MATLAB graph/cyclebasis: How can I extract labeling-independent “minimal loop units”?

I have two undirected graphs that represent the same connectivity (isomorphic up to node relabeling). When I call cyclebasis on each, I get different sets of cycles. I understand a cycle basis can depend on the spanning tree/edge order, but I want a labeling-invariant notion of “minimal loop units.”
Code
clc; clear; close all

node = [
2 7
2 3
3 4
4 5
5 6
6 1
1 4
1 5
3 6
1 3
5 7
6 7
2 6
5 8];

G = graph(node(:,1), node(:,2), []);
cycles = cyclebasis(G)

figure(1); plot(G,’Layout’,’force’);

node2 = [
1 2
2 3
3 4
4 1
1 5
5 6
2 4
6 7
3 6
2 5
3 7
4 7
2 6
6 8];

G2 = graph(node2(:,1), node2(:,2), []);
cycles2 = cyclebasis(G2)

figure(2); plot(G2,’Layout’,’force’);

Result:
cycles =
7×1 cell array
{[1 3 6 5]}
{[ 1 4 5]}
{[ 1 5 6]}
{[ 2 3 6]}
{[2 6 5 7]}
{[3 4 5 6]}
{[ 5 6 7]}

cycles2 =
7×1 cell array
{[ 1 2 4]}
{[ 1 2 5]}
{[ 2 3 4]}
{[ 2 3 6]}
{[2 3 7 6]}
{[2 4 7 6]}
{[ 2 5 6]}

Questions:
I know cyclebasis can vary with spanning tree/edge ordering. What’s the recommended way in MATLAB to obtain “minimal loop units” that do not depend on node labeling or edge input order? For example, in the above case, each cycle is a 3-node triangle, and there should be seven such cycles.I have two undirected graphs that represent the same connectivity (isomorphic up to node relabeling). When I call cyclebasis on each, I get different sets of cycles. I understand a cycle basis can depend on the spanning tree/edge order, but I want a labeling-invariant notion of “minimal loop units.”
Code
clc; clear; close all

node = [
2 7
2 3
3 4
4 5
5 6
6 1
1 4
1 5
3 6
1 3
5 7
6 7
2 6
5 8];

G = graph(node(:,1), node(:,2), []);
cycles = cyclebasis(G)

figure(1); plot(G,’Layout’,’force’);

node2 = [
1 2
2 3
3 4
4 1
1 5
5 6
2 4
6 7
3 6
2 5
3 7
4 7
2 6
6 8];

G2 = graph(node2(:,1), node2(:,2), []);
cycles2 = cyclebasis(G2)

figure(2); plot(G2,’Layout’,’force’);

Result:
cycles =
7×1 cell array
{[1 3 6 5]}
{[ 1 4 5]}
{[ 1 5 6]}
{[ 2 3 6]}
{[2 6 5 7]}
{[3 4 5 6]}
{[ 5 6 7]}

cycles2 =
7×1 cell array
{[ 1 2 4]}
{[ 1 2 5]}
{[ 2 3 4]}
{[ 2 3 6]}
{[2 3 7 6]}
{[2 4 7 6]}
{[ 2 5 6]}

Questions:
I know cyclebasis can vary with spanning tree/edge ordering. What’s the recommended way in MATLAB to obtain “minimal loop units” that do not depend on node labeling or edge input order? For example, in the above case, each cycle is a 3-node triangle, and there should be seven such cycles. I have two undirected graphs that represent the same connectivity (isomorphic up to node relabeling). When I call cyclebasis on each, I get different sets of cycles. I understand a cycle basis can depend on the spanning tree/edge order, but I want a labeling-invariant notion of “minimal loop units.”
Code
clc; clear; close all

node = [
2 7
2 3
3 4
4 5
5 6
6 1
1 4
1 5
3 6
1 3
5 7
6 7
2 6
5 8];

G = graph(node(:,1), node(:,2), []);
cycles = cyclebasis(G)

figure(1); plot(G,’Layout’,’force’);

node2 = [
1 2
2 3
3 4
4 1
1 5
5 6
2 4
6 7
3 6
2 5
3 7
4 7
2 6
6 8];

G2 = graph(node2(:,1), node2(:,2), []);
cycles2 = cyclebasis(G2)

figure(2); plot(G2,’Layout’,’force’);

Result:
cycles =
7×1 cell array
{[1 3 6 5]}
{[ 1 4 5]}
{[ 1 5 6]}
{[ 2 3 6]}
{[2 6 5 7]}
{[3 4 5 6]}
{[ 5 6 7]}

cycles2 =
7×1 cell array
{[ 1 2 4]}
{[ 1 2 5]}
{[ 2 3 4]}
{[ 2 3 6]}
{[2 3 7 6]}
{[2 4 7 6]}
{[ 2 5 6]}

Cart

Cart