Journal articles: 'Resolving connected dominating set'

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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-connected 2-resolving hop dominating sets in the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values.

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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 dominating set is secure. If for every , there exists such that is a resolving set, then the resolving set is secure. These four cardinality values are the metric dimension of , the connected metric dimension of , the secure metric dimension of , and the connected domination metric dimension of , respectively. They correspond to the cardinality of the smallest resolving set of , the minimal connected resolving set, the minimal secure resolving set, and the minimal connected domination resolving set. In this paper, we introduce the secure connected domination metric dimension of graphs. If each vertex in G can be uniquely recognized by its vector of distances to the vertices in Scddim, then every vertex set Scddim of a connected graph resolves G. If the subgraph induced by Scddim is a nontrivial connected subgraph of G, then the resolving set Scddim of G is connected. That resolving set is dominating if each vertex in G that is not an element of Scddim is a neighbor of some vertices in Scddim. If there is a v in D such that is a dominating set for any in then the dominating set is secure. If for every there exists such that is a resolving set, then the resolving set is secure. These four cardinality values are the metric dimension of $G$, the connected metric dimension of , the secure metric dimension of , and the connected domination metric dimension of G, respectively. They correspond to the cardinality of the smallest resolving set of , the minimal connected resolving set, the minimal secure resolving set, and the minimal connected domination resolving set. In this paper, we introduce the secure connected dominant metric dimension of some graphs such as triangular snake graph, path graph, star tree and alternate quadrilateral snake. In particular, we derive the explicit formulas for the subdivision of triangular snake graph, alternate triangular snake graph, total graph of cycle graph and bistar tree.

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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-movable resolving hop dominating set of G. This paper presents the characterization of the 1-movable resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding 1-movable resolving hop domination number.

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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 the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolvingdominating sets in these types of graphs and determine the exact values of their 1-movable strong resolving domination numbers.

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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 cardinality γr2R(G) is referred to as a γr2R-set in G. This study deals with the concept of restrained 2-resolving dominating set of a graph. It characterizes the restrained 2-resolving dominating set in the join, corona and lexicographic product of two graphs and determine the bounds or exact values of the restrained 2-resolving domination number of these graphs.

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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 with the concept of 2-resolving dominating set of a graph. It characterizes the 2-resolving dominating set in the join, corona and lexicographic product of two graphs and determine the bounds or exact values of the 2-resolving dominating number of these graphs.

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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 $G$. In this paper, we investigate the concept and study it for graphs resulting from some binary operations. Specifically, we characterize the 1-movable 2-resolving hop dominating sets in the join, corona and lexicographic products of graphs, and determine the bounds of the 1-movable 2-resolving hop domination number of each of these graphs.

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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 the join, corona, edge corona and lexicographic product of graphs, as well as their corresponding bounds or exact values.   

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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 graphs, as well as their corresponding bounds or exact values.   

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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 D is secure if for any u in V – D, there exists v in D such that (D – {v}) ∪ {u} is a dominating set. A resolving set R is secure if for any s ∈ V – R, there exists r ∈ R such that (R – {r}) ∪ {s} is a resolving set. The secure resolving domination number is defined, and its value is found for several classes of graphs. The characterization of graphs with specific secure resolving domination number is also done.

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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 characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number.

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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 set, and the cardinality of the minimal connected domination resolving set are the metric dimension of G, connected metric dimension of G, and connected domination metric dimension of G, respectively. We present the first attempt to compute heuristically the minimum connected dominant resolving set of graphs by a binary version of the equilibrium optimization algorithm (BEOA). The particles of BEOA are binary-encoded and used to represent which one of the vertices of the graph belongs to the connected domination resolving set. The feasibility is enforced by repairing particles such that an additional vertex generated from vertices of G is added to B, and this repairing process is iterated until B becomes the connected domination resolving set. The proposed BEOA is tested using graph results that are computed theoretically and compared to competitive algorithms. Computational results and their analysis show that BEOA outperforms the binary Grey Wolf Optimizer (BGWO), the binary Particle Swarm Optimizer (BPSO), the binary Whale Optimizer (BWO), the binary Slime Mould Optimizer (BSMO), the binary Grasshopper Optimizer (BGO), the binary Artificial Ecosystem Optimizer (BAEO), and the binary Elephant Herding Optimizer (BEHO).

