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Journal articles on the topic 'Minimum equitable domination set'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

ANITHA, A., S. ARUMUGAM, and MUSTAPHA CHELLALI. "EQUITABLE DOMINATION IN GRAPHS." Discrete Mathematics, Algorithms and Applications 03, no. 03 (2011): 311–21. http://dx.doi.org/10.1142/s1793830911001231.

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Let D be a dominating set of a graph G = (V, E). For v ∈ D, let n1(v) = |N(v) ∩ (V - D)| and for w ∈ V - D, let n2(w) = |N(w) ∩ D|. Then D is called an equitable dominating set of type 1 if |n1(v1) - n1(v2)| ≤ 1 for all v1, v2 ∈ D and is called an equitable dominating set of type 2 if |n2(w1) - n2(w2)| ≤ 1 for all w1, w2 ∈ V - D. The minimum cardinality of an equitable dominating set of G of type 1 (type 2) is called the 1-equitable (2-equitable) domination number of G and is denoted by γ eq1 (G)(γ eq2 (G)). If D is an equitable dominating set of type 1 and type 2, then D is called an equitabl

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Caay, Mark, Rodolfo Maza, Rafael Duarte, Sergii R. Lana, Bril Soriano, and April Janer. "Equitable isolate domination in graphs." Gulf Journal of Mathematics 19, no. 1 (2025): 337–46. https://doi.org/10.56947/gjom.v19i1.2603.

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A subset S ⊆ V(G) is said to an equitable isolate dominating set of a graph G if it is both equitable dominating set of G and isolate dominating set of G. The minimum cardinality of an equitable isolate dominating set is called equitable isolate domination number of G and is denoted by γe0 (G). An equitable isolate dominating set S of G is called γe0 -set of G. In this paper, the authors give characterizations of equitable isolate dominating set of some graphs and graphs obtained from the join and corona of two graphs. Furthermore, the equitable isolate domination numbers of these graphs is de

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Sumathi, P., and G. Alarmelumangai. "Locating Equitable Domination and Independence Subdivision Numbers of Graphs." Bulletin of Mathematical Sciences and Applications 9 (August 2014): 27–32. http://dx.doi.org/10.18052/www.scipress.com/bmsa.9.27.

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Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wєV-D, N(u)∩D ≠ N(w)∩D, |N(u)∩D| ≠ |N(w)∩D|. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of ed

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Rana, Sohel, and Sk Md Abu Nayeem. "A linear time algorithm for minimum equitable dominating set in trees." Proyecciones (Antofagasta) 40, no. 4 (2021): 805–14. http://dx.doi.org/10.22199/issn.0717-6279-4552.

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Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum

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Caay, Mark, and Andrew Hernandez. "Perfect Equitable Isolate Dominations in Graphs." European Journal of Pure and Applied Mathematics 17, no. 2 (2024): 969–78. http://dx.doi.org/10.29020/nybg.ejpam.v17i2.4963.

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A subset $S \subseteq V(G)$ is said to be a perfect equitable isolate dominating set of a graph $G$ if it is both perfect equitable dominating set of $G$ and isolate dominating set of $G$. The minimum cardinality of a perfect equitable isolate dominating set is called perfect equitable isolate domination number of $G$ and is denoted by $\gamma_{pe0}(G)$. A perfect equitable isolate dominating set $S$ of $G$ is called $\gamma_{pe0}$-set of $G$. In this paper, the authors give characterizations of a perfect equitable isolate dominating set of some graphs and graphs obtained from the join and cor

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M.S.Mahesh. "Some More Results On Neighborhood Connected Two Out Degree Equitable Domination Number Of Graphs." Advances in Nonlinear Variational Inequalities 28, no. 6s (2025): 522–28. https://doi.org/10.52783/anvi.v28.4337.

