Roughly speaking, VLSI CAD is to develop software algorithms to assist VLSI circuit designs. This objective itself implies that both Electrical Engineering (EE) and Computer Science (CS) background are requested. Since the ultimate goal is to enable efficient and high performance circuit design, obviously you have to understand VLSI circuit design. As for the CS part, both algorithm design and proficient programming skill are essential, because CAD research is somewhat equivalent to algorithm design and the effectiveness of the algorithms need to be verified through software implementations. Sometimes advanced algorithm designs require knowledgement on Applied Math such as linear programming, graph theory and combinatorial optimizations.
Ideally, it is preferred that you have background on all of the aforementioned fields: VLSI design, algorithms and applied math. However, this rarely happens for a junior graduate student. In practice, I wish you to have solid background on at least one of these fields. As long as you are diligent and intelligent, you will be successful!