Let $G=(V,E) $ be a series parallel graph (definition) with terminals $s,t \in V$
I am trying to prove the follwowing two statements.
Let $e \in E$ be an edge and $G'$ be the graph obtained by adding a parallel edge to e. Then $G$ is an sp-graph if and only if $G'$ is an sp graph
Let $e \in E$ be any edge $e=u-v$. The graph $G'$ obtained by 'splitting' the edge (adding another vertex x so we have the path u-x-v) is a sp-graph if and only if $G$ is an sp -graph
I don't know how to prove the general case ($e$ an arbitrary edge) but if we take $e=(s,t)$ then I add a parallel edge to $e$ by doing parallel composition with the "base"-graph (graph only containing the edge $(s,t$) or a series composition for 2)
How do I prove the general case ?
Would appreciate any tips/hints/help