Noor A'lawiah Abd Aziz, N. J. Rad, H. Kamarulhaili
Kammerling and Volkmann [J. Korean Math. Soc. 46 (2009), 1309–1318] introduced the concept of Roman k-domination in graphs. For a fixed positive integer k, a function f: V (G) → {0,1,2} is a Roman k-dominating function on G if every vertex valued 0 under f is adjacent to at least k vertices valued 2 under f. In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman k-domination by not-fixing k. We prove upper bounds for the new variant in cactus graphs and characterize cactus graph achieving equality for the given bound. We also present a probabilistic upper bound for this variant.
[J]。韩国的数学。Soc. 46(2009)， 1309-1318]在图中引入了罗马k-支配的概念。对于一个固定的正整数k，函数f: V (G)→{0,1,2}是G上的罗马k支配函数，如果f下每个值为0的顶点与f下至少k个值为2的顶点相邻。本文受图中的联盟概念的启发，通过不固定k重新讨论了罗马k支配的概念。我们证明了仙人掌图中新变种的上界，并刻画了仙人掌图在给定界内达到相等。我们也给出了这种变体的概率上界。
{"title":"Roman κ-domination revisited","authors":"Noor A'lawiah Abd Aziz, N. J. Rad, H. Kamarulhaili","doi":"10.1051/ro/2023072","DOIUrl":"https://doi.org/10.1051/ro/2023072","url":null,"abstract":"Kammerling and Volkmann [J. Korean Math. Soc. 46 (2009), 1309–1318] introduced the concept of Roman k-domination in graphs. For a fixed positive integer k, a function f: V (G) → {0,1,2} is a Roman k-dominating function on G if every vertex valued 0 under f is adjacent to at least k vertices valued 2 under f. In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman k-domination by not-fixing k. We prove upper bounds for the new variant in cactus graphs and characterize cactus graph achieving equality for the given bound. We also present a probabilistic upper bound for this variant.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80783281","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the era of big data, much of the data is susceptible to noise with heavy-tailed distribution. Fused Lasso can effectively handle high dimensional sparse data with strong correlation between two adjacent variables under known Gaussian noise. However, it has poor robustness to nonGaussian noise with heavy-tailed distribution. Robust fused Lasso with l1 norm loss function can overcome the drawback of fused Lasso when noise is heavy-tailed distribution. But the key challenge for solving this model is nonsmoothness and its nonseparability. Therefore, in this paper, we first deform the robust fused Lasso into an easily solvable form, which changes the three-block objective function to a two-block form. Then, we propose an accelerated proximal alternating direction method of multipliers (APADMM) with an additional update step, which is base on a new PADMM that changes the Lagrangian multiplier term update. Furthermore, we give the O(1/K) nonergodic convergence rate analysis of the proposed APADMM. Finally, numerical results show that the proposed new PADMM and APADMM have better performance than other existing ADMM solvers.
{"title":"An accelerated proximal alternating direction method of multipliers for robust fused Lasso","authors":"Yibao Fan, Youlin Shang, Zheng-Fen Jin, Jia Liu, Roxin Zhang","doi":"10.1051/ro/2023065","DOIUrl":"https://doi.org/10.1051/ro/2023065","url":null,"abstract":"In the era of big data, much of the data is susceptible to noise with heavy-tailed distribution. Fused Lasso can effectively handle high dimensional sparse data with strong correlation between two adjacent variables under known Gaussian noise. However, it has poor robustness to nonGaussian noise with heavy-tailed distribution. Robust fused Lasso with l1 norm loss function can overcome the drawback of fused Lasso when noise is heavy-tailed distribution. But the key challenge for solving this model is nonsmoothness and its nonseparability. Therefore, in this paper, we first deform the robust fused Lasso into an easily solvable form, which changes the three-block objective function to a two-block form. Then, we propose an accelerated proximal alternating direction method of multipliers (APADMM) with an additional update step, which is base on a new PADMM that changes the Lagrangian multiplier term update. Furthermore, we give the O(1/K) nonergodic convergence rate analysis of the proposed APADMM. Finally, numerical results show that the proposed new PADMM and APADMM have better performance than other existing ADMM solvers.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73662118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In supply chain management, the selection of supplier is of vital importance, and plays a valuable role in the performance of organizations. A hierarchical structure, with different levels in the selection of suppliers, is employed, wherein sequential decisions are made from the highest to the lowest level. Decision variables, called controlling factors, are divided into several categories. In the decision-making process, often because of the lack of confidence or uncertainty, It becomes hard for decision-makers to give explicit/crisp values to any parameter, resulting in uncertainty in the problem. In this paper, we address a multi-level supplier selection problem with fuzzy supply and demand. To avoid decision conflicts, superior or upper-level decision makers give tolerances that could be used as a possible relaxation. Thus, the problem is employed with fuzzy constraints. Based on a neutrosophic decision set, the Neutrosophic Compromise Programming Approach (NCPA) is used as a solution technique with the idea of indeterminacy degree as well as different objectives for membership and non-membership degree. Membership functions (Linear-type) are used to develop satisfactory solutions by fuzzily describing objective functions as well as controlling factors. A numerical illustration is provided to demonstrate the validity and appropriateness of NCPA.