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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 called a resolving set of G if r(u|W) ≠ r(υ|W) ∀ u, υ ∈ G. A subset Z of V (G) is called the resolving efficient dominating set of graph G if it is an efficient dominating set and r(u|Z) ≠ r(υ|Z) ∀ u, υ ∈ G. Suppose γre (G) denotes the minimum cardinality of the resolving efficient dominating set. In other word we call a resolving efficient domination number of graphs. We obtained γreG of some comb product graphs in this paper, namely Pm ⊲ Pn , Sm ⊲ Pn , and Km ⊲ Pn .

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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 resolving set (RS) of G. With this work, we explore different strategies to find the RCDS with minimum cardinality in complex networks where the vertices have different importances.

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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 if u is a strong arc. Now, again consider a subset F which is both a resolving and dominating set, then it is called a fuzzy resolving domination set (FRDS) and the smallest cardinality of this set is known as the fuzzy resolving domination number (FRDN) and it is denoted by Fγr(G) We have defined a few basic properties and theorems based on this FRDN and also developed an application for social network connection. Moreover, a few related statements and illustrations are discussed in order to strengthen the concept.

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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 known graphs in terms of their domination number to get some properties of dominant local metric dimension.

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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 rdim2-set in G. This study deals with the concept of restrained 2-resolving set of a graph. Itcharacterizes the restrained 2-resolving set in the join, corona and lexicographic product of two graphs and determine the bounds or exact values of the 2-resolving dominating number of these graphs.

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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 dominating broadcast functions [Formula: see text] of [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the set of vertices [Formula: see text] such that either [Formula: see text] lies on a [Formula: see text] geodesic or [Formula: see text] lies on a [Formula: see text] geodesic of [Formula: see text]. Let [Formula: see text] be a function and, for any [Formula: see text], let [Formula: see text]. We say that [Formula: see text] is a strong resolving function of [Formula: see text] if [Formula: see text] for every pair of distinct vertices [Formula: see text], and the strong metric dimension, [Formula: see text], of [Formula: see text] is the minimum of [Formula: see text] over all strong resolving functions [Formula: see text] of [Formula: see text]. For any connected graph [Formula: see text], we show that [Formula: see text]; we characterize [Formula: see text] satisfying [Formula: see text] equals two and three, respectively, and characterize unicyclic graphs achieving [Formula: see text]. For any tree [Formula: see text] of order at least three, we show that [Formula: see text], and characterize trees achieving equality. Moreover, for a tree [Formula: see text] of order [Formula: see text], we obtain the results that [Formula: see text] if [Formula: see text] is central, and that [Formula: see text] if [Formula: see text] is bicentral; in each case, we characterize trees achieving equality. We conclude this paper with some remarks and an open problem.

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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 developed.

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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 dominating set of flower graphs, gear graphs, and sunflower graphs.

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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 graphs and determine the bounds or exact values of their 1-movable strong resolving hop domination numbers.

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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 graph Fln . Through this paper, we will use the notations γst (G) and dim(G) which show the strong domination and dimension numbers.

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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 multiset basis of [Formula: see text]. If [Formula: see text] has a multiset basis, then its cardinality is called multiset dimension, denoted by [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is a dominating set for [Formula: see text] if every vertex of [Formula: see text] that is not in [Formula: see text] is adjacent to some vertex of [Formula: see text]. The minimum cardinality of the dominating set is a domination number, denoted by [Formula: see text]. A vertex set of some vertices in [Formula: see text] that is both resolving and dominating set is a resolving dominating set. The minimum cardinality of resolving dominating set is called resolving domination number, denoted by [Formula: see text]. In our paper, we investigate and establish sharp bounds of the resolving domination number of [Formula: see text] and determine the exact value of some family graphs.

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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 and is denoted by γgc(G). In this paper we characterize the classes of graphs for which γgc(G) + χ(G) = 2n-5 and 2n-6 of global connected domination number and chromatic number and characterize the corresponding extremal graphs.

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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 of the jump graph with other parameters of G.

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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 set of some special graphs, shadow of graphs, join, corona and Lexicographic product of two graphs are characterized. Also, the weakly connected domination number of the aforementioned graphs are determined. Keywords: weakly connected set, hop dominating set, hop domination number, weakly connected hop dominating set, and weakly connected hop domination number

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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 doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

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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 and weak arcs, dominating sets, perfect dominating sets, generalization of triple-connected total perfect dominating sets of fuzzy graphs, complete, connected, bipartite, cut node, tree, bridge and some other new notions of fuzzy graphs which are analyzed with a strong and weak nctpkD(G) set of fuzzy graphs. The order and size of the strong and weak nctpkD(G) fuzzy set are studied. Additionally, a few related theorems and statements are analyzed.

