Periodic π-vacancies in graphene superlattices (GSLs) provide a symmetry-based route to band gap opening in graphene by modifying the π-band dispersion, yet the symmetry conditions that determine whether a vacancy motif can open a band gap remain unclear. Here, we investigate periodic π-vacancy GSLs using a nearest-neighbor tight-binding model with one $p_z$ orbital per carbon site to identify the symmetry requirements for band gap opening. Here, π vacancies, resembling functionalized, substituted, or missing carbon sites, are modeled as site deletions in the π basis, and the corresponding hopping matrix elements to and from the deleted sites are set to zero. We focus on π-vacancy motifs with $C_2$ and $C_3$ point groups. A $3n\times 3n$ GSL, where $n=1,2,3,\cdots$ is the integer scaling factor multiplying the honeycomb primitive-cell vectors, is required to fold $K$ and $K^{\prime }$ to $\mathrm{\Gamma }$, which is one of the requirements for opening a band gap. For $C_3$-type vacancies, the Dirac cones are pinned at the high-symmetry points, thus remaining at $\mathrm{\Gamma }$ in the folded $3n$ GSLs. However, those $C_2$-type vacancies, which reduce the global point group of the GSL to $D_{2h}$ by preserving a pair of perpendicular mirror symmetries (${\sigma }_v\perp {\sigma }_d)$, can constrain the positions of Dirac cones at $\mathrm{\Gamma }$. The absence of the ${\sigma }_v$ and ${\sigma }_d$ planes allows the cones to shift around $\mathrm{\Gamma } \left(\pm \mathrm{\Delta }\mathbf{q},\pm \mathrm{\Delta }\mathbf{q}\right)$ in the $3n$ superlattice. The $C_3$-type vacancies open a relatively larger band gap than the $C_2$-type vacancies due to twofold degeneracy at $\mathrm{\Gamma }$, while the $C_2$-type vacancies leave the GSL with nondegenerate bands. The energy contours near the Dirac cone for $C_3$-symmetric vacancies remain isotropic, while the absence of $C_3$ generates anisotropic cones. The band gap openings in $3n$ superlattices scale linearly with defect concentration for each vacancy motif, with the $D_{6h}$ motif yielding the largest gap, reaching 314 meV at only 3.7% defect concentration. Disorder of the superlattice by missing vacancies or by vacancy displacements of one or two primitive lattice vectors of graphene breaks the $C_3$ symmetry, lifts the degeneracy, and reduces the gap toward the pristine graphene limit.

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• Band gap
• Dirac semimetal
• Graphene
• Superlattices
• Tight-binding model