  Connected sum and trivalent vertex sum are natural operations on genus 2
spatial graphs and, as with knots, tunnel number behaves in interesting ways
under these operations. We prove sharp Scharlemann-Schultens type bounds for
the tunnel number of a composite genus 2 spatial graph. For the tunnel number
of a composite Brunnian $\theta$-curve, our result implies that the tunnel
number is at least the number of summands, as in the knot case. We also prove a
version of a theorem of Morimoto for knots: the tunnel number of a composite
m-small genus 2 spatial graph is at least the sum of the tunnel numbers of the
factors. We also study lower bounds for the bridge number of composite genus 2
graphs. In particular, our results imply that for a Brunnian composite
$\theta$-curve having $m$ factors in its prime factorization, the bridge number
is at least $m+3/2$.
