Generalized Rough Sets Applied to Graphs Related to Urban Problems Mihai Rebenciuc, Simona Mihaela Bibic Abstract—Branch of modern mathematics, graphs represent in- struments for optimization and solving practical applications in various fields such as economic networks, engineering, network op- timization, the geometry of social action, generally, complex systems including contemporary urban problems (path or transport efficien- cies, biourbanism, & c.). In this paper is studied the interconnection of some urban network, which can lead to a simulation problem of a digraph through another digraph. The simulation is made univoc or more general multivoc. The concepts of fragment and atom are very useful in the study of connectivity in the digraph that is simulation - including an alternative evaluation of k- connectivity. Rough set approach in (bi)digraph which is proposed in premier in this paper contribute to improved significantly the evaluation of k-connectivity. This rough set approach is based on generalized rough sets - basic facts are presented in this paper. Keywords—(Bi)digraphs, rough set theory, systems of interacting agents, complex systems. I. I NTRODUCTION C OMPLEX systems represent sets of elements (agents) which are not identical and connected through various interactions (networks). Biourbanism [36] is the science that focuses on the study of the concept of an urban organism (or city regarded as an urban organism), considering it as a hypercomplex system in relation to its internal and external dynamics, as well as their mutual interactions. Also, biour- banism aims to reformulate the epistemological foundation of architecture and urbanism, in line with the science of complex dynamic systems. In general, this approach therefore links bi- ourbanism to the other sciences like life sciences (e.g., botany, biology, zoology, agriculture and food, microbiology, physi- ology, biochemistry, medical sciences) and integrated systems sciences (e.g., ecology, statistical mechanics, thermodynamics, operations research). Thus, the analysis of the evolutionary dynamics of a complex system can be described using network theory. From a mathematical point of view, network study (particular, urban networks) uses graph theory (one of the fundamental domains of discrete mathematics). In this respect, the networks are essential elements to understand the basic principles of other sciences, if these organizational principles are structured around of the mathematics of complexity, such as fractals and chaos theory. In terms of applicability in other areas can list some of them: biology, IT, economics, social sci- ences, urban planning. In some speciality works [33]–[36] has been demonstrated that the urban environment is an extremely Mihai Rebenciuc and Simona Mihaela Bibic are with the Depart- ment of Applied Mathematics, University POLITEHNICA of Bucharest, 313 Splaiul Independentei, RO-060042, Bucharest, Romania (e-mail: mi- hai.rebenciuc@upb.ro, simona.bibic@upb.ro). complex system that can be characterized by a large number of relations and interconnections that occur both between its components (agents) and between them and the external environment. For example, the network of streets and alleys whose interactions and connections determine the comfort’s level of urban neighborhoods, as well as its overlapping with other networks (energy, informational, social, economic flows, ecological, etc.). Graph theory [27], [33]–[35] is a tool for optimization and solving practical applications in all fields, such as representa- tion and study of economic and social networks, engineering, optimization of networks (goods and information transport systems), social action geometry, complex general systems, including contemporary urban issues (analysis of transporta- tion and distribution problems, biourbanism [36]), dynamic programming to determine the optimal policy, game theory, information theory (study of signs and codes). However, the critical behavior of classical graph theory [35] has been noted in solving existing problems (e.g., network security applica- tions). In this respect, it was introduced the notion of bigraph (an extension of the graph [33]), improved subsequently by bigraph with sharing [34]. Bigraphs represent a mathematical model for interacting systems of agents (ubiquitous systems based on placing and linking) and having the ability to indicate the position in space, displacement, and the agents’ intercon- nections. As an extrapolation of applications of (bi)digraphs is proposed rough set approach regarding to issues of possible uncertainty related to urban problems [1]–[4]. II. GENERALIZED ROUGH SETS:ANEW LOOK Remark 1. (A brief history) Rough sets (RS) - as an extension of classic (crisp) sets had an exponential development (by applications in various areas) in the quarter century between when opening [5] and testamentary time [6] - and continue today; this is illustrated by a handbook [7] and by series LNCS Transactions in RS [8] and LNAI RS and Knowledge Technologies [9] (a Pawlak dedication). Originally RS were defined in a classifying (parti- tioning) approximation space, i.e., in an equivalence relational structure, then - more general in a covering approximation space, i.e., a tolerance relational structure - up to a (ho- mogeneous) relational approximation space [5], [11], [10], [12], but in [10] Pawlak speaks about an alternative - a topological approximation space which is an idea resumed in other paper [13], [14]. A category approach to RS is made in [15]. RS interfered and interferes with fuzzy sets (FS) theory World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:11, No:9, 2017 392 International Scholarly and Scientific Research & Innovation 11(9) 2017 scholar.waset.org/1307-6892/10007960 International Science Index, Mathematical and Computational Sciences Vol:11, No:9, 2017 waset.org/Publication/10007960