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1

Hassan, Javier, and Sergio R. Canoy, Jr. "Grundy Total Hop Dominating Sequences in Graphs." European Journal of Pure and Applied Mathematics 16, no. 4 (2023): 2597–612. http://dx.doi.org/10.29020/nybg.ejpam.v16i4.4877.

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Let G = (V (G), E(G)) be an undirected graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Sˆ ={v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2G(vi) \Si−1j=1 N2G(vj ) ̸= ∅for every i ∈ {2, . . . , k}. If, in addition, Sˆ is a total hop dominating set of G, then S is a Grundy total hop dominating sequence. The maximum length of a Grundy total hop dominating sequence in a graph G, denoted by γth gr(G), is the Grundy total hop domination number of G. In this paper, we show that the Grundy total hop domination number of a graph G is between the total hop domination number and twice the Grundy hop domination number of G. Moreover, determine values or bounds of the Grundy total hop domination number of some graphs.
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2

Sivagnanam, C. "Neighborhood Total Domination and Colouring in Graphs." International Journal of Mathematics and Soft Computing 5, no. 1 (2015): 143. http://dx.doi.org/10.26708/ijmsc.2015.1.5.16.

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3

Wang, Kan, Changhong Lu, and Bing Wang. "Bounds on Neighborhood Total Domination Numberin Graphs." Bulletin of the Iranian Mathematical Society 45, no. 4 (2019): 1135–43. http://dx.doi.org/10.1007/s41980-018-0189-4.

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4

Henning, Michael A., and Nader Jafari Rad. "Bounds on neighborhood total domination in graphs." Discrete Applied Mathematics 161, no. 16-17 (2013): 2460–66. http://dx.doi.org/10.1016/j.dam.2013.05.014.

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5

Henning, Michael A., and Kirsti Wash. "Trees with large neighborhood total domination number." Discrete Applied Mathematics 187 (May 2015): 96–102. http://dx.doi.org/10.1016/j.dam.2015.01.037.

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6

RAD, NADER JAFARI. "A note on neighborhood total domination in graphs." Proceedings - Mathematical Sciences 125, no. 3 (2015): 271–76. http://dx.doi.org/10.1007/s12044-015-0241-8.

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7

Lee, Chuan-Min. "Exploring Dominating Functions and Their Complexity in Subclasses of Weighted Chordal Graphs and Bipartite Graphs." Mathematics 13, no. 3 (2025): 403. https://doi.org/10.3390/math13030403.

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Domination problems are fundamental problems in graph theory with diverse applications in optimization, network design, and computational complexity. This paper investigates {k}-domination, k-tuple domination, and their total domination variants in weighted strongly chordal graphs and chordal bipartite graphs. Specifically, the {k}-domination problem in weighted strongly chordal graphs and the total {k}-domination problem in weighted chordal bipartite graphs are shown to be solvable in O(n+m) time. For weighted proper interval graphs and convex bipartite graphs, we solve the k-tuple domination and total k-tuple domination problems in O(n2.371552log2(n)log(n/δ)), where δ is the desired accuracy. Furthermore, for weighted unit interval graphs, the k-tuple domination problem achieves a significant complexity improvement, reduced from O(nk+2) to O(n2.371552log2(n)log(n/δ)). These results are achieved through a combination of linear and integer programming techniques, complemented by totally balanced matrices, totally unimodular matrices, and graph-specific matrix representations such as neighborhood and closed neighborhood matrices.
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8

Sheikholeslami, Seyed Mahmoud, and Lutz Volkmann. "Outer independent total double Italian domination number." Computer Science Journal of Moldova 32, no. 1(94) (2024): 19–37. http://dx.doi.org/10.56415/csjm.v32.02.

