Should $A$ or $B$ be chosen as the principal chain for nomenclature? Why?

(A) The principal chain should be chosen such that it includes the substituent with more branching at the point of attachment.
Both $A$ and $B$ are alkyl groups with the same formula $C_{6}H_{13}$.
According to the $IUPAC$ rules for selecting the principal chain when two chains have equal length,the chain with more branching (or more complex branching) at the point of attachment is preferred.
In this case,the group $A$ has a quaternary carbon atom (a carbon bonded to four other carbons) at position $3$ of the substituent chain,which represents more branching compared to the structure of $B$.
Therefore,the chain containing $A$ should be selected as the principal chain.