I have an undirected graph and I need to reorder its vertices into a permutation that satisfies the following “prefix-neighbors” property:↵
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> Property: For every vertex v_i (with i > 0) in the permutation, if there exists any vertex v_j (with j < i) that is adjacent to v_i (i.e., (v_j, v_i) is an edge in E), then every vertex v_k with k < j must also be adjacent to v_i.↵
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In other words, if has any neighbors among the vertices that come before it in the ordering, then those neighbors must form a contiguous block starting from the very first vertex in the ordering.↵
↵
For example, consider a graph with vertices and edges: (0,1) (0,2) (1,3) (2,3)↵
↵
0↵
/ \↵
1 2↵
\ /↵
3↵
↵
↵
One valid ordering is :↵
↵
- Vertex 1: Placed first, so no condition applies.↵
- Vertex 2: Placed second; if it has any neighbor among vertices before it, then the very first vertex must be adjacent—but here it happens that 2 is not adjacent to 1, so the condition is not triggered.↵
- Vertex 0: Placed third; its neighbors among are 1 and 2. The earliest (lowest-index) neighbor is 1, which is at the beginning of the ordering.↵
- Vertex 3: Placed last; its neighbors among are 1 and 2 (its first neighbor is 1 at index 0), so the condition is satisfied.↵
↵
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My questions are:↵
↵
1. Algorithm & Complexity: Is it possible to compute such a permutation in time?↵
2. Approach:What algorithm would you recommend for this problem?↵
3. Uniqueness:Under what conditions is such an ordering unique, and how can I detect if multiple valid orders exist or if no valid ordering exists?↵
↵
Any algorithm sketches, insights, or sample code in Python or C++ would be greatly appreciated!
↵
> Property: For every vertex v_i (with i > 0) in the permutation, if there exists any vertex v_j (with j < i) that is adjacent to v_i (i.e., (v_j, v_i) is an edge in E), then every vertex v_k with k < j must also be adjacent to v_i.↵
↵
In other words, if has any neighbors among the vertices that come before it in the ordering, then those neighbors must form a contiguous block starting from the very first vertex in the ordering.↵
↵
For example, consider a graph with vertices and edges: (0,1) (0,2) (1,3) (2,3)↵
↵
0↵
/ \↵
1 2↵
\ /↵
3↵
↵
↵
One valid ordering is :↵
↵
- Vertex 1: Placed first, so no condition applies.↵
- Vertex 2: Placed second; if it has any neighbor among vertices before it, then the very first vertex must be adjacent—but here it happens that 2 is not adjacent to 1, so the condition is not triggered.↵
- Vertex 0: Placed third; its neighbors among are 1 and 2. The earliest (lowest-index) neighbor is 1, which is at the beginning of the ordering.↵
- Vertex 3: Placed last; its neighbors among are 1 and 2 (its first neighbor is 1 at index 0), so the condition is satisfied.↵
↵
↵
My questions are:↵
↵
1. Algorithm & Complexity: Is it possible to compute such a permutation in time?↵
2. Approach:What algorithm would you recommend for this problem?↵
3. Uniqueness:Under what conditions is such an ordering unique, and how can I detect if multiple valid orders exist or if no valid ordering exists?↵
↵
Any algorithm sketches, insights, or sample code in Python or C++ would be greatly appreciated!
