaid, Olayan Alharbi, Badr Almutairi, and Basma Mohamed. "COMPUTING THE SECURE CONNECTED DOMINANT METRIC DIMENSION PROBLEM OF CLASSES OF GRAPHS." Advances and Applications in Discrete Mathematics 42, no. 3 (2025): 219–33. https://doi.org/10.17654/0974165825015.

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This paper investigates the NP-hard problem of finding the lowest secure connected domination metric dimension of graphs. If each vertex in can be uniquely recognized by its vector of distances to the vertices in Scddim, then every vertex set Scddim of a connected graph resolves . If the subgraph induced by Scddim is a nontrivial connected subgraph of , then the resolving set Scddim of is connected. That resolving set is dominating if each vertex in that is not an element of Scddim is a neighbor of some vertices in Scddim. If there is a in such that is a dominating set for any in , then the do

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Monsanto, Gerald Bacon, and Helen M. Rara. "Resolving Restrained Domination in Graphs." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 829–41. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3985.

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Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S âŠ† V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

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Monsanto, Gerald Bacon, Penelyn L. Acal, and Helen M. Rara. "Strong Resolving Domination in the Lexicographic Product of Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 363–72. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4652.

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Let G be a connected graph. A subset S ⊆ V (G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u, v ∈ V (G), there exists a vertex w ∈ S such that u ∈ IG[v, w] or IG[u, w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the lexicographic product of graphs and determine the corresponding resolving domination number.

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Mohamad, Jerson, and Helen Rara. "1-Movable Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 418–29. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4671.

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Let G be a connected graph. A set W ⊆ V (G) is a resolving hop dominating set of G if W is a resolving set in G and for every vertex v ∈ V (G) \ W there exists u ∈ W such that dG(u, v) = 2. A set S ⊆ V (G) is a 1-movable resolving hop dominating set of G if S is a resolving hop dominating set of G and for every v ∈ S, either S \ {v} is a resolving hop dominating set of G or there exists a vertex u ∈ ((V (G) \ S) ∩ NG(v)) such that (S \ {v}) ∪ {u} is a resolving hop dominating set of G. The 1-movable resolving hop domination number of G, denoted by γ 1 mRh(G) is the smallest cardinality of a 1-

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Sumaoy, Helyn Cosinas, and Helen Rara. "On Movable Strong Resolving Domination in Graphs." European Journal of Pure and Applied Mathematics 15, no. 3 (2022): 1201–10. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4440.

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Let G be a connected graph. A strong resolving dominating set S is a 1-movable strong resolving dominating set of G if for every v ∈ S, either S \ {v} is a strong resolving dominating set or there exists a vertex u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a strong resolving dominating set of G. The minimum cardinality of a 1-movable strong resolving dominating set of G,denoted by γ1 msR(G) is the 1-movable strong resolving domination number of G. A 1-movable strong resolving dominating set with cardinality γ1msR(G) is called a γ1msR-set of G. In this paper, we study this concept and

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Monsanto, Gerald Bacon, Penelyn L. Acal, and Helen M. Rara. "On Strong Resolving Domination in the Join and Corona of Graphs." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 170–79. http://dx.doi.org/10.29020/nybg.ejpam.v13i1.3625.

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Let G be a connected graph. A subset S \subseteq V(G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u,v \in V(G), there exists a vertex w \in S such that u \in I_G[v,w] or v \in I_G[u,w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the join and corona of graphs and determine the bounds or exact values of the strong resolving domination number of these graphs.

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Monsanto, Gerald Bacon, Penelyn L. Acal, and Helen M. Rara. "On Strong Resolving Domination in the Join and Corona of Graphs." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 170–79. http://dx.doi.org/10.29020/nybg.ejpam.v1i1.3625.

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Let G be a connected graph. A subset S \subseteq V(G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u,v \in V(G), there exists a vertex w \in S such that u \in I_G[v,w] or v \in I_G[u,w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the join and corona of graphs and determine the bounds or exact values of the strong resolving domination number of these graphs.

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Cabaro, Jean Mansanadez, and Helen Rara. "Restrained 2-Resolving Dominating Sets in the Join, Corona and Lexicographic Product of Two Graphs." European Journal of Pure and Applied Mathematics 15, no. 3 (2022): 1047–53. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4451.

