  The L-move for classical braids extends naturally to trivalent braids. We
follow the L-move approach to the Markov Theorem, to prove a one-move
Markov-type theorem for trivalent braids. We also reformulate this L-Move
Markov theorem and prove a more algebraic Markov-type theorem for trivalent
braids. Along the way, we provide a proof of the Alexander's theorem analogue
for spatial trivalent graphs and trivalent braids.
