Profile and neighbourhood complexity for graphs excluding a subdivision

Investigate the profile and neighbourhood complexity of graphs excluding K_t as a subdivision, and obtain bounds substantially better than the currently available bound.

Background

The authors extend their discussion from graphs excluding a complete minor to graphs excluding K_t as a subdivision. Their method yields a bound of the form (r+2)^{(c_t r)^r}(c_t r)^r k for some constant c_t, but they explicitly regard this bound as far from optimal. They identify improved control of the distance VC-dimension as a potentially useful route to better profile-complexity bounds.