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Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set for G if, for any distinct vertices u, w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. A set S ⊆ V (G) is a restrained 2-resolving dominating set in G if S is a 2-resolving dominating set in G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2R-domination number of G, denoted by γr2R(G), is the smallest cardinality of a restrained 2-resolving dominating set in G. Any restrained 2-resolving dominating set of card

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Cabaro, Jean Mansanadez, and Helen Rara. "On 2-Resolving Dominating Sets in the Join, Corona and Lexicographic Product of Two Graphs." European Journal of Pure and Applied Mathematics 15, no. 3 (2022): 1417–25. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4426.

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Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set for G if, for any distinct vertices u, w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. A 2-resolving set S ⊆ V (G) which isdominating is called a 2-resolving dominating set or simply 2R-dominating set in G. The minimum cardinality of a 2-resolving dominating set in G, denoted by γ2R(G), is called the 2R-domination number of G. Any 2R-dominating set of cardinality γ2R(G) is then referred to as a γ2R-set in G. This study deals wit

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11

Mahistrado, Angelica Mae, and Helen Rara. "$1$-movable $2$-Resolving Hop Domination in Graph." European Journal of Pure and Applied Mathematics 16, no. 3 (2023): 1464–79. http://dx.doi.org/10.29020/nybg.ejpam.v16i3.4770.

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Let $G$ be a connected graph. A set $S$ of vertices in $G$ is a 1-movable 2-resolving hop dominating set of $G$ if $S$ is a 2-resolving hop dominating set in $G$ and for every $v \in S$, either $S\backslash \{v\}$ is a 2-resolving hop dominating set of $G$ or there exists a vertex $u \in \big((V (G) \backslash S) \cap N_G(v)\big)$ such that $\big(S \backslash \{v\}\big) \cup \{u\}$ is a 2-resolving hop dominating set of $G$. The 1-movable 2-resolving hop domination number of $G$, denoted by $\gamma^{1}_{m2Rh}(G)$ is the smallest cardinality of a 1-movable 2-resolving hop dominating set of $G$.

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Mahistrado, Angelica Mae, and Helen Rara. "Restrained 2-Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 286–303. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4665.

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    Let G be a connected graph. A set S ⊆ V (G) is a restrained 2-resolving hop dominating set of G if S is a 2-resolving hop dominating set of G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving hop domination number of G, denoted by γr2Rh(G) is the smallest cardinality of a restrained 2-resolving hop dominating set of G. This study aims to combine the concept of hop domination with the restrained 2-resolving sets of graphs. The main results generated in this study include the characterization of restrained 2-resolving hop dominating sets in

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13

Mohamad, Jerson Saguin, and Helen M. Rara. "On Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 14, no. 3 (2021): 1015–23. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.4055.

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A set S of vertices in a connected graph G is a resolving hop dominating set of G if S is a resolving set in G and for every vertex v âˆˆ V (G) \ S there exists u âˆˆ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the resolving hop domination number of G. This paper presents the characterizations of the resolving hop dominating sets in the join, corona and lexicographic product of two graphs and determines the exact values of their corresponding resolving hop domination number.

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Mahistrado, Angelica Mae, and Helen Rara. "On 2-Resolving Hop Dominating Sets in the Join, Corona and Lexicographic Product of Graphs." European Journal of Pure and Applied Mathematics 15, no. 4 (2022): 1982–97. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4585.

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    Let G be a connected graph. A set S of vertices in G is a 2-resolving hop dominating set of G if S is a 2-resolving set in G and for every vertex x ∈ V (G)\S there exists y ∈ S such that dG(x, y) = 2. The minimum cardinality of a set S is called the 2-resolving hop domination number of G and is denoted by γ2Rh(G). This study aims to combine the concept of hop domination with the 2-resolving sets of graphs. The main results generated in this study include the characterization of 2-resolving hop dominating sets in the join, corona and lexicographic product of two grap

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15

Subramanian, Hemalathaa, and Subramanian Arasappan. "Secure Resolving Sets in a Graph." Symmetry 10, no. 10 (2018): 439. http://dx.doi.org/10.3390/sym10100439.

