It is known that the uniquely decodable code pairs (C1, C2) for the two-user binary adder channel relates to the maximum independent set of a graph associated with a binary code. This paper formulates the independence number of a class of graphs associated with binary linear codes, and presents an algorithm of the maximum independent set for those graphs. Uniquely decodable code pairs (C1, C2)'s are produced, where C1 is a linear code and C2 is a maximum independent set of the graph associated with C1. For the given C1, the transmission rate of C2 is higher than that by Khachatrian, which is known as the best result as so far. This is not rather surprising because the code C2 is a maximum independent set in this paper but not be Khachatrian's.