Heat supplied dQ = increased in internal energy dU is true for

The First Law of Thermodynamics states $dQ = dU + dW$, where $dW = P\,dV$ for a simple compressible system.

When the heat supplied equals the increase in internal energy, $dQ = dU$, substitution yields $dU = dU + dW$, which implies $dW = 0$.

Since $dW = P\,dV$, and pressure $P \neq 0$, this requires $dV = 0$.

A process with constant volume is isochoric. For such processes, $dW = 0$ and consequently $dQ = dU$.

A constant temperature (isothermal) process has $dU = 0$ for an ideal gas, giving $dQ = dW$.

An adiabatic process has $dQ = 0$, so $dU = -dW$.

A constant pressure (isobaric) process has $dW = P\,dV \neq 0$, so $dQ \neq dU$.

Only the isochoric process satisfies $dQ = dU$.

Final answer: $\boxed{D}$