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Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u1, u2, …, uk} of V(G) is called a resolving set (locating set) if for any x ∈ V(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, x)), is different for different x. The minimum cardinality of a resolving set is called the dimension of G and is denoted by dim(G). A security concept was introduced in domination. A subset D of V(G) is called a dominating set of G if for any v in V – D, there exists u in D such that u and v are adjacent. A dominating set

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Mohamad, Jerson, and Helen Rara. "Strong Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 131–43. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4578.

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A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V (G) if v is in any shortest u-w path or if u is in any shortest v-w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W. A set S subset of V (G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characte

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17

Mohamed, Basma, Linda Mohaisen, and Mohamed Amin. "Binary Equilibrium Optimization Algorithm for Computing Connected Domination Metric Dimension Problem." Scientific Programming 2022 (October 6, 2022): 1–15. http://dx.doi.org/10.1155/2022/6076369.

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We consider, in this paper, the NP-hard problem of finding the minimum connected domination metric dimension of graphs. A vertex set B of a connected graph G = (V, E) resolves G if every vertex of G is uniquely identified by its vector of distances to the vertices in B. A resolving set B of G is connected if the subgraph B ¯ induced by B is a nontrivial connected subgraph of G. A resolving set is dominating if every vertex of G that does not belong to B is a neighbor to some vertices in B. The cardinality of the smallest resolving set of G, the cardinality of the minimal connected resolving se

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Dapena, Adriana, Daniel Iglesia, Francisco J. Vazquez-Araujo, and Paula M. Castro. "New Computation of Resolving Connected Dominating Sets in Weighted Networks." Entropy 21, no. 12 (2019): 1174. http://dx.doi.org/10.3390/e21121174.

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In this paper we focus on the issue related to finding the resolving connected dominating sets (RCDSs) of a graph, denoted by G. The connected dominating set (CDS) is a connected subset of vertices of G selected to guarantee that all vertices in the graph are connected to vertices in the CDS. The connected dominating set with minimum cardinality, or minimum CDS (MCDS), is an adequate virtual backbone for information interchange in a network. When distinct vertices of G have also distinct representations with respect to a subset of vertices in the MCDS, it is said that the MCDS includes a resol

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Kusumawardani, I., Dafik, E. Y. Kurniawati, I. H. Agustin, and R. Alfarisi. "On resolving efficient domination number of path and comb product of special graph." Journal of Physics: Conference Series 2157, no. 1 (2022): 012012. http://dx.doi.org/10.1088/1742-6596/2157/1/012012.

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Abstract We use finite, connected, and undirected graph denoted by G. Let V (G) and E(G) be a vertex set and edge set respectively. A subset D of V (G) is an efficient dominating set of graph G if each vertex in G is either in D or adjoining to a vertex in D. A subset W of V (G) is a resolving set of G if any vertex in G is differently distinguished by its representation respect of every vertex in an ordered set W. Let W = {w 1, w 2, w 3, …, wk } be a subset of V (G). The representation of vertex υ ∈ G in respect of an ordered set W is r(υ|W) = (d(υ, w 1),d(υ, w 2), …, d(υ, wk )). The set W is

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Vasuki, Manimozhi, Ramachandramoorthi Shanmugapriya, Miroslav Mahdal, and Robert Cep. "A Study on Fuzzy Resolving Domination Sets and Their Application in Network Theory." Mathematics 11, no. 2 (2023): 317. http://dx.doi.org/10.3390/math11020317.

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Consider a simple connected fuzzy graph (FG) G and consider an ordered fuzzy subset H = {(u1, σ(u1)), (u2, σ(u2)), ...(uk, σ(uk))}, |H| ≥ 2 of a fuzzy graph; then, the representation of σ − H is an ordered k-tuple with regard to H of G. If any two elements of σ − H do not have any distinct representation with regard to H, then this subset is called a fuzzy resolving set (FRS) and the smallest cardinality of this set is known as a fuzzy resolving number (FRN) and it is denoted by Fr(G). Similarly, consider a subset S such that for any u∈S, ∃v∈V − S, then S is called a fuzzy dominating set only

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Umilasari, Reni, Liliek Susilowati, Slamin, AFadekemi Janet Osaye, and Ilham Saifudin. "Some Properties of Dominant Local Metric Dimension." Statistics, Optimization & Information Computing 12, no. 6 (2024): 1912–20. http://dx.doi.org/10.19139/soic-2310-5070-2062.