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For any graph G, the out degree of a vertex v with respect to D is denoted by and it is defined by). A sub set D of V(G) is said to be two out degree equitable dominating set if D is dominating set and for any two vertices such that The minimum number of vertices in two out degree equitable dominating set is called two out degree equitable domination number and it is denoted by Based on the above concept of out degree we introduce a new domination parameter called neighborhood connected two out degree equitable domination number and study this domination number for some standard graphs In this

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M.S.Mahesh. "Some More Results on Neighborhood Connected Two Out Degree Equitable Domination Number of Graphs." Advances in Nonlinear Variational Inequalities 28, no. 5s (2025): 438–45. https://doi.org/10.52783/anvi.v28.3914.

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For any graph G, the out degree of a vertex v with respect to D is denoted by and it is defined by). A sub set D of V(G) is said to be two out degree equitable dominating set if D is dominating set and for any two vertices such that The minimum number of vertices in two out degree equitable dominating set is called two out degree equitable domination number and it is denoted by Based on the above concept of out degree we introduce a new domination parameter called neighborhood connected two out degree equitable domination number and study this domination number for some standard graphs In this

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Vaidya, S. K., and N. J. Kothari. "Some Results on Equi Independent Equitable Dominating Sets in Graph." Journal of Scientific Research 7, no. 3 (2015): 77–85. http://dx.doi.org/10.3329/jsr.v7i3.23412.

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A subset D of V(G) is called an equitable dominating set if for every ? ? V(G) - D, there exists a vertex u ? D such that u? ? E(G) and | deg(u) deg(?) | ? 1. A subset D of V(G) is called an equitable independent set if for any D of V(G), ? ? Ne(u) for all ? ? D - {u}where, Ne(u) = {? ? V(G)/? ? N(u), | deg(u) deg(?) | ? 1}. An equitable dominating set D is said to be an equi independent equitable dominating set if it is also an equitable independent set. The minimum cardinality of an equi independent equitable dominating set is called equi independent equitable domination number which is

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Ameenal Bibi, K., та P. Rajakumari. "Neighborhood Connected Two -Outdegree Equitable Domination of the ϑ-Obrazom Graphs of P_n,G_n and H_n". International Journal of Engineering & Technology 7, № 4.10 (2018): 399. http://dx.doi.org/10.14419/ijet.v7i4.10.21025.

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Let G=(V,E) be a connected graph. A two-outdegree equitable dominating set D of a graph G is called the neighborhood connected two-outdegree equitable dominating set (nc2oe-set) if the induced sub graph <N(D)> is connected. The minimum cardinality of the minimal neighborhood connected two-outdegree equitable dominating set is called neighborhood connected two-outdegree equitable domination number of G and is denoted by . In this paper, we initiated the a study of this parameter and obtained for the -obrazom graphs of and . Also, we investigated the two-outdegree equitable domination numb

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T. R., Thasneem, and Manju K. Menon. "Outdegree equitable domination number of certain graph operators." Proyecciones (Antofagasta) 43, no. 5 (2024): 1037–53. http://dx.doi.org/10.22199/issn.0717-6279-6275.

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For a vertex u in the dominating set D of a graph G, the number of edges from u to V-D is called the outdegree of u with respect to D, d°D, G(u). A dominating set D is called the outdegree equitable dominating set if the absolute value of the differences of outdegrees of any two vertices in D is at most one. The minimum cardinality of an outdegree equitable dominating set of G is called the outdegree equitable domination number of G, γoe(G). In this paper, we study the outdegree equitable domination number of certain graph operators such as complement, double graph, mycielskian and subdivision

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Vaidya, S. K., and A. D. Parmar. "Total Equitable Bondage Number of a Graph." Journal of Scientific Research 10, no. 3 (2018): 231–38. http://dx.doi.org/10.3329/jsr.v10i3.33940.

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If for any total dominating set D with ν ∈ V (G) − D there exists a vertex u ∈ D such that uν ∈ E (G) and |d(ν)−d(u)| ≤ 1 then D is called the total equitable dominating set. The minimum cardinality of the total equitable dominating set is called the total equitable domination number denoted by γet (G). The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G – E0) > γ(G). We introduced the concept of total equitable bondage number and proved several results.

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Banu Priya, S., A. Parthiban, and N. Srinivasan. "Equitable Power Domination Number of Mycielskian of Certain Graphs." International Journal of Engineering & Technology 7, no. 4.10 (2018): 842. http://dx.doi.org/10.14419/ijet.v7i4.10.26772.

