Reconfigurable intelligent surfaces (RISs) enable programmable control of the wireless propagation environment and are key enablers for future networks. Beyond-diagonal RIS (BD-RIS) architectures enhance conventional RIS by interconnecting elements through tunable impedance components, offering greater flexibility with higher circuit complexity. However, excessive interconnections between BD-RIS elements require multi-layer printed circuit board (PCB) designs, increasing fabrication difficulty. In this letter, we use graph theory to characterize the BD-RIS architectures that can be realized on double-layer PCBs, denoted as planar-connected RISs. Among the possible planar-connected RISs, we identify the ones with the most degrees of freedom, expected to achieve the best performance under practical constraints.

翻译：可重构智能表面（RIS）能够实现对无线传播环境的可编程控制，是未来网络的关键使能技术。超对角可重构智能表面（BD-RIS）架构通过可调阻抗元件互连各单元，增强了传统RIS的功能，以更高的电路复杂度提供了更大的灵活性。然而，BD-RIS单元间过多的互连需要多层印刷电路板（PCB）设计，增加了制造难度。本文中，我们利用图论来刻画可在双层PCB上实现的BD-RIS架构，称为平面互连RIS。在可能的平面互连RIS中，我们识别出具有最多自由度的架构，预期能在实际约束下实现最佳性能。