Dark matter halo clustering depends not only on halo mass, but also on other properties such as concentration and shape. This phenomenon is known broadly as assembly bias. We explore the dependence of assembly bias on halo definition, parametrized by
spherical overdensity parameter, $Delta$. We summarize the strength of concentration-, shape-, and spin-dependent halo clustering as a function of halo mass and halo definition. Concentration-dependent clustering depends strongly on mass at all $Delta$. For conventional halo definitions ($Delta sim 200mathrm{m}-600mathrm{m}$), concentration-dependent clustering at low mass is driven by a population of haloes that is altered through interactions with neighbouring haloes. Concentration-dependent clustering can be greatly reduced through a mass-dependent halo definition with $Delta sim 20mathrm{m}-40mathrm{m}$ for haloes with $M_{200mathrm{m}} lesssim 10^{12}, h^{-1}mathrm{M}_{odot}$. Smaller $Delta$ implies larger radii and mitigates assembly bias at low mass by subsuming altered, so-called backsplash haloes into now larger host haloes. At higher masses ($M_{200mathrm{m}} gtrsim 10^{13}, h^{-1}mathrm{M}_{odot}$) larger overdensities, $Delta gtrsim 600mathrm{m}$, are necessary. Shape- and spin-dependent clustering are significant for all halo definitions that we explore and exhibit a relatively weaker mass dependence. Generally, both the strength and the sense of assembly bias depend on halo definition, varying significantly even among common definitions. We identify no halo definition that mitigates all manifestations of assembly bias. A halo definition that mitigates assembly bias based on one halo property (e.g., concentration) must be mass dependent. The halo definitions that best mitigate concentration-dependent halo clustering do not coincide with the expected average splashback radii at fixed halo mass.
