Odd Degree Subgraph

In the Cow Republic, there are N towns and M undirected paths between them. The invaders are coming, and the cows want to hinder them by shutting down some of the paths. Your task is to choose a subset of the paths to keep open so that every town is connected to an odd number of open paths. If no such subset exists, output NO.

Note that the towns are numbered from 1 to N and the paths are given as pairs A_i and B_i (with 1 ≤ A_i, B_i ≤ N, A_i ≠ B_i). There are no duplicate paths. The republic might be disconnected.

A key observation is that since every kept path contributes 1 to the degree of two towns, the sum of all degrees in the kept subgraph equals 2×(number of kept paths) and must be even. Hence a necessary (and in fact sufficient) condition is that each connected component must contain an even number of towns.

If a solution exists, you should print YES followed by the number of kept paths K and then the indices (1-indexed, referring to the input order) of the kept paths (in any order) that yield the required odd degrees at every town. Otherwise, print NO.

It can be shown that a solution (if one exists) can be obtained by finding a spanning forest and then choosing a certain subset of the tree edges. One way to do this is to assign each town a deficit of 1 (meaning it needs an odd number of incident edges), and then perform a DFS in each connected component. For each tree edge from parent to child, if the child’s deficit is 1 then select that edge and flip the deficit of the parent. This yields a valid set of kept paths.

All formulas in this statement are rendered in ( \LaTeX ) (for example, the requirement for each town is ( \text{deg}(v) \equiv 1 \pmod{2} )).

inputFormat

The first line contains two integers N and M (1 ≤ N ≤ 50,000, 1 ≤ M ≤ 100,000).

Each of the following M lines contains two integers A_i and B_i describing an undirected path.

outputFormat

If a subset exists such that every town has an odd number of incident paths, print YES on the first line.

On the next line, print an integer K denoting the number of kept paths, and on the following line print K space‐separated integers corresponding to the indices (1-indexed) of the kept paths in any order.

If no such subset exists, output NO.

sample

4 4
1 2
1 3
2 3
3 4

YES
2
1 4