Finding the dense regions in a graph is an important problem in network analysis. Core decomposition and truss decomposition address this problem from two different perspectives. The former is a vertex-driven approach that assigns density indicators for vertices whereas the latter is an edge-driven technique that put density quantifiers on edges. Despite the algorithmic similarity between these two approaches, it is not clear how core and truss decompositions in a network are related. In this work, we introduce the vertex interplay (VI) and edge interplay (EI) plots to characterize the interplay between core and truss decompositions. Based on our observations, we devise CORE-TRUSSDD, an anomaly detection algorithm to identify the discrepancies between core and truss decompositions. We analyze a large and diverse set of real-world networks, and demonstrate how our approaches can be effective tools to characterize the patterns and anomalies in the networks. Through VI and EI plots, we observe distinct behaviors for graphs from different domains, and identify two anomalous behaviors driven by specific real-world structures. Our algorithm provides an efficient solution to retrieve the outliers in the networks, which correspond to the two anomalous behaviors. We believe that investigating the interplay between core and truss decompositions is important and can yield surprising insights regarding the dense subgraph structure of real-world networks.