  In this note we give two examples of partially commutative subgroups of
partially commutative groups. Our examples are counterexamples to the Extension
Graph Conjecture and to the Weakly Chordal Conjecture of Kim and Koberda,
\cite{KK}. On the other hand we extend the class of partially commutative
groups for which it is known that the Extension Graph Conjecture holds, to
include those with commutation graph containing no induced $C_4$ or $P_3$. In
the process, some new embeddings of surface groups into partially commutative
groups emerge.
