Branch vertex of a graph subdivision

A branch vertex of a graph subdivision is a retained original vertex in a specified graph subdivision. Its degree of a vertex in that graph subdivision equals its original degree of a vertex. It need not have degree of a vertex at least three: $B,C,D$ in the example are also branch vertices of a graph subdivision. When the target itself has vertices of degree of a vertex two, adjacency alone need not identify which vertices were original; the specified target mapping supplies that distinction.