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Let $G$ be a connected graph with vertex set $V$. Let $W_l$ be an ordered subset defined by $W_l=\{w_1,w_2,\dots,w_n\}\subseteq V(G)$. Then $W_l$ is said to be a dominant local resolving set of $G$ if $W_l$ is a local resolving set as well as a dominating set of $G$. A dominant local resolving set of $G$ with minimum cardinality is called the dominant local basis of $G$. The cardinality of the dominant local basis of $G$ is called the dominant local metric dimension of $G$ and is denoted by $Ddim_l(G)$. We characterize the dominant local metric dimension for any graph $G$ and for some commonly

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Cabaro, Jean Mansanadez, and Helen Rara. "Restrained 2-Resolving Sets in the Join, Corona and Lexicographic Product of Two Graphs." European Journal of Pure and Applied Mathematics 15, no. 3 (2022): 1229–36. http://dx.doi.org/10.29020/nybg.ejpam.v15i3.4427.

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Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set for G if, for any distinct vertices u, w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. A set S ⊆ V (G) is a restrained 2-resolving set in G if S is a 2-resolving set in G and S = V (G) or ⟨V (G)\S⟩ has no isolated vertex. The restrained 2-resolving number of G, denoted by rdim2(G), is the smallest cardinality of a restrained 2-resolving set in G. A restrained 2-resolving set of cardinality rdim2(G) is then referred to as an rdim

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Yi, Eunjeong. "Bounds on the sum of broadcast domination number and strong metric dimension of graphs." Discrete Mathematics, Algorithms and Applications 12, no. 01 (2020): 2050010. http://dx.doi.org/10.1142/s179383092050010x.

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Let [Formula: see text] be a connected graph of order at least two with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text]. A function [Formula: see text] is called a dominating broadcast function of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], and the broadcast domination number, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all dominati

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Shanmugapriya, Ramachandramoorthi, Perichetla Kandaswamy Hemalatha, Lenka Cepova, and Jiri Struz. "A Study of Independency on Fuzzy Resolving Sets of Labelling Graphs." Mathematics 11, no. 16 (2023): 3440. http://dx.doi.org/10.3390/math11163440.

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Considering a fuzzy graph G is simple and can be connected and considered as a subset H=u1,σu1,u2,σu2,…uk,σuk, |H|≥2; then, every two pairs of elements of σ−H have a unique depiction with the relation of H, and H can be termed as a fuzzy resolving set (FRS). The minimal H cardinality is regarded as the fuzzy resolving number (FRN), and it is signified by FrG. An independence set is discussed on the FRS, fuzzy resolving domination set (FRDS), and Fuzzy modified antimagic resolving set (FMARS). In this paper, we discuss the independency of FRS and FMARS in which an application has also been deve

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Prihandini, Rafiantika Megahniah, Nabilah Ayu Az-Zahra, Dafik, Antonius Cahya Prihandoko, and Robiatul Adawiyah. "Resolving Independent Dominating Set pada Graf Bunga, Graf Gear, dan Graf Bunga Matahari." Contemporary Mathematics and Applications (ConMathA) 5, no. 2 (2023): 64–78. http://dx.doi.org/10.20473/conmatha.v5i2.47046.

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Resolving independent dominating set is the development of metric dimension and independent dominating set. Resolving independent dominating sets is a concept which discusses about determining the minimum vertex on a graph provided that the vertex that becomes the dominating set can dominate the surrounding vertex and there are no two adjacent vertices dominator and also meet the requirement of metric dimension where each vertex in graph G must have a different representation which respect to the resolving independent dominating set . In this study, we examined the resolving independent domina

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Abragan, Armalene, and Helen M. Rara. "On 1-movable Strong Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 16, no. 2 (2023): 763–72. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4658.

