Kamal Hossain Gazi, S. Mondal, Banashree Chatterjee, Neha Ghorui, Arijit Ghosh, Debashis De
This research addresses the problem of restaurant locations ranking with applications for a cosmopolitan big city like Kolkata, India. A restaurant selection is based on occasions, spending capability, environment, location, comfort, quality of the food etc. In this research paper an exhaustive set of factors and sub-factors is taken into consideration to select and rank restaurants situated at different locations in the city of Kolkata with a population of around fifteen million. The ranking of restaurants depends on complex, conflicting qualitative attributes. In the paper Hexagonal Fuzzy Numbers (HFN) have been used to suitably depict the imprecise uncertain environment. HFN, its distance measure and defuzzification have been applied to deal with the hesitancy and impreciseness of the decision makers. Analytic Hierarchy Process (AHP) has been used as a Multi Criteria Decision Making (MCDM) tool to obtain factors and sub-factors weights. TOPSIS and COPRAS methods were used for ranking different restaurant locations. Using comparative analysis it is shown that HFN with the TOPSIS and COPRAS method gives better result than other fuzzy numbers. The sensitivity analysis portion also gives a direction for taking a suitable decision in different possible scenario.
{"title":"A new synergistic strategy for ranking restaurant locations: A decision-making approach based on the hexagonal fuzzy numbers","authors":"Kamal Hossain Gazi, S. Mondal, Banashree Chatterjee, Neha Ghorui, Arijit Ghosh, Debashis De","doi":"10.1051/ro/2023025","DOIUrl":"https://doi.org/10.1051/ro/2023025","url":null,"abstract":"This research addresses the problem of restaurant locations ranking with applications for a cosmopolitan big city like Kolkata, India. A restaurant selection is based on occasions, spending capability, environment, location, comfort, quality of the food etc. In this research paper an exhaustive set of factors and sub-factors is taken into consideration to select and rank restaurants situated at different locations in the city of Kolkata with a population of around fifteen million. The ranking of restaurants depends on complex, conflicting qualitative attributes. In the paper Hexagonal Fuzzy Numbers (HFN) have been used to suitably depict the imprecise uncertain environment. HFN, its distance measure and defuzzification have been applied to deal with the hesitancy and impreciseness of the decision makers. Analytic Hierarchy Process (AHP) has been used as a Multi Criteria Decision Making (MCDM) tool to obtain factors and sub-factors weights. TOPSIS and COPRAS methods were used for ranking different restaurant locations. Using comparative analysis it is shown that HFN with the TOPSIS and COPRAS method gives better result than other fuzzy numbers. The sensitivity analysis portion also gives a direction for taking a suitable decision in different possible scenario.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76069951","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}
Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.
{"title":"Optimality conditions and duality results for a robust bi-level programming problem","authors":"Shivani Saini, N. Kailey, I. Ahmad","doi":"10.1051/ro/2023026","DOIUrl":"https://doi.org/10.1051/ro/2023026","url":null,"abstract":"Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82379776","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}
Extending services and operations of organizations in the field of e-business or e-government sometimes requires the integration of business applications. However, sometimes due to challenges and risks, such as complex business processes reengineering, upstream organizations are reluctant to integrate their applications. This paper focuses on two critical questions; (1) How can organizations be encouraged to participate in integrating their business applications? (2) What is the amount of incentives required? In this study, cooperative game theory and the externalities of these systems have been considered to form a stable coalition between organizations for integrating their business applications. We provided an algorithm for determining the incentives to integrate the business applications with other organizations in this coalition. These incentives can be extended to various management issues for better decision-making such as economic aspects, public subsidies, and public participation. The results of experiments have shown that creating a coalition based on this strategy is always possible, and the benefits of organizations in the coalition rise with increasing service delivery in business applications.
