Pub Date : 2023-04-18DOI: 10.48550/arXiv.2304.09223
Seda Albayrak, J. Bell
A theorem of Cobham says that if $k$ and $ell$ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both $k$- and $ell$-automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse $k$-automatic subset of $mathbb{N}^d$ and a sparse $ell$-automatic subset of $mathbb{N}^d$ is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets.
Cobham的一个定理说，如果$k$和$ well $是两个相乘独立的自然数，那么同时是$k$-和$ well $-自动的自然数的子集最终是周期的。后来，Semenov给出了一个多维扩展。本文给出了稀疏自动集的Cobham-Semenov定理的一个定量版本，证明了$mathbb{N}^d$的稀疏$k$-自动子集与$mathbb{N}^d$的稀疏$ell$-自动子集的交集是有限的，其大小可以用接受这些集合的自动机的数据显式地有界。
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Pub Date : 2023-04-03DOI: 10.48550/arXiv.2304.00864
Searfino Cicerone, G. Stefano, Lara Drozek, J. Hedžet, S. Klavžar, I. Yero
If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)cap X subseteq {u,v}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.
如果$X$是图$G$顶点的子集，则$u$和$v$是$X$-可见的，如果存在最短的$u,v$-路径$P$使得$v (P)cap X subseteq {u,v}$。如果$X$中的每两个顶点都是$X$-可见的，则$X$是一个互可见集。$G$的互可见性数是$G$的最大互可见性集的基数，并且已经被研究过。本文介绍了各种互可见性问题，在这些问题的基础上要求顶点的自然对是X -可见的。这将产生总数、双数和外部相互可见性数。首先证明了这些图不变量之间的相互关系以及它们与经典互可视性数的关系，然后证明了这三个新引入的互可视性问题在计算上是困难的。根据这个结果，我们计算或绑定了包括网格图和环面在内的几个图类的值。我们通过这些参数值之间的一些相互比较来结束研究，这是基于我们对一些特定家庭的计算。
{"title":"Variety of mutual-visibility problems in graphs","authors":"Searfino Cicerone, G. Stefano, Lara Drozek, J. Hedžet, S. Klavžar, I. Yero","doi":"10.48550/arXiv.2304.00864","DOIUrl":"https://doi.org/10.48550/arXiv.2304.00864","url":null,"abstract":"If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)cap X subseteq {u,v}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"300 1","pages":"114096"},"PeriodicalIF":0.0,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90803211","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}
A. Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, Abhishek Sahu
For an undirected graph G , a pair of vertex disjoint subsets p A, B q is a pair of perfectly matched sets if each vertex in A (resp. B ) has exactly one neighbor in B (resp. A ). In the above, the size of the pair is | A | ( “ | B | ). Given a graph G and a positive integer k , the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G . This problem is known to be NP -hard on planar graphs and W[1] -hard on general graphs, when parameterized by k . However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k , and design FPT algorithms for: i) apex-minor-free graphs running in time 2 O p? k q ¨ n O p 1 q , and ii) K b,b -free graphs. We obtain a linear kernel for planar graphs and k O p d q -sized kernel for d -degenerate graphs. It is known that the problem is W[1] -hard on chordal graphs, in fact on split graphs, parameterized by k . We complement this hardness result by designing a polynomial-time algorithm for interval graphs. 2012 ACM Subject Classification Theory of computation Ñ Fixed parameter tractability
对于无向图G，一对顶点不相交的子集pa, B q是一对完全匹配的集合，如果a中的每个顶点(p。B)在B中恰好有一个邻居(p。一个)。在上面，这对的大小是| A | (" | B |)。给定一个图G和一个正整数k，完美匹配集问题是问G中是否存在一对大小至少为k的完美匹配集。当用k参数化时，已知该问题在平面图上为NP -hard，在一般图上为W[1] -hard。然而，对于受限图类问题的参数化复杂度，人们知之甚少。在这项工作中，我们研究了用k参数化的问题，并设计了FPT算法，用于:i)运行时间为2 O p?k q¨n O p 1q，和ii) k b,b个自由图。我们得到了平面图的线性核和d -退化图的k O p d q大小的核。已知问题是W[1]——在弦图上，实际上是在分裂图上，用k来参数化。我们通过设计一个区间图的多项式时间算法来补充这个困难的结果。2012 ACM学科分类计算理论Ñ固定参数可追溯性