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A set S is a 1-movable strong resolving hop dominating set of G if for every v ∈ S, either S\{v} is a strong resolving hop dominating set or there exists a vertex u ∈ (V (G)\S)∩NG(v) such that (S \ {v}) ∩ {u} is a strong resolving hop dominating set of G. The minimum cardinality of a 1-movable strong resolving hop dominating set of G is denoted by γ 1 msRh(G). In this paper, we obtained the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolving hop dominating sets in these types of

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Sumaoy, Helyn Cosinas, and Helen M. Rara. "On Restrained Strong Resolving Domination in Graphs." European Journal of Pure and Applied Mathematics 14, no. 4 (2021): 1367–78. http://dx.doi.org/10.29020/nybg.ejpam.v14i4.4112.

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A set S âŠ† V (G) is a restrained strong resolving dominating set in G if S is a strongresolving dominating set in G and S = V (G) or âŸ¨V (G) \ SâŸ© has no isolated vertex. The restrained strong resolving domination number of G, denoted by Î³rsR(G), is the smallest cardinality of a restrained strong resolving dominating set in G. In this paper, we present characterizations of the restrained strong resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the exact value of the restrained strong resolving domination number of each of these graphs.

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Humaizah, R., Dafik, A. I. Kristiana, I. H. Agustin, and E. Y. Kurniawati. "On the resolving strong domination number of some wheel related graphs." Journal of Physics: Conference Series 2157, no. 1 (2022): 012015. http://dx.doi.org/10.1088/1742-6596/2157/1/012015.

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Abstract This study aims to analyse the resolving strong dominating set. This concept combinations of two notions, they are metric dimension and strong domination set. By a resolving strong domination set, we mean a set D s ⊂ V(G) which satisfies the definition of strong dominating set as well as resolving set. The resolving strong domination number of graph G, denoted by γrst (G), is the minimum cardinality of resolving strong dominating set of G. In this paper, we determine the resolving strong domination number of some wheel related graphs, namely helm graph Hn , gear graph Gn , and flower

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G, Mahadevan, A. Selvam Avadayappan, and Twinkle Johns. "The Graphs Whose Sum of Global Connected Domination Number and Chromatic Number is 2n-5." Mapana - Journal of Sciences 11, no. 4 (2012): 91–98. http://dx.doi.org/10.12723/mjs.23.7.

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A subset S of vertices in a graph G = (V,E) is a dominating set if every vertex in V-S is adjacent to atleast one vertex in S. A dominating set S of a connected graph G is called a connected dominating set if the induced sub graph < S > is connected. A set S is called a global dominating set of G if S is a dominating set of both G and . A subset S of vertices of a graph G is called a global connected dominating set if S is both a global dominating and a connected dominating set. The global connected domination number is the minimum cardinality of a global connected dominating set of G an

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Alfarisi, Ridho, Dafik, and Arika Indah Kristiana. "Resolving domination number of graphs." Discrete Mathematics, Algorithms and Applications 11, no. 06 (2019): 1950071. http://dx.doi.org/10.1142/s179383091950071x.

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For a set [Formula: see text] of vertices of a graph [Formula: see text], the representation multiset of a vertex [Formula: see text] of [Formula: see text] with respect to [Formula: see text] is [Formula: see text], where [Formula: see text] is a distance between of the vertex [Formula: see text] and the vertices in [Formula: see text] together with their multiplicities. The set [Formula: see text] is a resolving set of [Formula: see text] if [Formula: see text] for every pair [Formula: see text] of distinct vertices of [Formula: see text]. The minimum resolving set [Formula: see text] is a m

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31

P, Bhaskarudu. "MATCHING DOMINATION IN GRAPHS." International Journal of Advances in Scientific Research and Engineering (ijasre 3, no. 4 (2017): 18–25. https://doi.org/10.5281/zenodo.576391.

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<em>A dominating set D is called a connected dominating set, if it induces a connected subgraph in G. Since a dominating set must contain atleast one vertex from every component of G, it follows that a connected dominating set for a graph G exists if and only if G is connected. The minimum of cardinalities of the connected dominating sets of G is called the connected domination number of G and is denoted by </em><em>(G). We have defined new parameter called the matching dominating set and the matching domination number.</em>

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Jasmine.S.E, Annie, and K. Ameenal Bibi. "Inverse Connected and Disjoint Connected Domination Number of a Jump Graph." International Journal of Engineering & Technology 7, no. 4.10 (2018): 585. http://dx.doi.org/10.14419/ijet.v7i4.10.21288.