{"title":"An incentive mechanism for integration of business applications between organizations","authors":"Rasool Esmaeilyfard, Atefe Salehi","doi":"10.1051/ro/2023024","DOIUrl":"https://doi.org/10.1051/ro/2023024","url":null,"abstract":"Extending services and operations of organizations in the field of e-business or e-government sometimes requires the integration of business applications. However, sometimes due to challenges and risks, such as complex business processes reengineering, upstream organizations are reluctant to integrate their applications. This paper focuses on two critical questions; (1) How can organizations be encouraged to participate in integrating their business applications? (2) What is the amount of incentives required? In this study, cooperative game theory and the externalities of these systems have been considered to form a stable coalition between organizations for integrating their business applications. We provided an algorithm for determining the incentives to integrate the business applications with other organizations in this coalition. These incentives can be extended to various management issues for better decision-making such as economic aspects, public subsidies, and public participation. The results of experiments have shown that creating a coalition based on this strategy is always possible, and the benefits of organizations in the coalition rise with increasing service delivery in business applications.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91055191","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}
Brands seeking for prominent positions in online consumer search may invest in two types of search engine marketing: paid search marketing (PSM) and search engine optimization (SEO). This paper investigates how firms should allocate marketing resources between PSM and SEO under different competition modes (interfirm and intrafirm competition). We develop an analytical model where within an interfirm competition, two brands compete for every step, while for the intrafirm competition, multi-brand firm coordinates the search engine marketing decisions for its brands, but delegates the price decisions to the brand managers. We find that the resource allocation decision may depend on the firm’s brand strength in organic search. Our analysis shows an increasing curve relation between resources allocated to PSM and the brand strength for interfirm competition and inverted-U shape for intrafirm competition. More specifically, as the brand strength goes up, for the interfirm competition, the resources allocated to PSM increase till 100%, while it should increase only when brand strength is sufficiently low, but decrease when brand strength is high for the intrafirm competition. This nontrivial result underscores the challenge facing a multi-brand firm in balancing between maximizing the search prominence for the category and minimizing the internal competition.
{"title":"Search engine marketing for different competition modes - Interfirm and intrafirm","authors":"Xueli Zhang, Defeng Yang, Can Li","doi":"10.1051/ro/2023021","DOIUrl":"https://doi.org/10.1051/ro/2023021","url":null,"abstract":"Brands seeking for prominent positions in online consumer search may invest in two types of search engine marketing: paid search marketing (PSM) and search engine optimization (SEO). This paper investigates how firms should allocate marketing resources between PSM and SEO under different competition modes (interfirm and intrafirm competition). We develop an analytical model where within an interfirm competition, two brands compete for every step, while for the intrafirm competition, multi-brand firm coordinates the search engine marketing decisions for its brands, but delegates the price decisions to the brand managers. We find that the resource allocation decision may depend on the firm’s brand strength in organic search. Our analysis shows an increasing curve relation between resources allocated to PSM and the brand strength for interfirm competition and inverted-U shape for intrafirm competition. More specifically, as the brand strength goes up, for the interfirm competition, the resources allocated to PSM increase till 100%, while it should increase only when brand strength is sufficiently low, but decrease when brand strength is high for the intrafirm competition. This nontrivial result underscores the challenge facing a multi-brand firm in balancing between maximizing the search prominence for the category and minimizing the internal competition.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87735658","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}
The main idea of this article is to characterize approximate proper efficiency that is a widely used optimality concept in multicriteria optimization problems that prevents solutions having unbounded trade-offs. We analyze a modification of approximate proper efficiency for problems with infinitely many objective functions. We obtain some necessary and sufficient optimality conditions for this modification of approximate proper efficiency. This modified version of approximation guarantees the general characterizations of approximate properly efficient points as solutions to weighted sum problems and modified weighted Tchebycheff norm problems, even if there is an infinite number of criteria. The provided proofs concerning the modified definition show that if the number of the objective functions is infinite, then these results become invalid under the primary definition of approximate proper efficiency.
{"title":"Identifying approximate proper efficiency in an infinite dimensional space","authors":"N. Hoseinpoor, M. Ghaznavi","doi":"10.1051/ro/2023019","DOIUrl":"https://doi.org/10.1051/ro/2023019","url":null,"abstract":"The main idea of this article is to characterize approximate proper efficiency that is a widely used optimality concept in multicriteria optimization problems that prevents solutions having unbounded trade-offs. We analyze a modification of approximate proper efficiency for problems with infinitely many objective functions. We obtain some necessary and sufficient optimality conditions for this modification of approximate proper efficiency. This modified version of approximation guarantees the general characterizations of approximate properly efficient points as solutions to weighted sum problems and modified weighted Tchebycheff norm problems, even if there is an infinite number of criteria. The provided proofs concerning the modified definition show that if the number of the objective functions is infinite, then these results become invalid under the primary definition of approximate proper efficiency.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77501502","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}
The two-machine flowshop scheduling problem with the performance measure of total tardiness is addressed. This performance measure is essential since meeting deadlines is a crucial part of scheduling and a major concern for some manufacturing systems. The processing times on both machines are uncertain variables and within some lower and upper bounds. This is due to uncertainty being an integral part of some manufacturing settings, making it impossible to predict processing times in advance. To the best of the author’s knowledge, this problem is addressed for the first time in this paper. A dominance relation is established and nineteen algorithms, which incorporate the established dominance relation, are presented. These algorithms are extensively evaluated through randomly generated data for different numbers of jobs and four different distributions, representing both symmetric and non-symmetric distributions. Computational experiments show that the presented algorithms perform extremely well when compared with a random solution. In particular, the best of the considered 19 algorithms reduces the error of the random solution by 99.99% and the error of the worst algorithm (among the 19 algorithms) by 99.96%. The results are confirmed by a test of hypothesis and this algorithm is recommended.