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Let be a simple graph with vertex set and edge set . A set is called a power dominating set (PDS), if every vertex is observed by some vertices in by using the following rules: (i) if a vertex in is in PDS, then it dominates itself and all the adjacent vertices of and (ii) if an observed vertex in has adjacent vertices and if of these vertices are already observed, then the remaining one non-observed vertex is also observed by in . A power dominating set in is said to be an equitable power dominating set (EPDS), if for every there exists an adjacent vertex such that the difference between the

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Vaidya, S. K., and A. D. Parmar. "Some More Results on Total Equitable Bondage Number of A Graph." Journal of Scientific Research 11, no. 3 (2019): 303–9. http://dx.doi.org/10.3329/jsr.v11i3.40573.

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The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G-E0) > γ (G). An equitable dominating set D is called a total equitable dominating set if the induced subgraph < D > has no isolated vertices. The total equitable domination number γte(G) of G is the minimum cardinality of a total equitable dominating set of G. If γte(G) ≠ |V(G)| and <G-E0> contains no isolated vertices then the total equitable bondage number bte(G) of a graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γte(G-E0) &g

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G. Navamani. "Effects of Subdivision on Fair Domination in Cyclic Structures." Advances in Nonlinear Variational Inequalities 28, no. 4s (2025): 184–95. https://doi.org/10.52783/anvi.v28.3236.

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A fair dominating set (FDS) is a dominating set of a graph , such that all vertices in are dominated by the same number of vertices of . The fair domination number (FDN) is the minimum cardinality of a fair dominating set of , denoted by The domination subdivision number (FDSN) , represents the smallest number of edges in that needs to be subdivided (with each edge being subdivided at most once) to increase the domination number of the graph. The fair domination subdivision number, (or ) is the minimum number of edge subdivisions needed to be applied to the graph to increase (or decrease) the

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G. Navamani. "Optimizing Subdivisions on Fair Domination in Petersen Graph Structures." Advances in Nonlinear Variational Inequalities 28, no. 3s (2024): 20–29. https://doi.org/10.52783/anvi.v28.2845.

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A fair dominating set (FDS) J is a dominating set of a graph H, such that all vertices in V \J are dominated by the same number of vertices of J. The fair domination number (FDN) is the minimum cardinality of a fair dominating set of H, denoted by γ_fd (H). The domination subdivision number (FDSN) 〖Sd〗_γ (H), represents the smallest number of edges in H that needs to be subdivided (with each edge being subdivided at most once) to increase the domination number of the graph. The fair domination subdivision number, 〖Sd〗_(γ_fd)^+ (H) (or 〖Sd〗_(γ_fd)^- (H)) is the minimum number of edge subdivisio

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Esakkimuthu, A., and S. Mari Selvam. "Equitable Color Class Domination Number of Honeycomb Networks Graph and Inverse Equitable Color Class Domination Number of a Graph." Indian Journal Of Science And Technology 17, no. 36 (2024): 3800–3810. http://dx.doi.org/10.17485/ijst/v17i36.2458.

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Objectives: In this study, we find a new graph domination parameter named “Inverse Equitable Color Class Domination” and discuss the equitable color class domination number of honeycomb networks graph. Methods: Let be a connected graph. Assume a group containing colors. Let be an equitably colorable function. A dominating subset of is called an equitable color class dominating set if the number of dominating nodes in each color class is equal. The least possible cardinality of an equitable color class dominating set of is the equitable color class domination number itself. It is indicated by .

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A, Esakkimuthu, and Mari Selvam S. "Equitable Color Class Domination Number of Honeycomb Networks Graph and Inverse Equitable Color Class Domination Number of a Graph." Indian Journal of Science and Technology 17, no. 36 (2024): 3800–3810. https://doi.org/10.17485/IJST/v17i36.2458.

