Inverter based microgrids are becoming a viable and attractive choice for future power distribution systems with substantial renewable penetration. The control architectures of such microgrids are currently designed to mimic conventional power systems with droop-based control, coupling active power to frequency and reactive power to voltage. However, the dynamic behavior of low-voltage microgrids is very different from large-scale power systems. In particular, the electromagnetic network dynamics plays an unexpectedly important role despite its very small timescales. This makes simplified swing-type equations inappropriate for microgrid stability assessment requiring an explicit consideration of line currents dynamics. In the present work we elucidate the role of network dynamics in such micro-grids and uncover the major factors affecting microgrids stability. We then present a systematic approach for proper accounting for the network dynamics by a special model-order reduction procedure. Based on this reduced-order model we derive a set of stability certificates each based only on local parameters, namely the settings of each pair of interconnected inverters and parameters of the corresponding interconnection line. This set of constraints establishes a natural foundation for plug-and-play interconnection standards. Possible applications of the derived stability criteria are discussed ranging from microgrid network planning to multi-microgrid interconnection/reconfiguration decision tools.