Determining if vertices lie within a set vertices
In my algorithm, I am finding graphs at different thresholds. Each graph G = (V,E). These are undirected graphs found using breadth first search. I would like to determine if the vertices of another graph G' = (V',E') lie within graph G. I am unfamiliar with graph algorithms so please let me know if you would like to see code or a more thorough explanation.
For example, If I have a graph G1 which is a square with 'corner' vertices (among others, but reduced for simplicity) of {(1,1), (1,6), (6,6), (6,1)}, then a smaller square G2 defined by corner vertices {(2,2), (2,5), (5,5), (5,2)} would lie within G1. The third graph G3 defined by corners {(3,3), (3,4), (4,4),(4,3)}. My algorithm produces the following figure for this configuration:
A square thresholded at 2, surrounded by t=1, surrounded by t=0. (I need to fix the edges but the vertices are correct)
My algorithm works on the following matrix:
import numpy as np
A = np.zeros((7,7))
#A[A<1] = -1
for i in np.arange(1,6):
for j in np.arange(1,6):
A[i,j] = 1
for i in np.arange(2,5):
for j in np.arange(2,5):
A[i,j] = 2
for i in np.arange(3,4):
for j in np.arange(3,4):
A[i,j] = 3
print(A)
To create three graphs, the first at threshold 2, the second at threshold 1, the third at threshold 0.
v1 = [[(3.0, 2.25), (3.0, 3.75), (2.25, 3.0), (3.75, 3.0)]]
v2 = [[(2.0, 1.333333), (1.333333, 3.0), (1.333333, 2.0), (1.333333, 4.0), (2.0, 4.666667), (3.0, 4.666667), (4.0, 4.666667), (4.666667, 4.0), (4.666667, 3.0), (4.666667, 2.0), (4.0, 1.333333), (3.0, 1.333333)]]
v3 = [[(1.0, 0.5), (0.5, 2.0), (0.5, 1.0), (0.5, 3.0), (0.5, 4.0), (0.5, 5.0), (1.0, 5.5), (2.0, 5.5), (3.0, 5.5), (4.0, 5.5), (5.0, 5.5), (5.5, 5.0), (5.5, 4.0), (5.5, 3.0), (5.5, 2.0), (5.5, 1.0), (5.0, 0.5), (4.0, 0.5), (3.0, 0.5), (2.0, 0.5)]]
And edge lists:
e1 = [[[2.25, 3.0], [3.0, 2.25]], [[3.0, 3.75], [2.25, 3.0]], [[3.0, 2.25], [3.75, 3.0]], [[3.0, 3.75], [3.75, 3.0]]]
e2 = [[[1.333333, 2.0], [2.0, 1.333333]], [[1.333333, 3.0], [1.333333, 2.0]], [[1.333333, 4.0], [1.333333, 3.0]], [[2.0, 4.666667], [1.333333, 4.0]], [[2.0, 1.333333], [3.0, 1.333333]], [[2.0, 4.666667], [3.0, 4.666667]], [[3.0, 1.333333], [4.0, 1.333333]], [[3.0, 4.666667], [4.0, 4.666667]], [[4.0, 1.333333], [4.666667, 2.0]], [[4.666667, 3.0], [4.666667, 2.0]], [[4.666667, 4.0], [4.666667, 3.0]], [[4.0, 4.666667], [4.666667, 4.0]]]
e3 = [[[0.5, 1.0], [1.0, 0.5]], [[0.5, 2.0], [0.5, 1.0]], [[0.5, 3.0], [0.5, 2.0]], [[0.5, 4.0], [0.5, 3.0]], [[0.5, 5.0], [0.5, 4.0]], [[1.0, 5.5], [0.5, 5.0]], [[1.0, 0.5], [2.0, 0.5]], [[1.0, 5.5], [2.0, 5.5]], [[2.0, 0.5], [3.0, 0.5]], [[2.0, 5.5], [3.0, 5.5]], [[3.0, 0.5], [4.0, 0.5]], [[3.0, 5.5], [4.0, 5.5]], [[4.0, 0.5], [5.0, 0.5]], [[4.0, 5.5], [5.0, 5.5]], [[5.0, 0.5], [5.5, 1.0]], [[5.5, 2.0], [5.5, 1.0]], [[5.5, 3.0], [5.5, 2.0]], [[5.5, 4.0], [5.5, 3.0]], [[5.5, 5.0], [5.5, 4.0]], [[5.0, 5.5], [5.5, 5.0]]]
Again, this gives graphs that look like this
This is the real data that I am working on. More complicated shapes.
Here, for example, I have a red shape inside of a green shape. Ideally, red shapes would lie within red shapes. They would be grouped together in one object (say an array of graphs).
The graphs are connected in a clockwise fashion. I really don't know how to describe it, but perhaps the graphs in the link show this. There's a bug on two of the lines (as you can see in the first plot, in the top right corner), but the vertices are correct.
Hope this helps! I can attach a full workable example, but it would include my whole algorithm and be pages long, with many functions! I basically want to use either input either g1, g2, and g3 into a function (or e1, e2, and e3). The function would tell me that g3 is contained with g2, which is contained within g1.
Your problem really does not have much to do with networks. Fundamentally, you are trying to determine if a point is inside a region described by an ordered list of points. The simplest way to this is to create matplotlib Path which has a contains_point method (there is also a 'contains_points` method to test many points simultaneously).
#!/usr/bin/env python
"""
Determine if a point is within the area defined by a path.
"""
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.path import Path
from matplotlib.patches import PathPatch
point = [0.5, 0.5]
vertices = np.array([
[0, 0],
[0, 1],
[1, 1],
[1, 0],
[0, 0] # NOTE the repetition of the first vertex
])
path = Path(vertices, closed=True)
print(path.contains_point(point))
# True
# plot to check visually
fig, ax = plt.subplots(1,1)
ax.add_patch(PathPatch(path))
ax.plot(point[0], point[1], 'ro')
Note that if a point is directly on the path, it is not inside the path. However, contains_point supports a radius argument that allows you to add an increment to the extent of the area. Whether you need a positive or negative increment depends on the ordering of the points. IIRC, radius shifts the path left in direction of the path but don't quote me on that.