Question:

how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?

Fixing the size of the subgraph is trivial; also some upper bounds on the number of edges between the elements of $W$ may be devised quite easily. I can't however come up with a provably necessary and sufficient condition.