Why have you marked the line as Not Connected.
But MarkT, his chip is powered with red and black from the battery, and the Arduino is powered on the usb. It's the (missing) ground between the 2 that OP is asking about.
Well I guess I should stick to connecting all grounds.
In this case yes, but be careful of such a sweeping statement. If you have one of those opto-isolatable relay boards, the ones with the JD-Vcc jumper, and you use external power for the relay coils, you want to keep the grounds un-connected.