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Let D be the minimum connected dominating set of a jump graph . If of contains a connected dominating set , then is called the inverse connected dominating set of the jump graph . The minimum cardinality of an inverse connected dominating set is the inverse connected domination number of the jump graph, denoted by. The disjoint connected domination number, of the jump graph , is the minimum cardinality of the union of two disjoint connected dominating set of . In this paper we have established bounds, exact values of and graph theoretic relations between the inverse connected domination number

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33

Hamja, Jamil, Imelda S. Aniversario, and Catherina I. Merca. "Weakly Connected Hop Domination in Graphs Resulting from Some Binary Operations." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 454–64. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4587.

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Let G = (V(G),E(G)) be a simple connected graph. A set S ⊆ V(G) is a weakly connected hop dominating set of G if for every q ∈ V \ S, there exists r ∈ S such that dG(q,r) = 2, the subgraph weakly induced by S, denoted by ⟨S⟩w = ⟨NG[S],Ew⟩ where Ew = {qr ∈ E(G) : q ∈ S or r∈S } is connected and S is a dominating set of G. The minimum cardinality of a weakly connected hop dominating set of G is called weakly connected hop domination number and is denoted by γwh(G). In this paper, the authors show and explore the concept of weakly connected hop dominating set. The weakly connected hop dominating

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Enriquez, Enrico L., and Albert D. Ngujo. "Clique doubly connected domination in the join and lexicographic product of graphs." Discrete Mathematics, Algorithms and Applications 12, no. 05 (2020): 2050066. http://dx.doi.org/10.1142/s1793830920500664.

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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] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique do

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35

Monsanto, Gerald Bacon, and Helan Rara. "Resolving Domination in Graphs under Some Binary Operations." European Journal of Pure and Applied Mathematics 16, no. 1 (2023): 18–28. http://dx.doi.org/10.29020/nybg.ejpam.v16i1.4643.

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In this paper, we investigate the concept of resolving dominating set in a graph. In particular, we characterize the resolving dominating sets in the join, corona and lexicographic product of two graphs and determine the resolving domination number of these graphs.

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Elavarasan, Krishnasamy, Tharmalingam Gunasekar, Lenka Cepova, and Robert Cep. "Study on a Strong and Weak n-Connected Total Perfect k-Dominating set in Fuzzy Graphs." Mathematics 10, no. 17 (2022): 3178. http://dx.doi.org/10.3390/math10173178.

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In this paper, the concept of a strong n-Connected Total Perfect k-connected total perfect k-dominating set and a weak n-connected total perfect k-dominating set in fuzzy graphs is introduced. In the current work, the triple-connected total perfect dominating set is modified to an n-connected total perfect k-dominating set nctpkD(G) and number γnctpD(G). New definitions are compared with old ones. Strong and weak n-connected total perfect k-dominating set and number of fuzzy graphs are obtained. The results of those fuzzy sets are discussed with the definitions of spanning fuzzy graphs, strong

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Cyman, Joanna, Magdalena Lemańska, and Joanna Raczek. "On the doubly connected domination number of a graph." Open Mathematics 4, no. 1 (2006): 34–45. http://dx.doi.org/10.1007/s11533-005-0003-4.

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AbstractFor a given connected graph G = (V, E), a set $$D \subseteq V(G)$$ is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

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Alfarisi, Ridho, Sharifah Kartini Said Husain, Liliek Susilowati, and Arika Indah Kristiana. "Dominant Mixed Metric Dimension of Graph." Statistics, Optimization & Information Computing 12, no. 6 (2024): 1826–33. http://dx.doi.org/10.19139/soic-2310-5070-1925.

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For $k-$ordered set $W=\{s_1, s_2,\dots, s_k \}$ of vertex set $G$, the representation of a vertex or edge $a$ of $G$ with respect to $W$ is $r(a|W)=(d(a,s_1), d(a,s_2),\dots, d(a,s_k))$ where $a$ is vertex so that $d(a,s_i)$ is a distance between of the vertex $v$ and the vertices in $W$ and $a=uv$ is edge so that $d(a,s_i)=min\{d(u,s_i),d(v,s_i)\}$. The set $W$ is a mixed resolving set of $G$ if $r(a|W)\neq r(b|W)$ for every pair $a,b$ of distinct vertices or edge of $G$. The minimum mixed resolving set $W$ is a mixed basis of $G$. If $G$ has a mixed basis, then its cardinality is called mix

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Et. al., S. Muthammai,. "Nonsplit Neighbourhood Tree Domination Number In Connected Graphs." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 9 (2021): 3237–44. http://dx.doi.org/10.17762/turcomat.v12i9.5444.

