Help identifying pls. What is this ‘weed’?

Hello!

I would love some help identifying this weird weed/root of something maybe?

I am in Bendigo, and my rental’s little front yard which struggles to maintain any lawn of sorts but does have this strange weed/root system running through it. I have dug all out that I could, but these roots do seem to be continuing under our terrace home but not into our back yard.

Does anyone know what this is? Is digging out what I can find enough?

Have included photos of my little yard, and pics of the weed I have pulled out plus the roots that are still there but run under my rose bush so don’t want to try to get any more out.

Thanks in advance 🙌🙏