{"title":"Minimizing total tardiness in a two-machine flowshop with uncertain and bounded processing times","authors":"Muberra Allahverdi","doi":"10.1051/ro/2023023","DOIUrl":"https://doi.org/10.1051/ro/2023023","url":null,"abstract":"The two-machine flowshop scheduling problem with the performance measure of total tardiness is addressed. This performance measure is essential since meeting deadlines is a crucial part of scheduling and a major concern for some manufacturing systems. The processing times on both machines are uncertain variables and within some lower and upper bounds. This is due to uncertainty being an integral part of some manufacturing settings, making it impossible to predict processing times in advance. To the best of the author’s knowledge, this problem is addressed for the first time in this paper. A dominance relation is established and nineteen algorithms, which incorporate the established dominance relation, are presented. These algorithms are extensively evaluated through randomly generated data for different numbers of jobs and four different distributions, representing both symmetric and non-symmetric distributions. Computational experiments show that the presented algorithms perform extremely well when compared with a random solution. In particular, the best of the considered 19 algorithms reduces the error of the random solution by 99.99% and the error of the worst algorithm (among the 19 algorithms) by 99.96%. The results are confirmed by a test of hypothesis and this algorithm is recommended.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81401876","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}
This paper considers the SLK/DIF due window assignment methods for single machine scheduling problems with general job-dependent positional effect, where the actual processing time of a job is subject to a job-dependent and positional effect. Production schedulers may decide to outsource/reject some jobs by paying the corresponding costs. Based on whether to accept the tardiness jobs, we consider two different objectives under slack and different due window assignment, respectively. The first objective function is to minimize the weighted sum of earliness, tardiness, due window starting time, due window size and outsourcing costs, while the second objective function is to minimize the cost function that includes earliness, due window starting time, due window size and outsourcing costs. We study the structural properties of two due window assignment methods and develop polynomial-time solution algorithms for the considered problems. A numerical example proves the advantage of outsourcing decision and the distinction of two different objectives.
{"title":"Single machine SLK/DIF due window assignment problem with general job-dependent positional effect and outsourcing operations","authors":"Chun-Lai Liu","doi":"10.1051/ro/2023020","DOIUrl":"https://doi.org/10.1051/ro/2023020","url":null,"abstract":"This paper considers the SLK/DIF due window assignment methods for single machine scheduling problems with general job-dependent positional effect, where the actual processing time of a job is subject to a job-dependent and positional effect. Production schedulers may decide to outsource/reject some jobs by paying the corresponding costs. Based on whether to accept the tardiness jobs, we consider two different objectives under slack and different due window assignment, respectively. The first objective function is to minimize the weighted sum of earliness, tardiness, due window starting time, due window size and outsourcing costs, while the second objective function is to minimize the cost function that includes earliness, due window starting time, due window size and outsourcing costs. We study the structural properties of two due window assignment methods and develop polynomial-time solution algorithms for the considered problems. A numerical example proves the advantage of outsourcing decision and the distinction of two different objectives.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80289142","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 study we develop a decision support tool to help carriers in defining their bids when participating in transportation combinatorial auctions. In this way, they can avoid many of their empty movements and ensure additional business through the spot market. We propose four algorithms that quantify the synergy as a key concept to measure the complementarity between the carrier’s existing network and the contracts to be included in the bundle to be submitted as a bid. The algorithms are validated on a real-life application and our results emphasized their superiority with respect to previous studies.
