Is "Reachable Object" really an NP-complete problem?
A problem $P$ is NP-complete if:


$P$ is NP-hard and
$P \in \textbf{NP}$.


The authors give a proof of item number 1. Item number 2 is probably apparent (and should be clear to the paper's audience). For the proof of item number 1, you only need a (many-one) reduction from some NP-complete problem (e.g., SAT) to $P$; there is no need to construct a reduction in the opposite direction.