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: Let G = (V, E) be a connected graph. A subset D of V is called a dominating set of G if N[D] = V. The minimum cardinality of a dominating set of G is called the domination number of G and is denoted by g(G). A dominating set D of a graph G is called a tree dominating set (tr - set) if the induced subgraph áDñ is a tree. The tree domination number γtr(G) of G is the minimum cardinality of a tree dominating set. A tree dominating set D of a graph G is called a neighbourhood tree dominating set (ntr - set) if the induced subgraph áN(D)ñ is a tree. The neighbourhood tree domination number γntr(G

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Anandha Selvam, D., and M. Davamani Christober. "VERY EXCELLENT DOMINATING WEAKLY CONNECTED SET DOMINATING SETS." Advances in Mathematics: Scientific Journal 9, no. 12 (2020): 11141–46. http://dx.doi.org/10.37418/amsj.9.12.94.

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Anandha Selvam, D., and M. Davamani Christober. "Dominating weakly connected set dominating bridge independent graphs." Malaya Journal of Matematik S, no. 1 (2019): 4–6. http://dx.doi.org/10.26637/mjm0s01/0002.

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Shalini, V., and Indra Rajasingh. "Inverse Domination in X-Trees and Sibling Trees." European Journal of Pure and Applied Mathematics 17, no. 2 (2024): 1082–93. http://dx.doi.org/10.29020/nybg.ejpam.v17i2.5038.

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A set $D$ of vertices in a graph $G$ is a dominating set if every vertex not in $D$ is adjacent to at least one vertex in $D$. The minimum cardinality of a dominating set in $G$ is called the domination number and is denoted by $\gamma(G)$. Let $D$ be a minimum dominating set of $G$. If $V-D$ contains a dominating set say $D^{'}$ of $G$, then $D^{'}$ is called an inverse dominating set with respect to $D$. The inverse domination number $\gamma^{'}(G)$ is the cardinality of a minimum inverse dominating set of $G$. A dominating set $D$ is called a connected dominating set or an independent domin

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Doren, Mylyn L., and Rene E. Leonida. "Doubly connected 2-domination in graphs." Applied Mathematical Sciences 19, no. 5 (2025): 229–36. https://doi.org/10.12988/ams.2025.919255.

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For a nontrivial connected graph $G$, a set $C \subseteq V(G)$ is a doubly connected 2-dominating set in $G$ if it is 2-dominating and both $\left\langle C\right\rangle$ and $\left\langle V(G) \backslash C\right\rangle$ are connected. The doubly connected 2-domination number $\gamma_{2cc}(G)$ of $G$, is the smallest cardinality of a doubly connected 2-dominating set in $G$. We prove some properties of the doubly connected 2-dominating set and give bounds to the doubly connected 2-domination number. Also, a connected graph with a given order and doubly connected $2$-domination number is constru

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Xaviour, X. Lenin, and S. Robinson Chellathurai. "Connected Geodetic Global Domination Number of a Graph." Journal of Electronics,Computer Networking and Applied Mathematics, no. 11 (September 1, 2021): 31–40. http://dx.doi.org/10.55529/jecnam.11.31.40.

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A set S of vertices in a connected graph {G=(V,E)} is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbour in D. A geodetic dominating set S is both a geodetic and a dominating set. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. The geodetic global domination number (geodetic domination number) is the minimum cardinality of a geodetic global dominating set (geodetic dominating s

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Dalal, Aseem. "On graphs with no proper perfect dominating set." Tamkang Journal of Mathematics 44, no. 4 (2013): 359–64. http://dx.doi.org/10.5556/j.tkjm.44.2013.975.

