  Recent works have explored the properties of L\'evy flights with resetting in
one-dimensional domains and have reported the existence of phase transitions in
the phase space of parameters which minimizes the Mean First Passage Time
(MFPT) through the origin [Phys. Rev. Lett. 113, 220602 (2014)]. Here we show
how actually an interesting dynamics, including also phase transitions for the
minimization of the MFPT, can also be obtained without invoking the use of
L\'evy statistics but for the simpler case of random walks with exponentially
distributed flights of constant speed. We explore this dynamics both in the
case of finite and infinite domains, and for different implementations of the
resetting mechanism to show that different ways to introduce resetting
consistently lead to a quite similar dynamics. The use of exponential flights
has the strong advantage that exact solutions can be obtained easily for the
MFPT through the origin, so a complete analytical characterization of the
system dynamics can be provided. Furthermore, we discuss in detail how the
phase transitions observed in random walks with resetting are closely related
to several ideas recurrently used in the field of random search theory, in
particular to other mechanisms proposed to understand random search in space as
mortal random-walks or multi-scale random-walks. As a whole we corroborate that
one of the essential ingredients behind MFPT minimization lies in the
combination of multiple movement scales (whatever its origin).