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If $G$ is a graph with vertex set $V(G)$, then let $N[u]$ be the closed neighborhood of the vertex $u\in V(G)$. A total double Italian dominating function (TDIDF) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ satisfying (i) $f(N[u])\ge 3$ for every vertex $u\in V(G)$ with $f(u)\in\{0,1\}$ and (ii) the subgraph induced by the vertices with a non-zero label has no isolated vertices. A TDIDF is an outer-independent total double Italian dominating function (OITDIDF) on $G$ if the set of vertices labeled $0$ induces an edgeless subgraph. The weight of an OITDIDF is the sum of its function values over all vertices, and the outer independent total double Italian domination number $\gamma_{tdI}^{oi}(G)$ is the minimum weight of an OITDIDF on $G$. In this paper, we establish various bounds on $\gamma_{tdI}^{oi}(G)$, and we determine this parameter for some special classes of graphs.
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9

Jha, Anupriya, D. Pradhan, and S. Banerjee. "Algorithm and hardness results on neighborhood total domination in graphs." Theoretical Computer Science 840 (November 2020): 16–32. http://dx.doi.org/10.1016/j.tcs.2020.05.002.

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10

Lu, Changhong, Bing Wang та Kan Wang. "Algorithm complexity of neighborhood total domination and $$(\rho ,\gamma _{nt})$$ ( ρ , γ n t ) -graphs". Journal of Combinatorial Optimization 35, № 2 (2017): 424–35. http://dx.doi.org/10.1007/s10878-017-0181-6.

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11

Arumugam, S., and K. Raja Chandrasekar. "Linear time algorithm for dominator chromatic number of trestled graphs." Discrete Mathematics, Algorithms and Applications 11, no. 06 (2019): 1950066. http://dx.doi.org/10.1142/s1793830919500666.

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A dominator coloring (respectively, total dominator coloring) of a graph [Formula: see text] is a proper coloring [Formula: see text] of [Formula: see text] such that each closed neighborhood (respectively, open neighborhood) of every vertex of [Formula: see text] contains a color class of [Formula: see text] The minimum number of colors required for a dominator coloring (respectively, total dominator coloring) of [Formula: see text] is called the dominator chromatic number (respectively, total dominator chromatic number) of [Formula: see text] and is denoted by [Formula: see text] (respectively, [Formula: see text]). In this paper, we prove that the dominator coloring problem and the total dominator coloring problem are solvable in linear time for trestled graphs.
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12

Shanmugam, Megala, Mohanapriya Nagaraj, Karthika Ravichandran, and Abirami Kamaraj. "On dominator and total dominator coloring of duplication corresponding corona of path, pan, complete and sunlet graphs." Open Journal of Discrete Applied Mathematics 8, no. 2 (2025): 18–31. https://doi.org/10.30538/psrp-odam2025.0113.

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A dominator coloring of a graph \(\mathscr{G}\) is a proper coloring where each vertex of \(\mathscr{G}\) is within the closed neighborhood of at least one vertex from each color class. The minimum number of color classes required for a dominator coloring of \(\mathscr{G}\) is termed the dominator chromatic number. Additionally, a total dominator coloring of a graph \(\mathscr{G}\) is a proper coloring in which every vertex dominates at least one color class other than its own. The minimum number of color classes needed for a total dominator coloring of \(\mathscr{G}\) is known as the total dominator chromatic number. In this paper, our objective is to derive findings concerning dominator and total dominator coloring of the duplication corresponding corona of specific graphs.
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13

Zhang, Yameng, and Xia Zhang. "On the fractional total domatic numbers of incidence graphs." Mathematical Modelling and Control 3, no. 1 (2023): 73–79. http://dx.doi.org/10.3934/mmc.2023007.