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A set of vertices in a graph is perfect dominating if every vertex outside the set is adjacent to exactly one vertex in the set, and is neighborhood connected if the subgraph induced by its open neighborhood is connected. In any graph the full set of vertices is perfect dominating, and in every connected graph the full set of vertices is neighborhood connected. It is shown that(i) in a connected graph, if the only neighborhood connected perfect dominating set is the full set of vertices, then the full set of vertices is also the only perfect dominating set; and (ii) if $ r \ge 3 $ and $ n_1, \

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S. Shirkol, Shailaja, Preeti B. Jinagouda, and A. R. Desai. "Doubly-Connected Dominating Energy of Graphs." Journal of University of Shanghai for Science and Technology 23, no. 09 (2021): 712–23. http://dx.doi.org/10.51201/jusst/21/09525.

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A connected dominating set D is said to be doubly-connected dominating set if the subgraph induced by the set V − D is connected. In this paper, we have defined a matrix called the doubly connected dominating matrix and obtained the the corresponding spectra and energy. Further, we have obtained the chemical applicability of the doubly connected energy followed by the mathematical properties.

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Prasanna, A., та N. Mohamedazarudeen. "Connected 𝐷 - Eccentric Domination in Graphs". Indian Journal Of Science And Technology 17, № 36 (2024): 3776–80. http://dx.doi.org/10.17485/ijst/v17i36.2672.

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Objectives: To introduce connected -eccentric point set, connected -eccentric number, connected -eccentric dominating set, connected -eccentric domination number in a graph and related concepts. Methods: -distance in graphs are used to find the connected -eccentric number and connected -eccentric domination number in graphs. Findings: The new term connected -eccentric domination in graphs are used in varies field like to construct least number of cell phone tower in low cost and traffic signal also. Novelty: Using the idea -distance, eccentricity in a graphs, the connected -eccentric dominatin

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Therese Sunitha Mary. "Split domsaturation and Some New Parameters." Proyecciones (Antofagasta) 43, no. 6 (2024): 1361–72. https://doi.org/10.22199/issn.0717-6279-6296.

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Let G be any connected graph. The split domination number γs (G) of G is the minimum cardinality of a split dominating set. The split domsaturation number dss (G) of a graph G is the least positive integer k such that every vertex of G lies in a split dominating set of cardinality k. A split dominating set S ⊆ V (G) is said to be connected split dominating set if < S > is connected. The minimum cardinality of all connected split dominating sets of G is called the connected split omination number of G and is denoted by γcs (G). The uniform split domination number γus (G) of a graph G is t

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Mahadevan, G., P. Niveditha, and C. Sivagnanam. "Outer Triple Connected Corona Domination Number of Graphs." Indian Journal Of Science And Technology 17, SPI1 (2024): 136–43. http://dx.doi.org/10.17485/ijst/v17sp1.250.

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Background/ Objective: Given a graph G, a dominating set is said to be corona dominating set if every vertex such that or there exist a vertex if then . A corona dominating set is said to be an outer triple connected corona dominating set if any three vertices in lie on a path. The minimum cardinality taken over all the outer triple connected corona dominating sets of is called outer triple connected corona dominating number and it is denoted by . The study aims to find the outer triple connected corona domination number of some graphs. Method: To obtain outer triple connected corona dominatio

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Hamja, Jamil, Imelda S. Aniversario, and Helen M. Rara. "On Weakly Connected Closed Geodetic Domination in Graphs Under Some Binary Operations." European Journal of Pure and Applied Mathematics 15, no. 2 (2022): 736–52. http://dx.doi.org/10.29020/nybg.ejpam.v15i2.4356.

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Let G be a simple connected graph. For S ⊆ V (G), the weakly connected closed geodetic dominating set S of G is a geodetic closure IG[S] which is between S and is the set of all vertices on geodesics (shortest path) between two vertices of S. We select vertices of Gsequentially as follows: Select a vertex v1 and let S1 = {v1}. Select a vertex v2 ̸= v1 and let S2 = {v1, v2}. Then successively select vertex vi ∈/ IG[Si−1] and let Si = {v1, v2, ..., vi} for i = 1, 2, ..., k until we select a vertex vk in the given manner that yields IG[Sk] = V (G). Also, the subgraph weakly induced ⟨S⟩w by S is c

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