Puzzling Grid Embeddings

Grid Recognition: Classical and Parameterized Computational Perspectives

“…Since {N i,j (5), N i,j (6)} ∈ E(G) we get that f (N i,j (6)) = (12(i − 1) + 7, 8(j − 1) + 5) or f (N i,j (6)) = (12(i − 1) + 9, 8(j − 1) + 5) or f (N i,j (6)) = (12(i − 1) + 8, 8(j − 1) + 4). If f (N i,j (6)) = (12(i − 1) + 7, 8(j − 1) + 5) we get that one of N i,j (7), N i,j (8), N i,j (9) is embedded to (12(i − 1) + 7, 8(j − 1) + 6), but one of N i,j (3), N i,j (4) must be embedded there, so we get a contradiction. Similarly, we get a contradiction also if f (N i,j (6)) = (12(i − 1) + 9, 8(j − 1) + 5).…”

“…Similarly, we get a contradiction also if f (N i,j (6)) = (12(i − 1) + 9, 8(j − 1) + 5). Now, if f (N i,j (6)) = (12(i − 1) + 8, 8(j − 1) + 4), then one of N i,j (7), N i,j (8), N i,j (9) is embedded at (12(i − 1) + 9, 8(j − 1) + 4). From Lemma 4.12 we get that f ((12(i − 1) + 10, 8(j − 1) + 4)) = (12(i − 1) + 10, 8(j − 1) + 4) and (12(i − 1) + 10, 8(j − 1) + 4) has four neighbors that N i,j (7), N i,j (8), N i,j (9) are not among them.…”

“…Shortly afterwards, it has been proven to be NP-hard even on binary trees [33]. On the positive side, there is research on practical algorithms for this problem [7]. The related upward planarity testing and rectilinear planarity testing problems are also known to be NP-hard [32], as well as HV-planarity testing even on graphs of maximum degree 3 [22].…”

Sobre Imersão de Árvores em Grades

“…Por conveniência, permitiremos confundir o conceito de caminho no grafo com sua representação no modelo. A imersão de grafos em grade já foi considerada com diferentes perspectivas [1,2,5,7,8,9]. Uma mudança de direção de um caminho em um ponto da grade é chamada de dobra e o ponto da grade no qual uma dobra ocorre é chamado de ponto de dobra.…”

On Embedding Trees in Grids

“…The problem of grid embedding is that of drawing a graph G onto a rectangular twodimensional grid (called simply grid) such that each vertex v ∈ V (G) corresponds to a grid point (an intersection of a horizontal and a vertical grid line) and the edges of G correspond to paths of the grid. Grid embedding of graphs has been considered with different perspectives [Beck and Storandt 2020, Liu et al 1998, Schnyder 1990]. In [Liu et al 1998], linear-time algorithms are described for embedding planar graphs having their edges drawn as non-intersecting paths in the grid, such that the maximum number of bends of any edge is minimized, as well as the total number of bends.…”