{"title":"Parameterized Complexity of Perfectly Matched Sets","authors":"A. Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, Abhishek Sahu","doi":"10.2139/ssrn.4289703","DOIUrl":"https://doi.org/10.2139/ssrn.4289703","url":null,"abstract":"For an undirected graph G , a pair of vertex disjoint subsets p A, B q is a pair of perfectly matched sets if each vertex in A (resp. B ) has exactly one neighbor in B (resp. A ). In the above, the size of the pair is | A | ( “ | B | ). Given a graph G and a positive integer k , the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G . This problem is known to be NP -hard on planar graphs and W[1] -hard on general graphs, when parameterized by k . However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k , and design FPT algorithms for: i) apex-minor-free graphs running in time 2 O p? k q ¨ n O p 1 q , and ii) K b,b -free graphs. We obtain a linear kernel for planar graphs and k O p d q -sized kernel for d -degenerate graphs. It is known that the problem is W[1] -hard on chordal graphs, in fact on split graphs, parameterized by k . We complement this hardness result by designing a polynomial-time algorithm for interval graphs. 2012 ACM Subject Classification Theory of computation Ñ Fixed parameter tractability","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"208 1","pages":"113861"},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76425477","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}
{"title":"Combining decomposition approaches for the Maximum Weight Stable Set problem","authors":"A. Brandstädt, R. Mosca","doi":"10.2139/ssrn.4055778","DOIUrl":"https://doi.org/10.2139/ssrn.4055778","url":null,"abstract":"","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"593 1","pages":"113914"},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78514389","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}
Pub Date : 2023-03-10DOI: 10.7494/csci.2023.24.2.4871
Roopam Sadh, Rajeev Kumar
Currently, Machine Learning is being significantly used in almost all of the research domains. However, its applicability in survey research is still in its infancy. We in this paper, attempt to highlight the applicability of Machine Learning in survey research while working on two different aspects in parallel. First, we introduce a pattern-based transformation method for ordinal survey data. Our purpose behind developing such a transformation method is twofold. Our transformation facilitates easy interpretation of ordinal survey data and provides convenience while applying standard Machine Learning approaches. Second, we demonstrate the application of various classification techniques over real and transformed ordinal survey data and interpret their results in terms of their suitability in survey research. Our experimental results suggest that Machine Learning coupled with the Pattern Recognition paradigm has a tremendous scope in survey research.