{"title":"Neutrosophic programming approach to multilevel decision-making model for supplier selection problem in a fuzzy situation","authors":"A. Adhami, Anas Melethil, F. Ahmad","doi":"10.1051/ro/2023064","DOIUrl":"https://doi.org/10.1051/ro/2023064","url":null,"abstract":"In supply chain management, the selection of supplier is of vital importance, and plays a valuable role in the performance of organizations. A hierarchical structure, with different levels in the selection of suppliers, is employed, wherein sequential decisions are made from the highest to the lowest level. Decision variables, called controlling factors, are divided into several categories. In the decision-making process, often because of the lack of confidence or uncertainty, It becomes hard for decision-makers to give explicit/crisp values to any parameter, resulting in uncertainty in the problem. In this paper, we address a multi-level supplier selection problem with fuzzy supply and demand. To avoid decision conflicts, superior or upper-level decision makers give tolerances that could be used as a possible relaxation. Thus, the problem is employed with fuzzy constraints. Based on a neutrosophic decision set, the Neutrosophic Compromise Programming Approach (NCPA) is used as a solution technique with the idea of indeterminacy degree as well as different objectives for membership and non-membership degree. Membership functions (Linear-type) are used to develop satisfactory solutions by fuzzily describing objective functions as well as controlling factors. A numerical illustration is provided to demonstrate the validity and appropriateness of NCPA.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73306121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fuzzy fractional signomial programming problem is a relatively new optimization problem. In real world problems, some variables may vacillate because of various reasons. To tackle these vacillating variables, vagueness is considered in form of fuzzy sets. In this paper, a nonlinear fuzzy fractional signomial programming problem is considered with all its coefficients in objective functions as well as constraints are fuzzy numbers. Two solution approaches are developed based on signomial geometric programming comprising nearest interval approximation with parametric interval valued functions and fuzzy α-cut with min-max approach. To demonstrate the proposed methods, two illustrative numerical examples are solved and the results are comparatively discussed showing its feasibility and effectiveness.
{"title":"Nonlinear fuzzy fractional signomial programming problem: A fuzzy geometric programming solution approach","authors":"Sudipta Mishra, R. Ota, Suvasis Nayak","doi":"10.1051/ro/2023063","DOIUrl":"https://doi.org/10.1051/ro/2023063","url":null,"abstract":"Fuzzy fractional signomial programming problem is a relatively new optimization problem. In real world problems, some variables may vacillate because of various reasons. To tackle these vacillating variables, vagueness is considered in form of fuzzy sets. In this paper, a nonlinear fuzzy fractional signomial programming problem is considered with all its coefficients in objective functions as well as constraints are fuzzy numbers. Two solution approaches are developed based on signomial geometric programming comprising nearest interval approximation with parametric interval valued functions and fuzzy α-cut with min-max approach. To demonstrate the proposed methods, two illustrative numerical examples are solved and the results are comparatively discussed showing its feasibility and effectiveness.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76151211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new integer programming formulation for the Fractional Chromatic Number Problem. The formulation is based on representatives of stable sets. In addition, we present a Lagrangian heuristic from a Lagrangian relaxation of this formulation to obtain a good feasible solution for the problem. Computational experiments are presented to evaluate and compare the upper and lower bounds provided by our approach.
