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Stability analysis of interconnected systems with finite as well as arbitrary interconnection delays is considered in this study. For the purpose, subsystems interconnected with finite delays are grouped to form larger subsystems with intraconnection delays. However, these larger subsystems are interconnected with arbitrary delays among themselves. By considering a suitable Lyapunov-Krasovskii functional, a stability criterion is derived in the linear matrix inequality (LMI) framework to study simultaneous delay-independent (for the interconnection delays) and delay-dependent (for the intraconnection delays) stability of the reformed interconnected system. This proposed criterion is useful when one attempts to estimate the tolerable bounds of the intraconnection delays. Finally, a numerical example is considered to illustrate the effectiveness of the proposed criterion.