{"title":"Transformation and Classification of Ordinal Survey Data","authors":"Roopam Sadh, Rajeev Kumar","doi":"10.7494/csci.2023.24.2.4871","DOIUrl":"https://doi.org/10.7494/csci.2023.24.2.4871","url":null,"abstract":"Currently, Machine Learning is being significantly used in almost all of the research domains. However, its applicability in survey research is still in its infancy. We in this paper, attempt to highlight the applicability of Machine Learning in survey research while working on two different aspects in parallel. First, we introduce a pattern-based transformation method for ordinal survey data. Our purpose behind developing such a transformation method is twofold. Our transformation facilitates easy interpretation of ordinal survey data and provides convenience while applying standard Machine Learning approaches. Second, we demonstrate the application of various classification techniques over real and transformed ordinal survey data and interpret their results in terms of their suitability in survey research. Our experimental results suggest that Machine Learning coupled with the Pattern Recognition paradigm has a tremendous scope in survey research.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85043639","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}
Pub Date : 2023-03-10DOI: 10.7494/csci.2023.24.2.4840
Son Van Nguyen, S. T. Nguyen, Anh Pham Thi Hong, Thao Thu Hoang, Ta Minh Thanh
To date, facial recognition has been one of the most intriguing, interesting research topics over years. It requires some specific face-based algorithms such as facial detection, facial alignment, facial representation, and facial recognition as well; however, all of these algorithms derive from heavy deep learning architectures that cause limitations for development, scalability, flawed accuracy, and deployment into publicity with mere CPU servers. It also calls for large datasets containing hundreds of thousands of records for training purposes. In this paper, we propose a full pipeline for an effective face recognition application which only uses a small Vietnamese celebrity dataset and CPU for training that can solve the leakage of data and the need for GPU devices. It is based on a face vector-to-string tokens algorithm then saves face’s properties into Elasticsearch for future retrieval, so the problem of online learning in Facial Recognition is also tackled. Comparison with another popular algorithm on the dataset, our proposed pipeline not only outweighs the accuracy counterpart, but it also achieves a very speedy time inference for a real-time face recognition application.
{"title":"Hybrid End-to-End Approach Integrating Online Learning with Face-identification System","authors":"Son Van Nguyen, S. T. Nguyen, Anh Pham Thi Hong, Thao Thu Hoang, Ta Minh Thanh","doi":"10.7494/csci.2023.24.2.4840","DOIUrl":"https://doi.org/10.7494/csci.2023.24.2.4840","url":null,"abstract":"To date, facial recognition has been one of the most intriguing, interesting research topics over years. It requires some specific face-based algorithms such as facial detection, facial alignment, facial representation, and facial recognition as well; however, all of these algorithms derive from heavy deep learning architectures that cause limitations for development, scalability, flawed accuracy, and deployment into publicity with mere CPU servers. It also calls for large datasets containing hundreds of thousands of records for training purposes. In this paper, we propose a full pipeline for an effective face recognition application which only uses a small Vietnamese celebrity dataset and CPU for training that can solve the leakage of data and the need for GPU devices. It is based on a face vector-to-string tokens algorithm then saves face’s properties into Elasticsearch for future retrieval, so the problem of online learning in Facial Recognition is also tackled. Comparison with another popular algorithm on the dataset, our proposed pipeline not only outweighs the accuracy counterpart, but it also achieves a very speedy time inference for a real-time face recognition application.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85661986","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}
Pub Date : 2023-03-10DOI: 10.7494/csci.2023.24.2.4783
Mokrane Kemiche, Malika Sadou
Currently, Handwritten Character Recognition (HCR) technology has become an interesting and immensely useful technology. It has been explored with highperformance in many languages. However, a few HCR systems are proposed for the Amazigh (Berber) language. Furthermore, the validation of any Amazighhandwritten recognition system remains a major challenge due to no availability of a robust Amazigh database. To address this problem, we first created two new datasets for Tifinagh and Amazigh Latin characters, by extending the well-known EMNIST database with the Amazigh alphabet. And then, we have proposed a handwritten character recognition system, which is based on a deep convolutional neural network to validate the created datasets. The proposed CNN has been trained and tested on our created datasets, and the experimental tests show that it achieves satisfactory results in terms of accuracy and recognition efficiency.
