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?