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. 



LP Constraints for Connected Subgraphs of Fixed Size
It looks like Section 3 in [Algorithms for the Maximum Weight Connected k
-Induced Subgraph Problem](https://www.algorithmics.informatik.uni-mainz.de/files/2016/05/Algorithms-for-the-Maximum-Weight-Connected-k-Induced-Subgraph-Problem.pdf) (Ernst Althaus, Markus Blumenstock, Alexej Disterhoft, Andreas Hildebrandt and Markus Krupp; COCOA 2014) contains several possible ILP models. 