{"title":"A parallel lagrangian heuristic for the fractional chromatic number of a graph","authors":"P. Araújo, Ricardo C. Corrêa, Manoel B. Campêlo","doi":"10.1051/ro/2023062","DOIUrl":"https://doi.org/10.1051/ro/2023062","url":null,"abstract":"We propose a new integer programming formulation for the Fractional Chromatic Number Problem. The formulation is based on representatives of stable sets. In addition, we present a Lagrangian heuristic from a Lagrangian relaxation of this formulation to obtain a good feasible solution for the problem.\u0000Computational experiments are presented to evaluate and compare the upper and lower bounds provided by our approach.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73282074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An edge in a graph G = (V, E) is said to ev-dominate the vertices incident to it as well as the vertices adjacent to these incident vertices. A subset F ⊆ E is an edge-vertex dominating set (or simply, ev-dominating set) if every vertex is ev-dominated by at least one edge of F. The ev-domination number γev(G) is the minimum cardinality of a ev-dominating set of G. An ev-dominating set is independent if its edges are independent. The independent ev-domination number iev(G) is the minimum cardinality of an independent ev-dominating set and the upper independent ev-domination number βev(G) is the maximum cardinality of a minimal independent ev-dominating set of G. In this paper, we show that for every nontrivial tree T, γev(T) = iev(T) ≤ γ(T) ≤ βev(T), where γ(T) is the domination number of T. Moreover, we provide a characterization of all trees T with iev(T) = βev(T), which we call well ev-covered trees, as well as a characterization of all trees T with γev(T) = iev(T) = γ(T).
{"title":"Well ev-covered trees","authors":"Razika Boutrig, M. Chellali","doi":"10.1051/ro/2023088","DOIUrl":"https://doi.org/10.1051/ro/2023088","url":null,"abstract":"An edge in a graph G = (V, E) is said to ev-dominate the vertices incident to it as well as the vertices adjacent to these incident vertices. A subset F ⊆ E is an edge-vertex dominating set (or simply, ev-dominating set) if every vertex is ev-dominated by at least one edge of F. The ev-domination number γev(G) is the minimum cardinality of a ev-dominating set of G. An ev-dominating set is independent if its edges are independent. The independent ev-domination number iev(G) is the minimum cardinality of an independent ev-dominating set and the upper independent ev-domination number βev(G) is the maximum cardinality of a minimal independent ev-dominating set of G. In this paper, we show that for every nontrivial tree T, γev(T) = iev(T) ≤ γ(T) ≤ βev(T), where γ(T) is the domination number of T. Moreover, we provide a characterization of all trees T with iev(T) = βev(T), which we call well ev-covered trees, as well as a characterization of all trees T with γev(T) = iev(T) = γ(T).","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88130277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Samuel E. Silva, C. Ribeiro, Uéverton dos Santos Souza
A graph is chordal if all its cycles of length greater than or equal to four contain a chord, i.e., an edge connecting two nonconsecutive vertices of the cycle. Given a graph G = (V, E), the chordal completion problem consists in finding the minimum set of edges to be added to G to obtain a chordal graph. It has applications in sparse linear systems, database management and computer vision programming. In this article, we developed a biased random-key genetic algorithm (BRKGA) for solving the chordal completion problem, based on the strategy of manipulating permutations that represent perfect elimination orderings of triangulations. Computational results show that the proposed heuristic improve the results of the constructive heuristics fill-in and min-degree. We also developed a strategy for injecting externally constructed feasible solutions coded as random keys into the initial population of the BRKGA that significantly improves the solutions obtained and may benefit other implementations of biased random-key genetic algorithms.
