Abstract
The extension of bipolar fuzzy graph is bipolar fuzzy incidence graph (BFIG) which gives the information regarding the effect of vertices on the edges. In this paper, the concept of matching in bipartite BFIG and also for BFIG is introduced. Some results and theorems of fuzzy graphs are also extended in BFIGs. The number of operations in BFIGs such as augmenting paths, matching principal numbers, relation between these principal numbers and maximum matching principal numbers are being investigated which are helpful in the selection of maximum most allied applicants for the job and also to get the maximum outcome with minimum loss (due to any controversial issues among the employees of a company). Some characteristics of maximum matching principal numbers in BFIG are explained which are helpful for solving the vertex and incidence pair fuzzy maximization problems. Lastly, obtained maximum matching principal numbers by using the matching concept to prove its applicability and effectiveness for the applications in bipartite BFIG and also for the BFIG.