Suppose I am given distinct nodes $a_1,a_2,.., a_l, r$ and several $a_i$-$r$ paths $P_i$ in a planar graph $G$.
I wish to construct a tree $T$ connecting $a_1,a_2,.., a_l, r$ that minimizes the maximum over all $a_i$, the number of nodes bounded by the cycle formed by $P_i$ and the $a_i$-$r$ path in $T$
how minimal can I make this quantity (number of nodes bounded by the cycle formed by $P_i$ and the $a_i$-$r$ path in $T$) relative to the maximum number of nodes that lie inside a cycle of $P_i \cup P_j$?
