I have an undirected graph and I need to reorder its vertices into a permutation that satisfies the following “prefix-neighbors” property:

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 [1, 2, 0, 3]:

• 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!

Suppose, I have given N distinct pairs(x, y) of elements. Choose k pairs from these elements every time. Count the chosen ways when number of distinct value is equal to R in the chosen pairs.

Constraints: N = 100 K >= 50 k <= R <= 2K.

Example: Given, N = 6, k = 3, r = 4.

Pairs: a. (1, 2) b. (1, 3) c. (2, 4) d. (2, 5) e. (3, 4) f. (4, 5)

Now if we choose pairs (a, b, e), the distinct values are:- S = {1, 2, 3, 4} Number of Distinct values |S| = 4 and |S| == R. So, this is a valid combination.

If we choose pairs (c, d, e}, the distinct values are:- S = {2, 3, 4, 5} Number of Distinct values |S| = 4 and |S| == R. So, this is a valid combination.

But, if we choose pairs (a, b, f), the distinct values are:- S = {1, 2, 3, 4, 5} Number of Distinct values |S| = 5 and |S| > R. So, this is not a valid combination.

In this example, there is more valid and invalid combinations. I have to count the valid combination for bigger datasets. Is there any combination formula or Dynamic Programming solution for this problem. An algorithm would be very helpful.