{"title":"Fast synergy approximation in transportation procurement with combinatorial auctions","authors":"M. E. Keskin, C. Triki, Adel Elomri","doi":"10.1051/ro/2023022","DOIUrl":"https://doi.org/10.1051/ro/2023022","url":null,"abstract":"In this study we develop a decision support tool to help carriers in defining their bids when participating in transportation combinatorial auctions. In this way, they can avoid many of their empty movements and ensure additional business through the spot market. We propose four algorithms that quantify the synergy as a key concept to measure the complementarity between the carrier’s existing network and the contracts to be included in the bundle to be submitted as a bid. The algorithms are validated on a real-life application and our results emphasized their superiority with respect to previous studies.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90819641","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}
S. Kosari, J. Amjadi, M. Chellali, S. M. Sheikholeslami
An independent Roman dominating function (IRD-function) on a graph $G$ is a function $f:V(G)rightarrow{0,1,2}$ satisfying the conditions that (i) every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number $i_{R}(G)$ of $G$ is the minimum weight of an IRD-function on $G$. In this paper, we initiate the study of the independent Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of set of edges $Fsubseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on $b_{iR}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iR}(T)leq3,$ while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph.
{"title":"Independent Roman bondage of graphs","authors":"S. Kosari, J. Amjadi, M. Chellali, S. M. Sheikholeslami","doi":"10.1051/ro/2023017","DOIUrl":"https://doi.org/10.1051/ro/2023017","url":null,"abstract":"An independent Roman dominating function (IRD-function) on a graph $G$ is a\u0000\u0000function $f:V(G)rightarrow{0,1,2}$ satisfying the conditions that (i) every\u0000\u0000vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which\u0000\u0000$f(v)=2$, and (ii) the set of all vertices assigned non-zero\u0000\u0000values under $f$ is independent. The weight of an IRD-function is\u0000\u0000the sum of its function values over all vertices, and the independent Roman\u0000\u0000domination number $i_{R}(G)$ of $G$ is the minimum weight of an\u0000\u0000IRD-function on $G$. In this paper, we initiate the study of the independent\u0000\u0000Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least\u0000\u0000one component of order at least three, defined as the smallest size of set of\u0000\u0000edges $Fsubseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing\u0000\u0000that the decision problem associated with the independent Roman\u0000\u0000bondage problem is NP-hard for bipartite graphs.\u0000\u0000Then various upper bounds on $b_{iR}(G)$ are established as well\u0000\u0000as exact values on it for some special graphs. In particular, for trees $T$\u0000\u0000of order at least three, it is shown that $b_{iR}(T)leq3,$\u0000\u0000while for connected planar graphs the upper bounds are in terms of\u0000\u0000the maximum degree with refinements depending on the girth of the graph.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86951116","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}
For a graph H, a graph G is H-saturated if it contains no copy of H as a (not necessarily induced) subgraph, but the addition of any edge missing from G creates a copy of H in the resultant graph. The connected saturation number sat'(n,H) is defined as the minimum number of edges in H-saturated connected graphs on n vertices. In this paper we consider the (Pk∪K3)-saturated connected graphs on n vertices and focus on the determination of sat'(n,Pk∪K3). We prove that n+2≤sat'(n,Pk∪ K3)≤n+(3k-6)/2 for n>(3k+6)/2 with k ≥ 4 and characterize the extremal graphs at which the upper bounds are attained. Moreover, the exact values of sat'(n,Pk∪ K3) are determined with k ∈{2,3,4}.
{"title":"Minimizing the number of edges in (Pk ∪ K3)-saturated connected graphs","authors":"Yuying Li, Kexiang Xu","doi":"10.1051/ro/2023018","DOIUrl":"https://doi.org/10.1051/ro/2023018","url":null,"abstract":"For a graph H, a graph G is H-saturated if it contains no copy of H as a (not necessarily induced) subgraph, but the addition of any edge missing from G creates a copy of H in the resultant graph. The connected saturation number sat'(n,H) is defined as the minimum number of edges in H-saturated connected graphs on n vertices. In this paper we consider the (Pk∪K3)-saturated connected graphs on n vertices and focus on the determination of\u0000 sat'(n,Pk∪K3). We prove that n+2≤sat'(n,Pk∪ K3)≤n+(3k-6)/2 for n>(3k+6)/2 with k ≥ 4 and characterize the extremal graphs at which the upper bounds are attained. Moreover, the exact values of sat'(n,Pk∪ K3) are determined with k ∈{2,3,4}.","PeriodicalId":20872,"journal":{"name":"RAIRO Oper. Res.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84012511","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：