The emergence of the Internet of Things (IoT) is introducing more and more services and applications such as smart cities. For some services, the availability of elements and the connectivity between them are necessary. The robust functioning of a fragile and often dynamic system in IoT needs strong monitoring which is investigated in this paper. IPv6 Routing Protocol for Low Power and Lossy Networks (RPL) is the standardized routing protocol over IP-connected IoT networks. In tandem with RPL, we propose a new solution which aims to achieve distributed monitoring with minimal computational complexity. The main objective is to increase robustness in IoT via monitoring the links in the Destination Oriented Directed Acyclic Graphs (DODAG) constructed by RPL. Although RPL has a reactive repairing mechanism to tackle robustness in connectivity in case of node failures, we found that for real-time critical applications it is important to use proactive approaches for preventing faults and making recovery for connectivity faster. The problem can be modeled as a Vertex Cover Problem (VCP) on the DODAG. Such problem is a well known NP-hard optimization problem. The monitoring should be simple and energy aware. We demonstrate that the monitor placement in our case is only Fixed-Parameter Tractable, and also polynomial-time solvable, which is the best case.