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Journal articles on the topic 'Resolving dominating set'

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Published: 3 June 2025
Last updated: 31 July 2025

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1

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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3

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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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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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 ⊆ 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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7

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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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 ⊆ 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 ⟨V (G) \ S⟩ has no isolated vertex. The restrained strong resolving domination number of G, denoted by γrsR(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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9

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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10

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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11

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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12

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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13

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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14

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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15

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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16

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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17

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 ∈ V (G) \ S there exists u ∈ 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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18

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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19

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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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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21

Abragan, Armalene, and Helen Rara. "Restrained Strong Resolving Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 15, no. 4 (2022): 1472–81. http://dx.doi.org/10.29020/nybg.ejpam.v15i4.4484.

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A set S ⊆ V (G) is a restrained strong resolving hop dominating set in G if for every v ∈ V (G)\S, there exists w ∈ S such that dG(v, w) = 2 and S = V (G) or V (G)\S has no isolated vertex. The smallest cardinality of such a set, denoted by γrsRh(G), is called the restrained strong resolving hop domination number of 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 restrained strong resolving hop dominating sets in these types of graphs and determine the bounds or exact
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22

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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23

Sherra, John, and Badekara Sooryanarayana. "Unique Metro Domination of a Ladder." Mapana - Journal of Sciences 15, no. 3 (2016): 55–64. http://dx.doi.org/10.12723/mjs.38.6.

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A dominating set $D$ of a graph $G$ which is also a resolving set of $G$ is called a metro dominating set. A metro dominating set $D$ of a graph $G(V,E)$ is a unique metro dominating set (in short an UMD-set) if $|N(v) \cap D| = 1$ for each vertex $v\in V-D$ and the minimum cardinality of an UMD-set of $G$ is the unique metro domination number of $G$. In this paper, we determine unique metro domination number of $P_n\times P_2$.
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24

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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25

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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26

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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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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Ervani, R. S. R., Dafik, I. M. Tirta, R. Alfarisi, and R. Adawiyah. "On resolving total dominating set of sunlet graphs." Journal of Physics: Conference Series 1832, no. 1 (2021): 012020. http://dx.doi.org/10.1088/1742-6596/1832/1/012020.

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Umilasari, Reni, Liliek Susilowati, S. Slamin, and Savari Prabhu. "On the Dominant Local Resolving Set of Vertex Amalgamation Graphs." CAUCHY: Jurnal Matematika Murni dan Aplikasi 7, no. 4 (2023): 597–607. http://dx.doi.org/10.18860/ca.v7i4.18891.

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Basically, the new topic of the dominant local metric dimension which be symbolized by Ddim_l (H) is a combination of two concepts in graph theory, they were called the local metric dimension and dominating set. There are some terms in this topic that is dominant local resolving set and dominant local basis. An ordered subset W_l is said a dominant local resolving set of G if W_l is dominating set and also local resolving set of G. While dominant local basis is a dominant local resolving set with minimum cardinality. This study uses literature study method by observing the local metric dimensi
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30

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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31

Retnowardani, Dwi Agustin, Liliek Susilowati, Dafik, and Kamal Dliou. "Distance k-domination and k-resolving domination of the corona product of graphs." Statistics, Optimization & Information Computing 13, no. 1 (2024): 72–87. https://doi.org/10.19139/soic-2310-5070-2101.

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For two simple graphs $G$ and $H$, the corona product of $G$ and $H$ is the graph obtained by adding a copy of $H$ for every vertex of $G$ and joining each vertex of $G$ to its corresponding copy of $H$. For $k \geq 1$, a set of vertices $D$ in a graph $G$ is a distance $k$-dominating set if any vertex in $G$ is at a distance less or equal to $k$ from some vertex in $D$. The minimum cardinality overall distance $k$-dominating sets of $G$ is the distance $k$-domination number, denoted by $\gamma_k(G)$. The metric dimension of a graph is the smallest number of vertices required to distinguish al
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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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33

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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Prihandini, R. M., M. R. Rahmadani, and Dafik Dafik. "ANALYSIS OF RESOLVING EFFICIENT DOMINATING SET AND ITS APPLICATION SCHEME IN SOLVING ETLE PROBLEMS." BAREKENG: Jurnal Ilmu Matematika dan Terapan 18, no. 3 (2024): 1615–28. http://dx.doi.org/10.30598/barekengvol18iss3pp1615-1628.

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This research focuses on the analysis of Resolving Efficient Dominating Set (REDS) and its application in solving Electronic Traffic Law Enforcement (ETLE) problems using the Spatial Temporal Graph Neural Network (STGNN). Resolving Efficient Dominating Set (REDS) is a concept in graph theory that studies a set of points in a graph that efficiently monitors other points. It involves ensuring that each point v ∈ V (G) - D is dominated by exactly one point in D, with no adjacent points in D, and the representation of point v ∈ V (G) concerning D is not the same, which is termed as a resolving eff
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35

Shi, Xiaolong, Maryam Akhoundi, A. A. Talebi, and Masome Mojahedfar. "A Study on Regular Domination in Vague Graphs with Application." Advances in Mathematical Physics 2023 (May 20, 2023): 1–9. http://dx.doi.org/10.1155/2023/7098134.