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Abstract <strong>Objectives:</strong> In this study, we find a new graph domination parameter named “Inverse Equitable Color Class Domination” and discuss the equitable color class domination number of honeycomb networks graph. <strong>Methods:</strong> Let be a connected graph. Assume a group containing colors. Let be an equitably colorable function. A dominating subset of is called an equitable color class dominating set if the number of dominating nodes in each color class is equal. The least possible cardinality of an equitable color class dominating set of is th

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Ahamed, T. Rabeeh, S. Ismail Mohideen, and R. Jahir Hussain. "Complementary Nil Equitable Domination in Fuzzy Graphs." Indian Journal Of Science And Technology 17, no. 38 (2024): 3942–46. http://dx.doi.org/10.17485/ijst/v17i38.2819.

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Objectives: To demonstrate an inventive showcase prominent as complementary nil equitable dominating set, complementary nil equitable domination value in fuzzy graphs (FGs), and pertinent innovations. Methods: Strong arc in FGs ideas are used to perceive complementary nil equitable domination value. Findings: Modern concepts of complementary nil equitable dominance in FGs will lead to a number of applications, including the creation of cell phone towers in remote places, traffic light signals, and Google Maps. Novelty: Employing the ideas of equitable and strong arc in FGs, the complementary n

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T, Rabeeh Ahamed, Ismail Mohideen S, and Jahir Hussain R. "Complementary Nil Equitable Domination in Fuzzy Graphs." Indian Journal of Science and Technology 17, no. 38 (2024): 3942–46. https://doi.org/10.17485/IJST/v17i38.2819.

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Abstract <strong>Objectives:</strong> To demonstrate an inventive showcase prominent as complementary nil equitable dominating set, complementary nil equitable domination value in fuzzy graphs (FGs), and pertinent innovations.<strong> Methods:</strong> Strong arc in FGs ideas are used to perceive complementary nil equitable domination value. <strong>Findings:</strong> Modern concepts of complementary nil equitable dominance in FGs will lead to a number of applications, including the creation of cell phone towers in remote places, traffic light signals, and Google Maps.

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Senthilkumar, B., Yanamandram B. Venkatakrishnan, and H. Naresh Kumar. "DOMINATION AND EDGE DOMINATION IN TREES." Ural Mathematical Journal 6, no. 1 (2020): 147. http://dx.doi.org/10.15826/umj.2020.1.012.

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Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with domination number equal to twice edge domination nu

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Hosamani, Sunilkumar M. "Degree Equitable Regular Set Domination in Graphs." National Academy Science Letters 41, no. 6 (2018): 379–83. http://dx.doi.org/10.1007/s40009-018-0651-0.

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Caay, Mark, and Marjun Durog. "On some independent equitable domination of graphs." Gulf Journal of Mathematics 11, no. 1 (2021): 57–64. http://dx.doi.org/10.56947/gjom.v11i1.668.

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One of the most interesting facts about the study of connectivity and working is in the area of domination. By this, we determine the independent equitable domination number of some graphs and some results. Moreover, we investigate graphs with the existence of independent equitable dominating set.

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A., ESAKKIMUTHU, MARI SELVAM S., S. VASANTHI., and D. GNANAVIJI. "VARIATION OF EQUITABLE COLOR CLASS DOMINATION NUMBER AFTER THE LINK REMOVAL IN GRAPHS." Xi'an Shiyou Daxue Xuebao (Ziran Kexue Ban)/Journal of Xi'an Shiyou University, Natural Sciences Edition 66, no. 08 (2023): 180–90. https://doi.org/10.5281/zenodo.8289140.

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<strong>Abstract</strong> Let  be a connected graph. Assume a group  containing colors. Let  be an equitably colorable function. A dominating subset  of  is called an equitable color class dominating set if the number of dominating nodes in each color class is equal. The least possible cardinality of an equitable color class dominating set of  is called the equitable color class domination number itself. It is indicated by . In this paper, we study the changing and unchanging of   Domination number after the link removal.

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Banu Priya, S., and N. Srinivasan. "Equitable Power Domination Number of Certain Graphs." International Journal of Engineering & Technology 7, no. 4.10 (2018): 349. http://dx.doi.org/10.14419/ijet.v7i4.10.20933.