{"title":"Deep Convolutional Neural Network using a New Dataset for Berber Language","authors":"Mokrane Kemiche, Malika Sadou","doi":"10.7494/csci.2023.24.2.4783","DOIUrl":"https://doi.org/10.7494/csci.2023.24.2.4783","url":null,"abstract":"Currently, Handwritten Character Recognition (HCR) technology has become an interesting and immensely useful technology. It has been explored with highperformance in many languages. However, a few HCR systems are proposed for the Amazigh (Berber) language. Furthermore, the validation of any Amazighhandwritten recognition system remains a major challenge due to no availability of a robust Amazigh database. To address this problem, we first created two new datasets for Tifinagh and Amazigh Latin characters, by extending the well-known EMNIST database with the Amazigh alphabet. And then, we have proposed a handwritten character recognition system, which is based on a deep convolutional neural network to validate the created datasets. The proposed CNN has been trained and tested on our created datasets, and the experimental tests show that it achieves satisfactory results in terms of accuracy and recognition efficiency.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73698951","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}
Pub Date : 2023-03-10DOI: 10.7494/csci.2023.24.2.4685
H. Bendjenna, Asma Aoun Allah, A. Meraoumia
The feature extraction step is a major and crucial step in analyzing and understanding raw data as it has a considerable impact on the system accuracy. Unfortunately, despite the very acceptable results obtained by many handcrafted methods, they can have difficulty representing the features in the case of large databases or with strongly correlated samples. In this context, we proposed a new, simple and lightweight method for deep feature extraction. Our method can be configured to produce four different deep features, each controlled to tune the system accuracy. We have evaluated the performance of our method using a multispectral palmprint based biometric system and the experimental results, using the CASIA database, have shown that our method has high accuracy compared to many current handcrafted feature extraction methods and many well known deep learning based methods.
{"title":"Learning-Free Deep Features for Multispectral Palm-Print Classification","authors":"H. Bendjenna, Asma Aoun Allah, A. Meraoumia","doi":"10.7494/csci.2023.24.2.4685","DOIUrl":"https://doi.org/10.7494/csci.2023.24.2.4685","url":null,"abstract":"The feature extraction step is a major and crucial step in analyzing and understanding raw data as it has a considerable impact on the system accuracy. Unfortunately, despite the very acceptable results obtained by many handcrafted methods, they can have difficulty representing the features in the case of large databases or with strongly correlated samples. In this context, we proposed a new, simple and lightweight method for deep feature extraction. Our method can be configured to produce four different deep features, each controlled to tune the system accuracy. We have evaluated the performance of our method using a multispectral palmprint based biometric system and the experimental results, using the CASIA database, have shown that our method has high accuracy compared to many current handcrafted feature extraction methods and many well known deep learning based methods.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80836386","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}
Pub Date : 2023-03-06DOI: 10.7494/csci.2023.24.1.4605
Suhail Ahmad, A. H. Mir
Software Defined Networks (SDN) advocates segregation of network control logic, forwarding functions and management applications into different planes to achieve network programmability, automated and dynamic flow control in next generation networks. It promotes deployment of novel and augmented network management functions to have flexible, robust, scalable and cost-effective network deployments. All these features introduce new research challenges and require secure communication protocols among the segregated network planes. This manuscript focuses on the security issue of southbound interface which operates between the SDN control and data plane. We have highlighted the security threats associated with an unprotected southbound interface and the issues related with the existing TLS based security solution. A lightweight blockchain based decentralized security solution is proposed for southbound interface to secure the resources of logically centralized SDN controllers and distributed forwarding devices from opponents. The proposed mechanism can operate in multi-domain SDN deployment and can be used with wide range of network controllers and data plane devices. In addition to it, the proposed security solution is analyzed in terms of security features, communication and reauthentication overhead.
