Before revision:  We show that for any k \geq 0, the class of B_k-CPG graphs is strictly contained in the class of B_{k+1}-CPG graphs even within the class of planar graphs, thus implying that there exists no k \geq 0 such that every planar CPG graph is B_k-CPG. -hard forB_k-CPG with k=1,2, and for planar B_k-CPG with k \geq 3. } Additionally, we examine the computational complexity of several graph problems restricted to CPG graphs.
After revision:  We show that for any k \geq 0, the class of B_k-CPG graphs is strictly contained in the class of B_{k+1}-CPG graphs even within the class of planar graphs, thus implying that there exists no k \geq 0 such that every planar CPG graph is B_k-CPG. We further show that Recognition is \mathsf{NP-hard forB_k-CPG with k=1,2, and for planar B_k-CPG with k \geq 3. } Additionally, we examine the computational complexity of several graph problems restricted to CPG graphs.