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Let 12G"> be a graph with vertex set 12V"> , a set 12Sâٹ†V"> is said to be a power dominating set (PDS), if every vertex 12uâˆˆV-S"> is observed by some vertices in 12S"> using the following rules: (i) if a vertex 12v"> in 12G"> is in PDS, then it dominates itself and all the adjacent vertices of 12v"> and (ii) if an observed vertex 12v"> in 12G"> has 12k&gt;1"> adjacent vertices and if 12k-1"> of these vertices are already observed, then the remaining one non-observed vertex will also be observed by 12v"> in 12G"> . The degree 12d(v)"> of a ver

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Alex, Annie, and Sangeetha V. "Secure equitable subdivision number of graphs." Gulf Journal of Mathematics 17, no. 2 (2024): 216–25. http://dx.doi.org/10.56947/gjom.v17i2.2049.

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A dominating set S of a graph G(V, E) is called a secure dominating set, if for each vertex x belongs to V(G) \ S there exists at least one vertex y in S such that y belongs to the neighbourhood of x and a vertex y can be replaced by a vertex x without losing domination property. A secure dominating set S is a secure equitable dominating set if for each x in V(G) \ S and the vertex y in S, which replaces x, we have |deg(x)-deg(y)| is less than or equal to 1. In this paper, we initiate the study of a new domination parameter, the secure equitable subdivision number of graphs. Also, we introduce

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Wardani, Ayu Anisa, Lucia Ratnasari, Robertus Heri Soelistyo Utomo, and Siti Khabibah. "Inverse Domination Number and Inverse Total Domination of Sierpinski Star Graph." International Journal of Mathematics and Statistics Studies 12, no. 3 (2024): 71–79. http://dx.doi.org/10.37745/ijmss.13/vol12n37179.

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Given a graph G=(V(G),E(G)) consisting of the set of vertices V(G) and the set of edges E(G). For example, D(G) is a domination set of graph G with minimum cardinality, if V(G)-D(G) contains a domination set D^(-1) (G), then D^(-1) (G) is called the inverse domination set of graph G. The minimum cardinality of the inverse domination set of the graph G is called the inverse domination number, denoted by γ^(-1) (G). If D_t (G) is the total domination set of the graph G with minimal cardinality, and V(G)-D_t (G) contains the total domination set D_t^(-1) (G), then D_t^(-1) (G) is called the inver

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

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

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Tamizh Chelvam, T., and T. Asir. "Graphs with Constant Sum of Domination and Inverse Domination Numbers." International Journal of Combinatorics 2012 (August 27, 2012): 1–7. http://dx.doi.org/10.1155/2012/831489.

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A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of gr

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Nakanishi, Misa. "Generalized domination structure in cubic graphs." Journal of Combinatorial Mathematics and Combinatorial Computing 124 (March 19, 2025): 731–36. https://doi.org/10.61091/jcmcc124-48.

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The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is 2-connected and the length of each formed induced cycle is 0 mod 3. We label every three vertices in the induced cycles of length 0 mod 3. Then there is a way of labeling in which the set of all labeled vertices is the minimum dominating set of the resulting graph, and is contained in the minimum dominating set of the original graph. We also consider the remain

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Lia, Febby Desy, Nilamsari Kusumastuti, and Fransiskus Fran. "BILANGAN INVERS DOMINASI TOTAL PADA GRAF BUNGA DAN GRAF TRAMPOLIN." EPSILON: JURNAL MATEMATIKA MURNI DAN TERAPAN 16, no. 1 (2022): 1. http://dx.doi.org/10.20527/epsilon.v16i1.5160.

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Given a simple, finite, undirected and contains no isolated vertices graph , with is the set of vertices in and is the set of edges in . The set is called the dominating set in if for every vertex of is adjacent to at least one vertex in . The set is called the total dominating set in graph if for every vertex in is adjacent to at least one vertices in . If is the total domination set with minimum cardinality of the graph and contains another total domination set, for example , then is called the inverse set of total domination respect to . The minimum cardinality of an inverse set of total do

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Salasalan, Gemma Puebla, and Sergio R. Canoy, Jr. "Revisiting Domination, Hop Domination, and Global Hop Domination in Graphs." European Journal of Pure and Applied Mathematics 14, no. 4 (2021): 1415–28. http://dx.doi.org/10.29020/nybg.ejpam.v14i4.4144.

