Low-Complexity Planar Beyond-Diagonal RIS Architecture Design Using Graph Theory

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.