{"title":"A biased random-key genetic algorithm for the chordal completion problem","authors":"Samuel E. Silva, C. Ribeiro, Uéverton dos Santos Souza","doi":"10.1051/ro/2023081","DOIUrl":"https://doi.org/10.1051/ro/2023081","url":null,"abstract":"A graph is chordal if all its cycles of length greater than or equal to four contain a chord, i.e., an edge connecting two nonconsecutive vertices of the cycle. Given a graph G = (V, E), the chordal completion problem consists in finding the minimum set of edges to be added to G to obtain a chordal graph. It has applications in sparse linear systems, database management and computer vision programming. In this article, we developed a biased random-key genetic algorithm (BRKGA) for solving the chordal completion problem, based on the strategy of manipulating permutations that represent perfect elimination orderings of triangulations. Computational results show that the proposed heuristic improve the results of the constructive heuristics fill-in and min-degree. We also developed a strategy for injecting externally constructed feasible solutions coded as random keys into the initial population of the BRKGA that significantly improves the solutions obtained and may benefit other implementations of biased random-key genetic algorithms.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74237401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose the first efficient multi parametric kernel function with logarithmic barrier term. A class of polynomial interior-point algorithms for P*(κ)-horizontal linear complementarity problem based on this kernel function, with parameters pi > 0 for all i ∈ 1, 2, , m, are presented. Then by using some simple analysis tools, we present a primal-dual interior point method (IPM) for P*(κ)-horizontal linear complementarity problems based on this kernel function. At the same time, we derive the complexity bounds small and large-update methods, respectively. In particular, if we take many different values of the parameters, we obtain the best known iteration bounds for the algorithms with large- and small-update methods are derived, namely, O((1 + 2κ)√n(log n)log n/ϵ) and O((1 + 2κ)√n log n/ϵ) respectively. We illustrate the performance of the proposed kernel function by some numerical results that are derived by applying our algorithm.
{"title":"An efficient multi parametric kernel function for large and small-update methods interior point algorithm for P*(κ)-horizontal linear complementarity problem","authors":"Mousaab Bouafia, Adnan Yassine","doi":"10.1051/ro/2023094","DOIUrl":"https://doi.org/10.1051/ro/2023094","url":null,"abstract":"In this paper, we propose the first efficient multi parametric kernel function with logarithmic barrier term. A class of polynomial interior-point algorithms for P*(κ)-horizontal linear complementarity problem based on this kernel function, with parameters pi > 0 for all i ∈ 1, 2, , m, are presented. Then by using some simple analysis tools, we present a primal-dual interior point method (IPM) for P*(κ)-horizontal linear complementarity problems based on this kernel function. At the same time, we derive the complexity bounds small and large-update methods, respectively. In particular, if we take many different values of the parameters, we obtain the best known iteration bounds for the algorithms with large- and small-update methods are derived, namely, O((1 + 2κ)√n(log n)log n/ϵ) and O((1 + 2κ)√n log n/ϵ) respectively. We illustrate the performance of the proposed kernel function by some numerical results that are derived by applying our algorithm.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86882042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Production lot-sizing techniques used by lean practitioners to lower waste inventories and increase production efficiency in the manufacturing industry, are the subject of this paper's speculation. Lean manufacturing aims to incorporate innovative tools into the manufacturing process to improve productivity and reduce processing time. In view of this, the model anticipates a flexible production rate based on labor, energy, and tool/die costs, to meet the demand while minimizing wastage. Moreover, a discrete investment in set-up costs is considered to lower the initial set-up cost since it is a critical component of smooth manufacturing operations. Further, it is found that price plays a significant role in stimulating a product’s demand; consequently, demand is presumed to be price-sensitive. Besides this, to reduce the carbon footprint in the production systems, two methods namely 'Carbon tax' and 'Cap-and-trade,' have been employed. The purpose of the developed model is to maximize total profit by jointly optimizing the production rate, selling price, and set-up cost. Numerical experiments are performed to validate the model findings. Results suggest that manufacturers' production time decreases simultaneously with the introduction of advanced labor and technologies. With respect to carbon policies, cap-and-trade policy performs better with an increase in total profit and a higher production rate as compared to that of carbon tax. Also, sensitivity analysis is performed to support the manufacturer in the decision-making process for ancillary benefits of the optimal policy.
