Tunnel number and bridge number of composite genus 2 spatial graphs

Connected sum and trivalent vertex 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 of a composite graph. For Brunnian $\theta$-curve, our result implies that is at least summands, knot case. also version theorem Morimoto knots: m-small graph numbers factors. study lower bridge graphs. In particular, results imply $\theta$-curve having $m$ factors its prime factorization, $m+3/2$.