Chordal Graphs and maximum independent sets For a chordal graph $G$ there is a clique tree such that its vertices corresponds to maximal cliques of $G$ and there is a edge between two vertices iff the intersection of the corresponding cliques are also their minimal vertex separator and for each vertex in the graph the cliques containing it, induces a subtree.

Now my questions are:- 


  1.Take a subpath of the tree of length 5 with the property that no
  vertex has degree more that 2 and intersection of all the maximal
  cliques is non empty. Does there exist a independent set of
  atleast size 3 in the subgraph induced by that vertices present in the
  maximal cliques taken in the path?
  
  2.If yes, then is the bound of path length and independent set size tight?
There are arbitrarily long sub-path whose induced subgraph contains no three independent vertices.  See the graph below for length 4:



Indeed, vertices in a sub-path induces an interval subgraph.  So you can use standard interval models (the endpoints of the interval for a vertex $v$ are the indices of leftmost and rightmost bags that contain $v$) to construct the tight example easily.  The graph given above have intervals:
[1,1], [1,2], [1,3], [2,4], [3,4],[4,4].  Likewise, you can construct for a subpath of length n: the basic idea is that each interval either starts from 1 or ends at n.