{"title":"Analysing a lean manufacturing inventory system with price-sensitive demand and carbon control policies","authors":"Lalremruati Lalremruati, A. Khanna","doi":"10.1051/ro/2023060","DOIUrl":"https://doi.org/10.1051/ro/2023060","url":null,"abstract":"Production lot-sizing techniques used by lean practitioners to lower waste inventories and increase production efficiency in the manufacturing industry, are the subject of this paper's speculation. Lean manufacturing aims to incorporate innovative tools into the manufacturing process to improve productivity and reduce processing time. In view of this, the model anticipates a flexible production rate based on labor, energy, and tool/die costs, to meet the demand while minimizing wastage. Moreover, a discrete investment in set-up costs is considered to lower the initial set-up cost since it is a critical component of smooth manufacturing operations. Further, it is found that price plays a significant role in stimulating a product’s demand; consequently, demand is presumed to be price-sensitive. Besides this, to reduce the carbon footprint in the production systems, two methods namely 'Carbon tax' and 'Cap-and-trade,' have been employed. The purpose of the developed model is to maximize total profit by jointly optimizing the production rate, selling price, and set-up cost. Numerical experiments are performed to validate the model findings. Results suggest that manufacturers' production time decreases simultaneously with the introduction of advanced labor and technologies. With respect to carbon policies, cap-and-trade policy performs better with an increase in total profit and a higher production rate as compared to that of carbon tax. Also, sensitivity analysis is performed to support the manufacturer in the decision-making process for ancillary benefits of the optimal policy.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91428547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nafiseh Ebrahimi, H. A. Ahangar, M. Chellali, S. M. Sheikholeslami
For an integer $kgeq1,$ a Roman ${k}$-dominating function (R${k}$DF) on a graph $G=(V,E)$ is a function $f:Vrightarrow{0,1,dots,k}$ such that for every vertex $vin V$ with $f(v)=0$, $sum_{uin N(v)}f(u)geq k$, where $N(v)$ is the set of vertices adjacent to $v.$ The weight of an R ${k}$DF is the sum of its function values over the whole set of vertices, and the Roman ${k}$-domination number $gamma_{{kR}}(G)$ is the minimum weight of an R${k}$DF on $G$. In this paper, we will be focusing on the case $k=3$, where trivially for every connected graphs of order $ngeq3,$ $3leq$ $gamma _{{kR}}(G)leq n.$ We characterize all connected graphs $G$ of order $ngeq3$ such that $gamma_{{3R}}(G)in{3,n-1,n},$ and we improve the previous lower and upper bounds. Moreover, we show that for every tree $T$ of order $ngeq3$, $gamma_{{3R}}(T)geqgamma(T)+2$, where $gamma(T)$ is the domination number of $T,$ and we characterize the trees attaining this bound.
{"title":"Graphs with small or large Roman {3}-domination number","authors":"Nafiseh Ebrahimi, H. A. Ahangar, M. Chellali, S. M. Sheikholeslami","doi":"10.1051/ro/2023058","DOIUrl":"https://doi.org/10.1051/ro/2023058","url":null,"abstract":"For an integer $kgeq1,$ a Roman ${k}$-dominating function (R${k}$DF) on a\u0000\u0000graph $G=(V,E)$ is a function $f:Vrightarrow{0,1,dots,k}$ such that for\u0000\u0000every vertex $vin V$ with $f(v)=0$, $sum_{uin N(v)}f(u)geq k$, where\u0000\u0000$N(v)$ is the set of vertices adjacent to $v.$ The weight of an R ${k}$DF is\u0000\u0000the sum of its function values over the whole set of vertices, and the Roman\u0000\u0000${k}$-domination number $gamma_{{kR}}(G)$ is the minimum weight of an\u0000\u0000R${k}$DF on $G$. In this paper, we will be focusing on the case $k=3$, where\u0000\u0000trivially for every connected graphs of order $ngeq3,$ $3leq$ $gamma\u0000\u0000_{{kR}}(G)leq n.$ We characterize all connected graphs $G$ of order\u0000\u0000$ngeq3$ such that $gamma_{{3R}}(G)in{3,n-1,n},$ and we improve the\u0000\u0000previous lower and upper bounds. Moreover, we show that for every tree $T$ of\u0000\u0000order $ngeq3$, $gamma_{{3R}}(T)geqgamma(T)+2$, where $gamma(T)$ is the\u0000\u0000domination number of $T,$ and we characterize the trees attaining this bound.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80793133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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