  We study theta characteristics of hyperelliptic metric graphs of genus $g$
with no bridge edges. These graphs have a harmonic morphism of degree two to a
metric tree that can be lifted to morphism of degree two of a hyperelliptic
curve $X$ over $K$ to the projective line, with $K$ an algebraically closed
field of char$(K) \not =2$, complete with respect to a non-Archimedean
valuation, with residue field $k$ of char$(k)\not=2$. The hyperelliptic curve
has $2^{2g}$ theta characteristics. We show that for each effective theta
characteristics on the graph, $2^{g-1}$ even and $2^{g-1}$ odd theta
characteristics on the curve specialize to it; and $2^g$ even theta
characteristics on the curve specialize to the unique not effective theta
characteristics on the graph.
