Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem.

💡 Research Summary

The paper investigates two closely related families of geometric visibility graphs that arise from axis‑parallel visibility between unit squares placed in the plane. The first family, Unit Square Visibility graphs (USV), allows unit squares to be positioned at arbitrary real coordinates; the second family, Unit Square Grid Visibility graphs (USGV), restricts the lower‑left corner of each unit square to integer lattice points. The authors develop a rigorous formalism for visibility layouts, defining horizontal (→) and vertical (↓) visibility rectangles that connect two squares without intersecting any other square, and they translate these geometric relations into undirected graphs.

The combinatorial analysis begins with basic structural constraints that any USGV must satisfy. Lemma 1 shows that USGV is closed under edge and vertex deletion: removing a square or moving it away can eliminate the corresponding edges without affecting the remaining structure. Lemma 2 establishes three critical properties: (1) the maximum degree of a USGV graph is four (since a square can see at most one neighbor in each of the four cardinal directions); (2) any two distinct vertices share at most two common neighbors; and (3) if two vertices are adjacent, they have no common neighbor. These constraints immediately rule out many familiar dense graphs (e.g., K₁,₅, K₂,₃, K₅) from the class, while allowing all cycles of length at least four, K₁, K₂, K₁,ⱼ (j ≤ 4), and K₂,₂ as induced subgraphs.

A central contribution is the identification of USGV with the well‑studied class of rectilinear (or “rectilinear”) graphs. By mapping each unit square to its lattice point and turning each visibility into a straight horizontal or vertical segment, a USGV layout yields a rectilinear drawing. Conversely, any rectilinear drawing can be scaled by a factor of two and each vertex replaced by a unit square to obtain a (weak) USGV layout. The authors prove (Theorem 4) that any weak USGV layout can be transformed into a proper USGV layout, establishing an exact equivalence between USGV and rectilinear graphs. Consequently, all known results for rectilinear graphs (including NP‑hardness of recognition) transfer directly to USGV.

The paper then explores planarity. Although one might expect USGV graphs to be planar because of the grid‑based construction, the authors demonstrate that USGV contains non‑planar graphs. Figure 2 exhibits a subdivision of K₃,₃ that admits a USGV representation, and by Kuratowski’s theorem they conclude (Theorem 1) that USGV is not a subset of planar graphs. Moreover, they show that some planar graphs cannot be realized by any planar USGV layout; any layout for the graph in Figure 1(a) necessarily contains crossing visibilities (Proposition 1). Thus, planarity of the underlying graph and planarity of a specific layout are independent properties.

Regarding characterisation by forbidden subgraphs, Theorem 2 proves that USGV cannot be characterised by a finite set of forbidden induced subgraphs. This follows from the NP‑hardness of the recognition problem: if a finite forbidden set existed, USGV membership could be decided in polynomial time, contradicting hardness. In contrast, trees admit a simple characterisation (Theorem 3): a tree belongs to USGV if and only if its maximum degree does not exceed four. The authors also show (Theorem 4) that the weak version USGVᵂ coincides with USGV, because any weak representation can be “tightened” without introducing new edges.

The computational complexity results are a major focus. The recognition problems Rec(USV) and Rec(USGV) are defined as decision problems asking whether a given abstract graph admits a USV or USGV representation, respectively. For USGV, the authors rely on the known NP‑hardness of recognizing rectilinear graphs (citing