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Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In graph theory, a dominating set (DS) for a graph G ∗ = X , E is a subset S of the vertices X such that every vertex not in S is adjacent to at least one member of S . The concept of DS in FGs has received the attention of many researchers due to its many applications in various fields such as computer science and electronic networks. In this paper, we introduce the notion of ϵ 1 , ϵ 2 , 2 -Regular vague domina
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Wang, Xiaoding, Li Xu, Shuming Zhou, and Joseph Liu. "A Resolving Set based Algorithm for Fault Identification in Wireless Mesh Networks." JUCS - Journal of Universal Computer Science 21, no. (3) (2015): 384–405. https://doi.org/10.3217/jucs-021-03-0384.

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Abstract: Wireless Mesh Networks (WMNs) have emerged as a key technology for next-generation wireless networking. By adding some Long-ranged Links, a wireless mesh network turns into a complex network with the characteristic of small worlds. As a communication backbone, the high fault tolerance is a significant property in communication of WMNs. In this paper, we design a novel malfunctioned router detection algorithm, denoted by A-SRS, on searching resolving set based on private neighbor of dominating set. The A-SRS not only offers a highly efficient solution to position malfunctioned routers
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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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Rao, Yongsheng, Ruiqi Cai, Ali Asghar Talebi, and Masomeh Mojahedfar. "Some Properties of Double Domination in Vague Graphs with an Application." Symmetry 15, no. 5 (2023): 1003. http://dx.doi.org/10.3390/sym15051003.

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This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue “Advances in graph theory and Symmetry/Asymmetry” of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dom
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Dliou, Kamal, Adinda Putri Aziza, Dafik Dafik, Arika Indah Kristiana, and Dwi Agustin Retnowardani. "Analysis of Resolving Efficient Dominating Set and Its Application Scheme in Multi-Step Time Series Forecasting of pH and Soil Moisture in Horizontal Farming." CAUCHY: Jurnal Matematika Murni dan Aplikasi 10, no. 1 (2025): 106–16. https://doi.org/10.18860/cauchy.v10i1.29960.

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This research focuses on the analyzing the Resolving Efficient Dominating Set (REDS) and its application scheme in horizontal farming using the Spatial Temporal Graph Neural Network (STGNN). Soil moisture and pH are crucial factors that affect the growth and yield, as they directly impact productivity and plant health. In cases where soil moisture and pH are lacking, various types of companion planting need to be watered. In such planting systems, a central role is needed to monitor soil moisture and pH levels effectively. The placement of operators in this system requires the application of m
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S. Anitha and A. Francina Shalini. "Similarity Measure of Plithogenic Cubic Vague Sets: Examples and Possibilities." Neutrosophic Systems with Applications 11 (October 21, 2023): 39–47. http://dx.doi.org/10.61356/j.nswa.2023.81.

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The crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets are the extension of the plithogenic set, in which elements are characterized by the number of attributes and each attribute can assume many values. To achieve more accuracy and precise exclusion, a contradiction or dissimilarity degree is specified between each attribute and the values of the dominating attribute. A plithogenic cubic vague set is a combination of a plithogenic cubic set and a vague set. The key tool for resolving problems with pattern recognition and clustering analysis is the similarity measure. In this research,
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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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42

Adawiyah, Robiatul, Fitriyatul Mardiyah, Dafik Dafik, Ika Hesti Agustin, Excelsa Suli Wildhatul Jannah, and Marsidi Marsidi. "ANALYSIS OF REAL RELATIVE ASYMMETRY IN URBAN TRANSPORTATION NETWORK PROBLEMS USING SPACE SYNTAX, REDS, AND MACHINE LEARNING CONCEPTS." BAREKENG: Jurnal Ilmu Matematika dan Terapan 19, no. 3 (2025): 1865–78. https://doi.org/10.30598/barekengvol19iss3pp1865-1878.

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In the context of urban growth and increasing population density, urban transportation networks face significant challenges such as traffic congestion, infrastructure limitations, and traffic law violations. This study integrates three analytical approaches—Space Syntax, Resolving Efficient Dominating Set (REDS), and Graph Neural Networks (GNN)—to identify strategic locations for the deployment of mobile Electronic Traffic Law Enforcement (ETLE) units and to forecast potential traffic violations. The research focuses on Malang City, Indonesia, and utilizes spatial data and ETLE violation recor
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Rao, Yongsheng, Saeed Kosari, Zehui Shao, Ruiqi Cai, and Liu Xinyue. "A Study on Domination in Vague Incidence Graph and Its Application in Medical Sciences." Symmetry 12, no. 11 (2020): 1885. http://dx.doi.org/10.3390/sym12111885.

