How to cluster the elements of a list based on connectivity?

Question

Consider a list (coordpairs) of coordinate pairs as

np = 4;
coordpairs = Cases[Subsets[Tuples[Range[np],{2}],{2}], {{a_, b_}, {c_, d_}} /; 
                   Abs[c - a] < 2 && Abs[d - b] < 2];

and their corresponding connectivities (conn) as

Edit 2: (corrected connectivity)

conn = {1., 1., 0., 1., 0., 1., 1., 1., 0., 1., 1., 0., 1., 0., 1., 0., 1.,
        0., 1., 1., 0., 1., 1., 0., 0., 0., 0., 0., 0., 1., 0., 0., 0., 0., 
        0., 0., 0., 0., 1., 1., 1., 1.};

1 indicates that there exists a path between the two points and 0 otherwise.

Now I wish to find all the paths (specifically, the coordinates of the points through the paths). For example, the first group will consist of the coordinates

{{1, 1}, {1, 2}, {2, 1}, {1, 3}, {2, 2}, {2, 3}, {1, 4}, {2, 4}, {3, 1}, {3, 2}, {3, 3}}

Second group:

{{3, 4}, {4, 4}, {4, 3}, {4, 2}, {4, 1}}

Edit 1 These two groups are visualized in the image below:

How can I do this?

Edit 3:(for higher np values)

Both the answers work well for np = 4. However, for np = 8

I have,

Length@coordpairs = 210;

conn = {0., 1., 1., 0., 0., 0., 0., 1., 0., 1., 1., 1., 0., 1., 1., 1., 0., 1., 1., 1., 1., 1., 0., 1., 1., 0., 0., 1., 1., 1., 1., 1., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 1., 0., 0., 1., 0., 0., 1., 0., 1., 0., 1., 1., 1., 1., 1., 1., 1., 0., 1., 1., 0., 0., 0., 0., 0., 0., 0., 1., 1., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0., 1., 1., 1., 1., 1., 1., 1., 0., 1., 1., 0., 1., 0., 1., 0., 1., 1., 1., 1., 0., 1., 1., 0., 0., 0., 1., 0., 1., 0., 1., 1., 1., 1., 1., 1., 1., 0., 1., 1., 0., 1., 0., 0., 1., 1., 0., 1., 1., 0., 1., 1., 0., 0., 0., 1., 0., 1., 0., 1., 1., 0., 1., 1., 1., 1., 0., 1., 1., 1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 1., 0., 0., 1., 1., 1., 0., 1., 1., 0., 0., 1., 0., 0., 0., 1., 0., 0., 0., 1., 0., 0., 0., 1., 0., 0., 1., 0., 0., 0., 0., 0., 0., 0., 1., 1., 1., 1., 0., 1.};

Now

 comps = ConnectedComponents[
  UndirectedEdge @@@ Extract[coordpairs, Position[db, 1.]]];

gives me 4 connected components where

Length@comps[[1]] + Length@comps[[2]] + Length@comps[[3]] + 
 Length@comps[[4]]

results

62

while it should be 64. The two missing coordinates are

{{1, 2}, {6, 3}}

where is it going wrong?

Answer 1

I'm agree this comment. This is your graph as you say

g = Graph[MapThread[
    If[#2 == 1, UndirectedEdge @@ #, Nothing @@ #] &, {coordpairs, conn}]];
Graph[g, VertexCoordinates -> VertexList[g], VertexLabels -> "Name"]

I also think your graph just have one connected component. You can group those connected vertices by ConnectedComponents and ConnectedGraphComponents

Answer 2

I would do it like this:

vertices = Union@Catenate[coordpairs];

edges = UndirectedEdge @@@ Pick[coordpairs, conn, 1.];

graph = Graph[vertices, edges, VertexCoordinates -> vertices, 
  VertexLabels -> Automatic]

ConnectedComponents[graph]
(* {{{3, 2}, {2, 2}, {2, 3}, {3, 3}, {1, 2}, {1, 3}, {1, 1}, {1,
    4}, {2, 4}, {2, 1}, {3, 1}}, {{3, 4}, {4, 4}, {4, 3}, {4, 2}, {4, 
   1}}} *)