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<abstract><p>For a hypergraph $ H $ with vertex set $ X $ and edge set $ Y $, the incidence graph of hypergraph $ H $ is a bipartite graph $ I(H) = (X, Y, E) $, where $ xy\in E $ if and only if $ x\in X $, $ y\in Y $ and $ x\in y $. A total dominating set of graph $ G $ is a vertex subset that intersects every open neighborhood of $ G $. Let $ \mathscr{M} $ be a family of (not necessarily distinct) total dominating sets of $ G $ and $ r_{\mathscr{M}} $ be the maximum times that any vertex of $ G $ appears in $ \mathscr{M} $. The fractional domatic number $ G $ is defined as $ FTD(G) = \sup_{\mathscr{M}}\frac{|\mathscr{M}|}{r_{\mathscr{M}}} $. In 2018, Goddard and Henning showed that the incidence graph of every complete $ k $-uniform hypergraph $ H $ with order $ n $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq 2k\geq 4 $. We extend the result to the range $ n > k\geq 2 $. More generally, we prove that every balanced $ n $-partite complete $ k $-uniform hypergraph $ H $ has $ FTD(I(H)) = \frac{n}{n-k+1} $ when $ n\geq k $ and $ H\ncong K_n^{(n)} $, where $ FTD(I(K_n^{(n)})) = 1 $.</p></abstract>
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14

Iordanski, Mikhail A. "Dominant sets with neighborhood for trees." Modeling and Analysis of Information Systems 32, no. 1 (2025): 32–41. https://doi.org/10.18255/1818-1015-2025-1-32-41.

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The subset $V' \subset V(G)$ forms a dominant set of vertices of the graph $G$ with a neighborhood $ \varepsilon$ if for any vertex $v \in V \backslash V'$ there is a vertex $u \in V'$ such that the length of the shortest chain connecting these vertices $d(v,u)\leqslant \varepsilon$; $\delta_{\varepsilon}(G)$ is the number of vertices in the minimal $\varepsilon$-dominating set; $\delta_{\varepsilon}(G) = 1$ for $r(G)\leqslant \varepsilon \leqslant d(G)$; for $ \varepsilon < r(G)$ the numbers $\delta_{\varepsilon}(G) > 1$, but the calculation of $\delta_{1}(G)=\delta(G)$ is an NP-complete problem. The paper considers class of trees $t_{d}^{\rho}$ of diameter $d$ whose degrees of all internal vertices are equal to $\rho$. Constructive descriptions of trees $t \in t_{d}^{\rho}$ are given. Procedures for calculating the values $\delta_{\varepsilon}(t)$ in the range $1\leqslant \varepsilon < r (t)$ have been developed. Asymptotic estimates for $\delta_{\varepsilon}(t)$ and their share of the total number of vertices $t \in t_{d}^{\rho}$ are set at $d \to \infty$. Computational examples are given.
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15

Sinar, T. Ricky Hafidsyah, Feby Milanie, Cut Nuraini, Abdiyanto Abdiyanto, and Ihsan Azhari. "Analysis of Determining Service Center Systems towards the Development of The Eastern Part of Medan City." International Journal Papier Advance and Scientific Review 4, no. 4 (2023): 91–105. http://dx.doi.org/10.47667/ijpasr.v4i4.258.

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The issues in the research area include a high concentration of built-up residential areas with the potential for slums and warehousing activities dominating trade and service areas or residential areas, leading to congestion. This requires attention, considering that the Eastern Part of Medan City has a high built-up area, necessitating the provision of affordable infrastructure and basic services for both newcomers and existing residents in the city. Several development theories and concepts can assist in determining and conceptualizing development in the research area. The study focuses on examining the Sub-Central Service System of the City as a center serving sub-urban areas. Various methods are used in determining service centers, including the scalogram method, Marshall's centrality index, and rank-size rule, which will ultimately provide recommendations for the central service system. After issuing recommendations from these three analyses, the determination of the existing urban internal structure is conducted using a scoring method to identify the central service location. Calculations from multiple service center analysis methods are scored and summed to obtain a total score. The neighborhood with the smallest score has the highest hierarchy value, and vice versa. It is revealed that the Sub-Central Service System is present in all sub-districts in the Eastern Part of Medan City, meaning each sub-district in each neighborhood already has comprehensive facilities. Consequently, the four sub-districts in the Eastern Part of Medan City have the potential for development as they meet service needs and can be directed to become service centers serving their respective sub-districts.
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16

Dundar, Bayram. "A simulated annealing with graph-based search for the social-distancing problem in enclosed areas during pandemics." PLOS ONE 20, no. 2 (2025): e0318380. https://doi.org/10.1371/journal.pone.0318380.

