Legal fuzzy incidence blocks and legal flow reduction sets with application to water distribution networks

Directed fuzzy incidence graphs (DFIGs) are fuzzy incidence structures where each edge and incidence pair has a specific direction. In these graphs, the relationships are not symmetric, making it easier to identify the extent of interaction between nodes and arcs. The comprehensive investigation of connectivity in directed fuzzy incidence graphs holds the potential to provide solutions for a wide range of real-world problems including traffic flow optimization in one-way traffic networks and migration analysis of refugees across various countries. The main objective of this paper is to extend the connectivity concepts of directed fuzzy incidence graphs, which help to analyze various stochastic networks influenced by external factors. The major concepts discussed in this article are legal fuzzy incidence blocks, legal flow reduction sets, and the DFIG-version of Menger's theorem. Directed fuzzy incidence graphs having no legal flow reduction nodes are defined as legal fuzzy incidence blocks (LFI-blocks). Legal flow reduction nodes (LFR-nodes) are distinct elements, the elimination of which results in a reduction of the directed incidence connectivity between some other pair of nodes. Since LFI-blocks lack these elements, the removal of none of the nodes reduce the legal flow between other pair of nodes. Consequently, for each node under consideration, any other pair of nodes connected by at least one di-path must possess a widest legal di-path which avoids that specific node. LFI-blocks exhibit differing attributes in contrast to their counterparts in fuzzy graphs and fuzzy incidence graphs. This viewpoint motivated the authors to investigate these distinctions and assess the possibility of defining equivalent conditions for LFI-blocks, similar to the equivalent conditions established for fuzzy blocks and fuzzy incidence blocks. A key characteristic of legal fuzzy incidence blocks is their capacity to incorporate legal flow reduction links, even allowing for the presence of shared nodes among two legal flow reduction links. This contrasts with fuzzy graphs and fuzzy incidence graphs, where if two fuzzy bridges happen to share a common node, it results in that node being a fuzzy cutnode. Two equivalent conditions are proposed for a directed fuzzy incidence graph to become a legal fuzzy incidence block. It is shown that any two nodes in a legal fuzzy incidence block which are connected by at least one di-path, but not by a legal flow reduction link will have at least two internally disjoint widest legal di-paths joining them. To identify the maximum number of such internally disjoint widest legal di-paths, the notion of legal flow reduction set of nodes and arcs is introduced. These sets are characterized using widest legal di-paths. Moreover, the DFIG-version of Menger's theorem for nodes is established, which explains the relation between number of internally disjoint widest legal di-paths and cardinality of legal flow reduction sets in a directed fuzzy incidence graph. Also, an illustration of Menger's theorem is proposed for water distribution networks.