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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 mixed metric dimension, denoted by $dim_m(G)$. A set $W$ of vertices in $G$ is a dominating set for $G$ if every vertex of $G$ that is not in $W$ is adjacent to some vertex of $W$. The minimum cardinality of dominating set is domination number , denoted by $\gamma(G)$. A vertex set of some vertices in $G$ that is both mixed resolving and dominating set is a mixed resolving dominating set. The minimum cardinality of mixed resolving dominating set is called dominant mixed metric dimension, denoted by $\gamma_{mr}(G)$. In our paper, we will investigated the establish sharp bounds of the dominant mixed metric dimension of $G$ and determine the exact value of some family graphs.

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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) of G is the minimum cardinality of a neighbourhood tree dominating set. A tree dominating set D of a graph G is called a nonsplit tree dominating set (nstd - set) if the induced subgraph á V - D ñ is connected. The nonsplit tree domination number γnstd(G) of G is the minimum cardinality of a nonsplit tree dominating set. A neighbourhood tree dominating set D of G is called a nonsplit neighbourhood tree dominating set, if the induced subgraph áV(G) ‒ Dñ is connected. The nonsplit neighbourhood tree domination number γnsntr(G) of G is the minimum cardinality of a nonsplit neighbourhood tree dominating set of G. In this paper, bounds for γnsntr(G) and its exact values for some particular classes of graphs and cartesian product of some standard graphs are found.

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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 dominating set of $G$ according as the induced subgraph $\langle D \rangle$ is connected or independent in $G$. The minimum of the cardinalities of the connected dominating sets of $G$ or the independent dominating sets of $G$ is called the connected domination number $\gamma_{c} (G)$ or the independent domination number $\gamma_{i} (G)$ respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included.

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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 set) in G. In this paper we introduced and investigate the connected geodetic global domination number of certain graphs and some of the general properties are studied.

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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, \ldots ,n_r \ge 2 $, then in $K_{n_1,\ldots,n_r}$ the only perfect dominating set is the full set of vertices. Also, (iii) estimates are derived of how many edges can be removed from or added to $K_{n_1,\ldots ,n_r}$ while preserving the property described in (ii).

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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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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 the least positive integer k, such that every k-element split subset S of V is a dominating set in G. In this paper, we investigate several properties of these dominating sets.

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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 dominating set and its number in graphs are obtained. Some points, observation, bounds and theorem related to connected -eccentric domination set and its number in graphs are stated and proved. 2010 Mathematics Subject Classification: 05C12, 05C69 Keywords: D-eccentric node, D-eccentric node-set, D-eccentric dominating set, D-eccentric domination number, Connected D-eccentric dominating set, Connected D-eccentric domination number

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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 domination number say m by proving and . To prove for a graph G we find a outer triple connected corona dominating set of G with cardinality m and then to prove we prove by contradiction. Findings: We investigated the above parameter for some derived graphs of path, cycle and wheel graph. Novelty : Outer triple connected corona domination number is a new concept in which the conditions of corona domination and triple connected are linked together. Keywords: Corona Domination, Pendent Vertex, Support Vertex, Triple Connected, Isolated Vertex, Pendant vertes

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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 connected where ⟨S⟩w = ⟨N[S], Ew⟩ with Ew = {u, v ∈E(G) : u ∈ S or v ∈ S} and S is a dominating set of G. The minimum cardinality of weakly connected closed geodetic dominating set of G is denoted by γwcg(G). In this paper, the authors show and investigate the concept weakly connected closed geodetic dominating sets of some graphsand the join, corona, and Cartesian product of two graphs are characterized. The weakly connected closed geodetic domination numbers of these graphs are determined. Also, some relationships between weakly connected closed geodetic dominating set, weakly connected closed geodetic set, geodetic dominating set, and geodetic connected dominating set are established.

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Samuel, Libin Chacko, and Mayamma Joseph. "New results on connected dominating structures in graphs." Acta Universitatis Sapientiae, Informatica 11, no. 1 (2019): 52–64. http://dx.doi.org/10.2478/ausi-2019-0004.

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Abstract A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found.

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Journal articles on the topic 'Resolving connected dominating set'

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