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A set S âŠ† V (G) is a hop dominating set of G if for each v âˆˆ V (G) \ S, there exists w âˆˆ S such that dG(v, w) = 2. It is a global hop dominating set of G if it is a hop dominating set of both G and the complementÂ of G. The minimum cardinality of a hop dominating (global hop dominating) set of G, denoted by Î³h(G)(resp.Î³gh(G)), is called the hop domination (resp. global hop domination) number of G. In this paper, we give some realization results involving domination, hop domination, and global hop domination parameters. Also, we give a rectification of a result found in a recent paper o

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Dr. B.K. Keerthiga Priyatharsini and Mrs. G. Uma. "Leaf Domination of Graphs." International Journal of Scientific Research in Modern Science and Technology 3, no. 1 (2024): 07–12. http://dx.doi.org/10.59828/ijsrmst.v3i1.169.

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Let G be connected graph. A dominating set D in G is said to be a leaf dominating set if <D> must have at least one leaf vertex. The minimum cardinality of a leaf dominating set is said to be the leaf domination number, denoted by Yld (G). Any minimal dominating set with minimum cardinality is called the Yld set. In this paper, we determine the leaf domination number and calculate the leaf domination number for standard graphs.

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Krishnakumari, B., and Y. B. Venkatakrishnan. "Double domination and super domination in trees." Discrete Mathematics, Algorithms and Applications 08, no. 04 (2016): 1650067. http://dx.doi.org/10.1142/s1793830916500671.

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A vertex of a graph [Formula: see text] is said to dominate itself and all its neighbors. A double dominating set (DDS) of a graph [Formula: see text] is a set [Formula: see text] of vertices such that every vertex of [Formula: see text] is dominated by at least two vertices of [Formula: see text]. The double domination number of a graph [Formula: see text] is the minimum cardinality of a DDS of [Formula: see text]. For a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a super dominating set SDS if for every vertex of [Formula: see text] there exists an extern

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Pramanik, Tarasankar, Sukumar Mondal, and Madhumangal Pal. "Minimum 2-Tuple Dominating Set of an Interval Graph." International Journal of Combinatorics 2011 (December 26, 2011): 1–14. http://dx.doi.org/10.1155/2011/389369.

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The k-tuple domination problem, for a fixed positive integer k, is to find a minimum size vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The case when k=2 is called 2-tuple domination problem or double domination problem. In this paper, the 2-tuple domination problem is studied on interval graphs from an algorithmic point of view, which takes O(n2) time, n is the total number of vertices of the interval graph.

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Basavanagoud, B., and S. M. Hosamani. "Complete Cototal Domination Number of a Graph." Journal of Scientific Research 3, no. 3 (2011): 547–55. http://dx.doi.org/10.3329/jsr.v3i3.7744.

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Let be a graph. A set of a graph is called a total dominating set if the induced subgraph has no isolated vertices. The total domination number of G is the minimum cardinality of a total dominating set of G. A total dominating set D is said to be a complete cototal dominating set if the induced subgraph has no isolated vertices. The complete cototal domination number of G is the minimum cardinality of a complete cototal dominating set of G. In this paper, we initiate the study of complete cototal domination in graphs and present bounds and some exact values for . Also its relationship with oth

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

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

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

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

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Dijkstra, Lukas, Andrei Gagarin, and Vadim Zverovich. "Weighted domination models and randomized heuristics." Utilitas Mathematica 123 (June 26, 2025): 61–86. https://doi.org/10.61091/um123-05.

