Connecting vertex sets to walls

Abstract: Menger's theorem on $A$--$B$-paths and Gallai's theorem on $A$-paths are among the most useful results in structural graph theory. Many variants and extensions are known. We add to this line of research and prove results that relate the maximal number of vertex-disjoint paths between vertex sets and a wall to the minimum number of vertices meeting all these paths. We also include types of paths that start and end in the wall.