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During the pandemic, decision-makers offered many preventive policies to reduce the negative effects of the pandemic. The social distance rule in enclosed areas was implemented by educational institutions in any countries. In this study, we deal with the problem of assigning students to seats by considering the social distancing constraint and with objective of maximizing the total distance among the students. This problem is found to be similar to the Maximum Diversity Problem (MDP) in the literature. We name this new problem as Maximum Diversity Social Distancing problem (MDPs). A simulated annealing algorithm framework for MDPs (SA-MDPs) is proposed to identify an optimal or near-optimal solution within a reasonable computational time. A greedy random-based algorithm is presented to determine efficiently an initial feasible solution. The new neighborhood search procedure based on graph theory is introduced, in which the dominated, dominating, and nondominated seats are determined based on social distance. The proposed SA-MDPs is evaluated on classrooms with varying capacities and benchmarked against an off-the-shelf optimization solver. The computational tests demonstrated that the SA-MDP model consistently provided either proven optimal solutions or superior best-known solutions compared to a commercial solver, all within a reasonable CPU time.
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17

Parpan, V. I., N. V. Shumska, M. J. Rudeichuk-Kobzeva, and M. M. Mylenka. "Syntaxonomy of vegetation of Kalush hexachlorobenzene toxic waste landfill (Ivano-Frankivsk region)." Biosystems Diversity 24, no. 2 (2016): 364–70. http://dx.doi.org/10.15421/011648.

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The vegetation of a landfill of hexachlorobenzene toxic waste was studied. It is situated in the neighborhood of Kalush (Ivano-Frankivsk region) and has an area of 4.5 ha. As a result of damage to the containers, hazardous waste has contaminated the air, soil and aquifers at the test site and adjacent areas. During the period 2010–2012 measures were taken to recover and remove the mixture of toxic waste and contaminated soil from the landfill. In its place, unpolluted soil was brought to the landfill. Work was carried out to recultivate the territory. Nowadays natural succession of vegetation cover is observed. There is closed herbaceous cover in the western part of the landfill. The total projective herbaceous cover in the central and eastern parts varies from 10% to 60%. Vegetation composition of the landfill contains eight syntaxa of association rank that belong to seven alliances, six orders and five classes. Communities of the Phragmito-Magnocaricetea and Bolboschoenetea maritimi classes (ass. Typhetum laxmanii) grow in areas with excessive humidification. The central and eastern parts of the waste landfill are primarily occupied by halophytic communities of the Puccinellio distanti-Tripolietum vulgare association of the Asteretea tripolium class. Ruderal communities belong to three associations of the Artemisietea vulgaris class. These communities mainly occur in the periphery zone of Kalush landfill. Areas with a moderate moisture regime are occupied by ruderal communities of the Calamagrostietum epigeios association of the Agropyretea repentis class. The total number of vascular herbaceous plant species at the landfill is 119. The dominating groups are meadow, synanthropic and wetland species. The differentiation of vegetation cover is caused by heterogeneity of edaphic and hydrological conditions, also by different activity of succession processes.
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18

Adamış, Emel, and Fatih Pınarbaşı. "Unfolding visual characteristics of social media communication: reflections of smart tourism destinations." Journal of Hospitality and Tourism Technology 13, no. 1 (2022): 34–61. http://dx.doi.org/10.1108/jhtt-09-2020-0246.