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Fuzzy graphs (FGs), broadly known as fuzzy incidence graphs (FIGs), have been acknowledged as being an applicable and well-organized tool to epitomize and solve many multifarious real-world problems in which vague data and information are essential. Owing to unpredictable and unspecified information being an integral component in real-life problems that are often uncertain, it is highly challenging for an expert to illustrate those problems through a fuzzy graph. Therefore, resolving the uncertainty accompanying the unpredictable and unspecified information of any real-world problem can be don
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44

Muklisin, Ahmad, Arika Indah Kristiana, I. Made Tirta, and Dafik . "The development of RBL-STEM learning materials to improve the student’s forecasting thinking skills to solve resolving efficient dominating set problem." International Journal of Multidisciplinary Research and Growth Evaluation 4, no. 2 (2023): 194–99. http://dx.doi.org/10.54660/.ijmrge.2023.4.2.194-199.

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Forecasting thinking skills are the ability to apply the scientific method in understanding, predicting, discovering knowledge effectively problem. The Indicators of forecasting thinking skills include and identifying the solving characteristics of problems, using patterns recognition for prediction, expressing any possibility in situations that have not been observed. The thinking skills are not possed optimally by students. The aims of the research is to develop of RBL-STEM learning materials to improve the student's forecasting thinking skills to solve resolving efficient dominating set. Th
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45

Zulfatu Zainiyah, Arika Indah Kristiana, Slamin, and Dafik. "The development of RBL-STEM learning materials to improve the students’ forecasting skills in solving resolving efficient dominating set for hydroponic farming." World Journal of Advanced Research and Reviews 21, no. 1 (2024): 2233–41. http://dx.doi.org/10.30574/wjarr.2024.21.1.0217.

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Students' forecasting skills are currently still very low. This study aims to develop learning tools with the Riset-Based Learning (RBL) model and using the STEM (Science, Technology, Engineering, and Mathematics) approach to improve students' forecasting skills in solving resolving efficient dominating set (REDS) problems. The development of the RBL-STEM device was carried out using the 4D development model (define, design, develop, and disseminate). The developed learning tools meet valid criteria with a percentage of 92.3%, practical criteria with a percentage of 96.26%, and effective with
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46

Zulfatu, Zainiyah, Indah Kristiana Arika, Slamin, and Dafik. "The development of RBL-STEM learning materials to improve the students' forecasting skills in solving resolving efficient dominating set for hydroponic farming." World Journal of Advanced Research and Reviews 21, no. 1 (2024): 2233–41. https://doi.org/10.5281/zenodo.13367299.

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Students' forecasting skills are currently still very low. This study aims to develop learning tools with the Riset-Based Learning (RBL) model and using the STEM (Science, Technology, Engineering, and Mathematics) approach to improve students' forecasting skills in solving resolving efficient dominating set (REDS) problems. The development of the RBL-STEM device was carried out using the 4D development model (define, design, develop, and disseminate). The developed learning tools meet valid criteria with a percentage of 92.3%, practical criteria with a percentage of 96.26%, and effective with
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47

Dafik, Rasyid Ridlo Zainur, Humaizah Roizatul, Made Tirta I, and Nisviasari Rosanita. "The Analysis of the Implementation of Research-Based Learning with STEM Approach to Improving the Students' Metaliteracy in Solving the Resolving Strong Dominating Set Problem on traffic CCTV placement." International Journal of Current Science Research and Review 05, no. 08 (2022): 3106–17. https://doi.org/10.5281/zenodo.7010733.

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Abstract : Metaliteracy is urgently needed in the digital era, however, is still not widely owned by students. This metaliteracy requires high-level thinking skills to process various problems with various media sources, as well as it requires a collaborative environment. To achieve a good metaliteracy equipped with a higher thinking skill, we will implement research-based learning with a STEM approach in the learning process. This study uses a mixed-method by combining qualitative and quantitative methods. The subject of this study is the students of mathematics education as the candidates fo
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Aziza, M. O., Dafik, and A. I. Kristiana. "The analysis of the implementation of research-based learning on the students combinatorial thinking skills in solving a resolving perfect dominating set problem." Journal of Physics: Conference Series 1836, no. 1 (2021): 012057. http://dx.doi.org/10.1088/1742-6596/1836/1/012057.

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Hakim, R. A., Dafik, and I. M. Tirta. "The study of the implementation of research-based learning model to improve the students’ proving skills in dealing with the resolving efficient dominating set problem." Journal of Physics: Conference Series 1836, no. 1 (2021): 012059. http://dx.doi.org/10.1088/1742-6596/1836/1/012059.

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Muharromah M D, Kristiana A I, Slamin, and Dafik. "The development of RBL - STEM learning materials to improve students' computational thinking skills in solving convolutional neural network problems." World Journal of Advanced Research and Reviews 21, no. 1 (2024): 2373–81. http://dx.doi.org/10.30574/wjarr.2024.21.1.0219.

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In 21st century learning, computational thinking skills have become one of the essential competencies that need to be emphasised in the development of knowledge. To enhance computational thinking skills, research-based learning (RBL) with a science, technology, engineering and mathematics (STEM) approach, known as RBL-STEM, can be used. This study aims to explore RBL-STEM activities, describe the process and outcomes of developing RBL-STEM materials, and analyse data. In this research, the RBL-STEM framework is used to improve students' computational thinking skills in applying Convolutional N
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