Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is $NP$-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be $NP$-hard, which settles an open question.