Node-Weighted Network Design in Planar and Minor-Closed Families of Graphs

We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph $G=(V,E)$ and integer connectivity requirements $r(uv)$ for each unordered pair of nodes $uv$. The goal is to find a minimum weighted subgraph $H$ of $G$ such that $H$ contains $r(uv)$ disjoint paths between $u$ and $v$ for each node pair $uv$. Thr