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matrix - Cell-adjacency Graph of a Square Mesh


Consider the list of points


pts = {{1, 1}, {1, 2}, {2, 1}, {2, 2}}

I want to use them to define a 2x2 square mesh using VoronoiMesh, where each cell has two neighbours. Following the discussion in this question, consider the following code


mesh = VoronoiMesh[pts, ImageSize -> Medium];
conn = mesh["ConnectivityMatrix"[2, 1]];
adj = conn.Transpose[conn];
centers = PropertyValue[{mesh, 2}, MeshCellCentroid];
g = AdjacencyGraph[adj, PlotTheme -> "Scientific", 

   VertexCoordinates -> centers];
Show[mesh, g]

enter image description here


As one can see, unlike other meshes, this one does not seem to work exactly as I want, since the diagonal edge should not appear. Why is this happening? Any way of avoiding that edge and get


enter image description here


as one would expect from a square lattice?


Edit: As noticed in the comment section, some of the polygons seem to have sharing edges that are single points, which is enough for them to be considered neighbouring cells. This effect is unchanged with the size of the lattice. If I consider, for example, the points


pts = Flatten[Table[{i, j}, {i, 7}, {j, 5}], 1];


I get


enter image description here


Any ideas on how to solve this? Maybe omit the extra edge in a way that doesn't this or other non-square meshes. For example, considering a random VoronoiMesh, nothing seems to wrong, though it could, theoretically, go


enter image description here



Answer



We can delete the rows in our incidence matrix that correspond to these edges of length 0.


pts = Flatten[Table[{i, j}, {i, 7}, {j, 5}], 1];

mesh = VoronoiMesh[pts, ImageSize -> Medium];
conn = mesh["ConnectivityMatrix"[1, 2]];


lens = PropertyValue[{mesh, 1}, MeshCellMeasure];
$threshold = 0.;
keep = Pick[Range[MeshCellCount[mesh, 1]], UnitStep[Subtract[$threshold, lens]], 0];
conn = conn[[keep]];

adj = Transpose[conn].conn;
centers = PropertyValue[{mesh, 2}, MeshCellCentroid];
g = AdjacencyGraph[adj, PlotTheme -> "Scientific", VertexCoordinates -> centers];


Show[mesh, g]

enter image description here


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