{"title":"Securing Centralized SDN control with Distributed Blockchain Technology","authors":"Suhail Ahmad, A. H. Mir","doi":"10.7494/csci.2023.24.1.4605","DOIUrl":"https://doi.org/10.7494/csci.2023.24.1.4605","url":null,"abstract":"Software Defined Networks (SDN) advocates segregation of network control logic, forwarding functions and management applications into different planes to achieve network programmability, automated and dynamic flow control in next generation networks. It promotes deployment of novel and augmented network management functions to have flexible, robust, scalable and cost-effective network deployments. All these features introduce new research challenges and require secure communication protocols among the segregated network planes. This manuscript focuses on the security issue of southbound interface which operates between the SDN control and data plane. We have highlighted the security threats associated with an unprotected southbound interface and the issues related with the existing TLS based security solution. A lightweight blockchain based decentralized security solution is proposed for southbound interface to secure the resources of logically centralized SDN controllers and distributed forwarding devices from opponents. The proposed mechanism can operate in multi-domain SDN deployment and can be used with wide range of network controllers and data plane devices. In addition to it, the proposed security solution is analyzed in terms of security features, communication and reauthentication overhead.","PeriodicalId":23063,"journal":{"name":"Theor. Comput. Sci.","volume":"49 5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76573691","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}
By a well known theorem of Robbins, a graph $G$ has a strongly connected orientation if and only if $G$ is 2-edge-connected and it is easy to find, in linear time, either a cut edge of $G$ or a strong orientation of $G$. A result of Durand de Gevigny shows that for every $kgeq 3$ it is NP-hard to decide if a given graph $G$ has a $k$-strong orientation. Thomassen showed that one can check in polynomial time whether a given graph has a 2-strong orientation. This implies that for a given digraph $D$ we can determine in polynomial time whether we can reorient (=reverse) some arcs of $D=(V,A)$ to obtain a 2-strong digraph $D'=(V,A')$. This naturally leads to the question of determining the minimum number of such arcs to reverse before the resulting graph is 2-strong. In this paper we show that finding this number is NP-hard. If a 2-connected graph $G$ has no 2-strong orientation, we may ask how many of its edges we may orient so that the resulting mixed graph is still 2-strong. Similarly, we may ask for a 2-edge-connected graph $G$ how many of its edges we can orient such that the resulting mixed graph remains 2-arc-strong. We prove that when restricted to graphs satisfying suitable connectivity conditions, both of these problems are equivalent to finding the minimum number of edges we must double in a 2-edge-connected graph in order to obtain a 4-edge-connected graph. Using this, we show that all these three problems are NP-hard. Finally, we consider the operation of deorienting an arc $uv$ of a digraph $D$ meaning replacing it by an undirected edge between the same vertices. In terms of connectivity properties, this is equivalent to adding the opposite arc $vu$ to $D$. We prove that for every $ellgeq 3$ it is NP-hard to find the minimum number of arcs to deorient in a digraph $D$ in order to obtain an $ell$-strong digraph $D'$.
根据著名的罗宾斯定理，当且仅当$G$是2边连通时，图$G$具有强连通方向，并且在线性时间内很容易找到$G$的切边或$G$的强定向。Durand de Gevigny的结果表明，对于每个$kgeq 3$来说，决定给定图$G$是否具有$k$ -强取向是np困难的。Thomassen证明了可以在多项式时间内检验给定图是否具有2强取向。这意味着对于给定的有向图$D$，我们可以在多项式时间内确定是否可以重新定向(=反转)$D=(V,A)$的一些弧以获得2强有向图$D'=(V,A')$。这自然导致了一个问题，即在生成的图形为2强之前，确定要反转的这种弧线的最小数量。在本文中，我们证明了找到这个数是np困难的。如果一个2连通图$G$没有2强定向，我们可以问有多少条边可以定向，这样得到的混合图仍然是2强的。类似地，我们可能会问一个2边连通图$G$，我们可以定向多少条边，这样得到的混合图仍然是2弧强的。我们证明了当图满足适当的连通性条件时，这两个问题等价于求出在2边连通图中为了得到4边连通图而必须加倍的最小边数。利用这一点，我们证明这三个问题都是np困难的。最后，我们考虑对有向图$D$的弧$uv$进行定向操作，这意味着用相同顶点之间的无向边替换它。就连通性而言，这相当于将相反的弧$vu$添加到$D$。我们证明了对于每一个$ellgeq 3$，在一个有向图$D$中，为了得到一个$ell$ -强有向图$D'$，要找到最小的去向弧数是np困难的。
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