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We consider the minimum weight and smallest weight minimum-size dominating set problems in vertex-weighted graphs and networks. The latter problem is a two-objective optimization problem, which is different from the classic minimum weight dominating set problem that requires finding a dominating set of the smallest weight in a graph without trying to optimize its cardinality. In other words, the objective of minimizing the size of the dominating set in the two-objective problem can be considered as a constraint, i.e. a particular case of finding Pareto-optimal solutions. First, we show how to

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S, Purushothama. "Critical and stable pendant domination." Open Journal of Mathematical Sciences 6, no. 1 (2022): 187–91. http://dx.doi.org/10.30538/oms2022.0187.

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Let \(S\) be a dominating set of a graph \(G\). The set \(S\) is called a pendant dominating set of \(G\) if the induced subgraph of \(S\) contains a minimum of one node of degree one. The minimum cardinality of the pendant dominating set in \(G\) is referred to as the pendant domination number of \(G\), indicated by \(\gamma_{pe}(G)\). This article analyzes the effect of \(\gamma_{pe}(G)\) when an arbitrary node or edge is removed.

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Vinodh Kumar A. R. "Inverse Double Domination in Graphs." Tuijin Jishu/Journal of Propulsion Technology 45, no. 03 (2024): 1124–27. http://dx.doi.org/10.52783/tjjpt.v45.i03.7347.

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A vertex set is a double – dominating set, if each vertex is adjacent to at least two vertices in . The minimum cardinality of a double dominating set of G, denoted by is called the double domination number of G. A double dominating set S of a graph is an inverse double dominating set, if contains a double dominating set say of . Then is called an inverse double dominating set. The inverse double domination number of is the minimum cardinality of an inverse double dominating set. In this paper, we obtain exact value of for some standard graphs and also, we establish some general results on thi

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Y. B., Venkatakrishnan, Naresh Kumar Hari, and Natarajan Chidambaram. "Total domination and vertex-edge domination in trees." Proyecciones (Antofagasta) 38, no. 2 (2019): 295–304. https://doi.org/10.22199/issn.0717-6279-3573.

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A vertex v of a graph G = (V,E) is said to ve-dominate every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set if every edge of E is ve-dominated by at least one vertex of S. The minimum cardinality of a vertex-edge dominating set of G is the vertex-edge domination number γve(G) . In this paper we prove (γt(T)−ℓ+1)/2 ≤ γve(T) ≤(γt(T)+ℓ−1)/2 and characterize trees attaining each of these bounds.

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W, Anjaline, and Stanis Arul Mary A. "Irredundant complete cototal dominating set." Journal of Computational Mathematica 6, no. 2 (2022): 75–78. http://dx.doi.org/10.26524/cm149.

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The complete cototal domination set is said to be irredundant complete cototal dominating set if for each u ∈ S, NG [S − {u}] ≠ [S]. The minimum cardinality taken over all an irredundant complete dominating set is called an irredundant complete cototal domination number and is denoted by γircc(G). Here a new domination parameter called an irredundant complete cototal dominating set was introduced and the study of bounds of γircc(G) was initiated.

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Jayasimman, Paulraj, Jenitha G, and Kumaravel A. "The Energy Graph for Minimum Majority Domination." International Journal of Engineering & Technology 7, no. 4.10 (2018): 562. http://dx.doi.org/10.14419/ijet.v7i4.10.21281.

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In this article we have introduced Minimum majority domination energy graph. A set is called a majority dominating set if at least half of the vertices either in or adjacent to the vertices . That is , . The minimum cardinality of a majority dominating set is called majority domination number . We defined majority dominating matrix and its energy values for some classes of graphs. Also some boundaries of Energy value of graph G are obtained.

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BHAT, P. G., R. S. BHAT, and SUREKHA R. BHAT. "RELATIONSHIP BETWEEN BLOCK DOMINATION PARAMETERS OF A GRAPH." Discrete Mathematics, Algorithms and Applications 05, no. 03 (2013): 1350018. http://dx.doi.org/10.1142/s1793830913500183.