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Purpose This study aims to explore the visual social media (SM) (Instagram) communication and the visual characteristics of smart tourism destination (STD) communication from destination marketing/management organizations (DMOs) and user-generated content (UGC) perspectives, which refer to projected image and perceived image, respectively. Design/methodology/approach Three DMO official accounts of STDs (Helsinki, Gothenburg and Lyon) and corresponding official hashtags were selected for the sample and total 6,000 post data (1,000 × 6) were retrieved from Instagram. Visual communication content was examined with a netnographic design over a proposed four-level visual content framework using corresponding methodological approaches (thematic analysis, visual analysis, object detection and text mining) for each level. Findings Among the eight emerging themes dominating the images, communication of smart elements conveys far less than expected textual and visual signals from DMOs despite their smart status, and in turn, from UGC as well. UGC revealed three extra image themes regardless of smartness perception. DMOs tend to project and give voice to their standard metropolitan areas and neighborhoods while UGCs focus on food-related and emotional elements. The findings show a partial overlap between DMOs and UGCs, revealing discrepancies in objects contained in visuals, hashtags and emojis. Additionally, as a rare attempt, the proposed framework for visual content analysis showed the importance of integrated methods to investigate visual content effectively. Research limitations/implications The number of attributes in visual analysis and focusing on the observed elements in text content (text, hashtags and emojis) are the limitations of the study in terms of methodology. Originality/value Apart from the multiple integrated methods used over a netnographic design, this study differs from existing SM and smart destinations intersection literature by attempting to fill a gap in focusing on and exploring visual SM communication, which is scarce in tourism context, for the contents generated by DMOs and users.
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19

C.Sivagnanam. "Neighborhood Total 2-Domination in Graphs." November 30, 2014. https://doi.org/10.5281/zenodo.826659.

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The graph G = (V,E) we mean a finite, undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n and m respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [3] and Haynes et.al [5-6].
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20

Singhwal, Nitisha, and Palagiri Venkata Subba Reddy. "Total vertex-edge domination in graphs: Complexity and algorithms." Discrete Mathematics, Algorithms and Applications, November 15, 2021. http://dx.doi.org/10.1142/s1793830922500318.

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Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.
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21

Salim, Jeffrey, Glee Tampipi, Albert Quinones, and Rosalio Artes Jr. "Connected Total Domination Neighborhood Polynomials Over Distance-Reducing Operations." International Journal of Mathematics and Computer Science, 2025, 767–70. https://doi.org/10.69793/ijmcs/03.2025/salim.

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22

Foucaud, Florent, and Michael A. Henning. "Locating-Total Dominating Sets in Twin-Free Graphs: a Conjecture." Electronic Journal of Combinatorics 23, no. 3 (2016). http://dx.doi.org/10.37236/5147.

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A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $\gamma_t^L(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \ge 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $\gamma_t^L(G) \le \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $\gamma_t^L(G) \le \frac{3}{4}n$.
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23

Chen, Qin. "Algorithm aspect on total Roman $\{2\}$-domination number of Cartesian products of paths and cycles." RAIRO - Operations Research, September 3, 2023. http://dx.doi.org/10.1051/ro/2023121.

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A total Roman $\{2\}$-dominating function (TR2DF) on a graph $G$ with vertex set $V$ is a function $f: V\rightarrow \{0,1,2\}$ having the property that for every vertex $v$ with $f(v)=0$, $\sum_{u\in N(v)}f(u)\geq 2$, where $N(v)$ represents the open neighborhood of $v$, and the subgraph of $G$ induced by the set of vertices with $f(v)>0$ has no isolated vertex. The weight of a TR2DF $f$ is the value $w(f)=\sum_{v\in V} f(v)$, and the minimum weight of a TR2DF of $G$ is the total Roman $\{2\}$-domination number $\gamma_{tR2}(G)$. The total Roman $\{2\}$-domination problem (TR2DP) is to determine the value $\gamma_{tR2}(G)$. In this paper, we first propose an integer linear programming (ILP) formulation for the TR2DP. Furthermore, we apply the discharging approach to determine the total Roman $\{2\}$-domination number for some Cartesian products of paths and cycles.
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Mahmoodi, Akram, Maryam Atapour, and Sepideh Norouzian. "On the signed strong total Roman domination number of graphs." Tamkang Journal of Mathematics, July 29, 2022. http://dx.doi.org/10.5556/j.tkjm.54.2023.3907.

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Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominationnumber of special classes of graphs.
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Casado, Alejandra, Jesús Sánchez-Oro, and Anna Martínez-Gavara. "Heuristics for the weighted total domination problem." TOP, February 10, 2025. https://doi.org/10.1007/s11750-025-00695-1.

