  We prove a conjecture of Menasco and Zhang that if a tangle is completely
tubing compressible then it consists of at most two families of parallel
strands. This is related to problems of graphs in 3-manifold. A 1-vertex graph
$\Gamma$ in a 3-manifold $M$ with a genus 1 Heegaard splitting is standard if
it consists of one or two parallel sets of core curves lying in the Heegaard
splitting solid tori of $M$ in the standard way. The above conjecture then
follows from the theorem which says that a 1-vertex graph in $M$ is standard if
and only if the exteriors of all its nontrivial subgraphs are handlebodies.