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Two vertices u, w ∈ V, vv-dominate each other if they are incident on the same block. A set S ⊆ V is a vv-dominating set (VVD-set) if every vertex in V – S is vv-dominated by a vertex in S. The vv-domination number γvv = γvv(G) is the cardinality of a minimum VVD-set of G. Two blocks b1, b2 ∈ B(G) the set of all blocks of G, bb-dominate each other if there is a common cutpoint. A set L ⊆ B(G) is said to be a bb-dominating set (BBD set) if every block in B(G) – L is bb-dominated by some block in L. The bb-domination number γbb = γbb(G) is the cardinality of a minimum BBD-set of G. A vertex v an

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Annida, Khilwa, Siti Khabibah, Robertus Heri Soelistyo Utomo, and Lucia Ratnasari. "2-Distance and 3-Distance Domination Numbers of the Sierpinski Star Graph." Journal of Mathematics Research 16, no. 3 (2024): 49. http://dx.doi.org/10.5539/jmr.v16n3p49.

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The domination set D(G) in graph G=(V(G),E(G)) is a subset of the vertex set in graph G such that every vertex in V(G)\D(G) is adjacent to at least one vertex in D(G). The minimum cardinality of a domination set in graph G is called the domination number and is denoted as &gamma;(G). The set S_k (G) is called the k-distance domination set in graph G if every vertex v in V(G)\S_k (G) has a distance of less than or equal to k from at least one vertex in S_k (G). The minimum cardinality of a k-distance domination set in graph G is called the k-distance domination number and is denoted as &amp

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46

Almalki, Norah, and Pawaton Kaemawichanurat. "Domination and Independent Domination in Hexagonal Systems." Mathematics 10, no. 1 (2021): 67. http://dx.doi.org/10.3390/math10010067.

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A vertex subset D of G is a dominating set if every vertex in V(G)\D is adjacent to a vertex in D. A dominating set D is independent if G[D], the subgraph of G induced by D, contains no edge. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G, and the independent domination number i(G) of G is the minimum cardinality of an independent dominating set of G. A classical work related to the relationship between γ(G) and i(G) of a graph G was established in 1978 by Allan and Laskar. They proved that every K1,3-free graph G satisfies γ(G)=i(H). Hexagonal syst

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Dettlaff, Magda, Saeed Kosary, Magdalena Lemańska, and Seyed Sheikholeslami. "Weakly convex domination subdivision number of a graph." Filomat 30, no. 8 (2016): 2101–10. http://dx.doi.org/10.2298/fil1608101d.

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A set X is weakly convex in G if for any two vertices a,b ? X there exists an ab-geodesic such that all of its vertices belong to X. A set X ? V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number ?wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd?wcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the stu

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Pan, Zhuo, Peng Pan, and Chongshan Tie. "On the Relation Between the Domination Number and Edge Domination Number of Trees and Claw-Free Cubic Graphs." Mathematics 13, no. 3 (2025): 534. https://doi.org/10.3390/math13030534.

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For a connected graph G=(V,E), the dominating set in graph G is a subset of vertices F⊂V such that every vertex of V−F is adjacent to at least one vertex of F. The minimum cardinality of a dominating set of G, denoted by γ(G), is the domination number of G. The edge dominating set in graph G is a subset of edges S⊂E such that every edge of E−S is adjacent to at least one edge of S. The minimum cardinality of an edge dominating set of G, denoted by γ′(G), is the edge domination number of G. In this paper, we characterize all trees and claw-free cubic graphs with equal domination and edge domina

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

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

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S.B. Anupama, Y. B. Maralabhavi, and Venkanagouda M. Goudar. "Cototal edge domination number of a graph." Malaya Journal of Matematik 4, no. 02 (2016): 325–37. http://dx.doi.org/10.26637/mjm402/017.

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A set $F$ of a graph $G(V, E)$ is an edge dominating set if every edge in $E-F$ is adjacent to some edge in $F$. An edge domination number $\gamma^{\prime}(G)$ of $G$ is the minimum cardinality of an edge dominating set. An edge dominating set $F$ is called a cototal edge dominating set if the induced subgraph $\langle E-F\rangle$ doesnot contain isolated edge. The minimum cardinality of the cototal edge dominating set in $G$ is its domination number and is denoted by $\gamma_{c o t}^{\prime}(G)$. We investigate several properties of cototal edge dominating sets and give some bounds on the cot

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