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Abstract The weighted total domination problem (WTDP) belongs to the family of dominating set problems. Given an edge- and vertex- weighted graph, the WTDP consists in selecting a total dominating set D, such that the sum of vertices and edges weights of the subgraph induced by D plus, for each vertex not in D, the minimum weight of its edge to a vertex in D is minimized. A total dominating set D is a subset of the graph’s vertices, such that every vertex, including those in D, is at least adjacent to one vertex in D. This problem arises in many real-life applications closely related to covering and independent set problems; however, it remains computationally challenging due to its $$\mathcal{N}\mathcal{P}$$ N P -hardness. This work presents a variable neighborhood search (VNS) procedure to tackle the WTDP, and investigates the advantages and disadvantages of a multi-start strategy within VNS methodology. In addition, we develop a biased greedy randomized adaptive search procedure (Biased GRASP) that keeps adding elements once a feasible solution is found to produce high-quality initial solutions. We perform extensive numerical analysis to look into the influences of the algorithmic components and to disclose the contribution of the elements and strategies of our method. Finally, the empirical analysis shows that our proposal outperforms the state-of-art results, and the statistical analysis confirms the superiority of our proposal to find the best total dominating sets.
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Shahbazi, L., H. Abdollahzadeh Ahangar, R. Khoeilar, and S. M. Sheikholeslami. "Total k-rainbow reinforcement number in graphs." Discrete Mathematics, Algorithms and Applications, September 7, 2020, 2050101. http://dx.doi.org/10.1142/s1793830920501013.

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Let [Formula: see text] be an integer, and let [Formula: see text] be a graph. A k-rainbow dominating function (or [Formula: see text]RDF) of [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the family of all subsets of [Formula: see text] such that for very [Formula: see text] with [Formula: see text], the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] of [Formula: see text] is the value [Formula: see text]. A k-rainbow dominating function [Formula: see text] in a graph with no isolated vertex is called a total k-rainbow dominating function if the subgraph of [Formula: see text] induced by the set [Formula: see text] has no isolated vertices. The total k-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of the total [Formula: see text]-rainbow dominating function on [Formula: see text]. The total k-rainbow reinforcement number of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges that must be added to [Formula: see text] in order to decrease the total k-rainbow domination number. In this paper, we investigate the properties of total [Formula: see text]-rainbow reinforcement number in graphs. In particular, we present some sharp bounds for [Formula: see text] and we determine the total [Formula: see text]-rainbow reinforcement number of some classes of graphs including paths, cycles and complete bipartite graphs.
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27

Kapunac, Stefan, Aleksandar Kartelj, and Marko Djukanovic. "Variable Neighborhood Search for Weighted Total Domination Problem and its Application in Social Network Information Spreading." SSRN Electronic Journal, 2022. http://dx.doi.org/10.2139/ssrn.4155122.

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28

Kapunac, Stefan, Aleksandar Kartelj, and Marko Djukanović. "Variable neighborhood search for weighted total domination problem and its application in social network information spreading." Applied Soft Computing, May 2023, 110387. http://dx.doi.org/10.1016/j.asoc.2023.110387.

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Artes, Rosalio G. Jr. "CONNECTED TOTAL DOMINATING NEIGHBORHOOD POLYNOMIAL OF GRAPHS." Advances and Applications in Discrete Mathematics, May 23, 2023, 145–54. http://dx.doi.org/10.17654/0974165823042.

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Artes Jr., Rosalio, Cerina Villarta, Venerando Tenio, Milani Udal, and Angelito Rendiza. "Algebraic Representation of CTDS Over Uniform Network Interactions." International Journal of Mathematics and Computer Science, 2025, 791–95. https://doi.org/10.69793/ijmcs/03.2025/avtur.

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Abstract:
In this paper, we characterize the connected total dominating sets (CTDS) in graphs resulting from uniform network interactions. Moreover, we establish the polynomial representation of the resulting graph with respect to its CTDS and neighborhood systems. We have shown that the polynomial can be expressed as the product of a binomial expansion and exponential form.
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