this algorithm is just big brain, much better impl of what i had in mind and still dont fully understand the "son" thing, but the general principle is cool.

just stare at invariants harder i guess to get better at algo design :/ 

#include <bits/stdc++.h>
using namespace std;
#define rep(i,a,n) for (int i=a;i<n;i++)
#define per(i,a,n) for (int i=n-1;i>=a;i--)
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define all(x) (x).begin(),(x).end()
#define fi first
#define se second
#define SZ(x) ((int)(x).size())
typedef vector<int> VI;
typedef basic_string<int> BI;
typedef long long ll;
typedef pair<int,int> PII;
typedef double db;
mt19937 mrand(random_device{}()); 
const ll mod=1000000007;
int rnd(int x) { return mrand() % x;}
ll powmod(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
ll gcd(ll a,ll b) { return b?gcd(b,a%b):a;}
// head
const int N=101000;
int n,m,comp[N],par[N],sz[N],cnt,ins[N],vis[N];
VI e[N],son[N],ans[N];

void del(int u) {
    assert(comp[u] == 0); 
    assert(vis[u]);
	++cnt;
	function<void(int)> dfs=[&](int u) {
		comp[u]=cnt;
		ans[cnt].pb(u);

        // wth is son array? 
		for (auto v:son[u]) if (comp[v]==0) {
			dfs(v);
		}
	};
	dfs(u);
}

void dfs(int u) {
	ins[u]=1;
	vis[u]=1;
	for (auto v:e[u]) {
		if (!vis[v]) {
			son[u].pb(v);
			par[v]=u;
			dfs(v);
			if (comp[v]==0) sz[u]^=1;
		} else if (!ins[v]&&comp[v]==0) {
            // we've already processed this size in the dfs, 
            // but when removing components size doesn't change
            // unless ur the intermediate nodes on the path,
            // which do change, but we're deleting them all already so :/  

            assert(sz[v] == 1); 
            // if we visited it and it's not in a component, it must be "finalized" to have odd size,
            // and all its parents must be "finalized" to have odd size. 

			comp[v]=-1;
			int w=par[v];
			while (w!=u) {
				del(w);
				w=par[w];
			}
			son[u].pb(v);
			par[v]=u;
			comp[v]=0;
		}
	}
	sz[u]^=1;
	if (sz[u]==0) {
		del(u);
	}
	ins[u]=0;
}

int main() {
	scanf("%d%d",&n,&m);
	rep(i,0,m) {
		int u,v;
		scanf("%d%d",&u,&v);
		e[u].pb(v);
		e[v].pb(u);
	}
	dfs(0);
	printf("%d\n",cnt);
	rep(i,1,cnt+1) {
		printf("%d",SZ(ans[i]));
		for (auto u:ans[i]) printf(" %d",u);
		puts("");
	}
}

answer.code:1:1: error: expected unqualified-id before ‘this’
    1 | this algorithm is just big brain, much better impl of what i had in mind and still dont fully understand the "son" thing, but the general principle is cool.
      | ^~~~
In file included from /usr/include/c++/13/bits/stl_algobase.h:62,
                 from /usr/include/c++/13/algorithm:60,
                 from /usr/include/x86_64-linux-gnu/c++/13/bits/stdc++.h:51,
                 from answer.code:5:
/usr/include/c++/13/ext/type_traits.h:164:35: error: ‘constexpr const bool __gnu_cxx::__is_null_pointer’ redeclared as different kind of entity
  164 |   __is_null_pointer(std::nullptr_t)
      |                                   ^
/usr/include/c++/13/ext/type_traits.h:159:5: note: previous declaration ‘template<class _Type> constexpr bool __gnu_cxx::__is_null_pointer(_Type)’
  159 |     __is_null_pointer(_Type)
      |     ^~~~~~~~~~~~~~~~~
/usr/include/c++/13/ext/type_traits.h:164:26: error: ‘nullptr_t’ is not a member of ‘std’; did you mean ‘nullptr_t’?
  164 |   __is_null_pointer(std::nullptr_t)
      |                          ^~~~~~~~~
In file included from /usr/include/c++/13/cstddef:50,
                 from /usr/include/x86_64-linux-gnu/c++/13/bits/stdc++.h:41:
/usr/lib/gcc/x86_64-linux-gnu/13/include/stddef.h:443:29: note: ‘nullptr_t’ declared here
  443 |   typedef decltype(nullptr) nullptr_t;
      |                             ^~~~~~~~~
In file included from /usr/include/c++/13/bits/stl_pair.h:60,
                 from /usr/include/c++/13/bits/stl_algobase.h:64:
/usr/include/c++/13/type_traits:510:26: error: ‘std::size_t’ has not been declared
  510 |   template<typename _Tp, std::size_t _Size>
      |                          ^~~
/usr/include/c++/13/type_traits:511:25: error: ‘_Size’ was not declared in this scope
  511 |     struct is_array<_Tp[_Size]>
      |                         ^~~~~
/usr/include/c++/13/type_traits:511:31: error: template argument 1 is invalid
  511 |     struct is_array<_Tp[_Size]>
      |                               ^
/usr/include/c++/13/type_traits:617:33: error: ‘nullptr_t’ is not a member of ‘std’; did you mean ‘nullptr_t’?
  617 |     struct is_null_pointer<std::nullptr_t>
      |                                 ^~~~~~~~~
/usr/lib/gcc/x86_64-linux-gnu/13/include/stddef.h:443:29: note: ‘nullptr_t’ declared here
  443 |   typedef decltype(nullptr) nullptr_t;
      |                             ^~~~~~~~~
/usr/include/c++/13/type_traits:617:42: error: template argument 1 is invalid
  617 |     struct is_null_pointer<std::nullptr_t>
      |                                          ^
/usr/include/c++/13/type_traits:621:48: error: template argument 1 is invalid
  621 |     struct is_null_pointer<const std::nullptr_t>
      |                                                ^
/usr/include/c++/13/type_traits:625:51: error: template argument 1 is invalid
  625 |     struct is_null_pointer<volatile std::nullptr_t>
      |                                                   ^
/usr/include/c++/13/type_traits:629:57: error: template argument 1 is invalid
  629 |     struct is_null_pointer<const volatile std::nullptr_t>
      |                                                         ^
/usr/include/c++/13/type_traits:1348:37: error: ‘size_t’ is not a member of ‘std’; did you mean ‘size_t’?
 1348 |     : public integral_constant<std::size_t, alignof(_Tp)>
      |                                     ^~~~~~
/usr/lib/gcc/x86_64-linux-gnu/13/include/stddef.h:214:23: note: ‘size_t’ declared here
  214 | typedef __SIZE_TYPE__ size_t;
      |                       ^~~~~~
/usr/include/c++/13/type_traits:1348:57: error: template argument 1 is invalid
 1348 |     : public integral_constant<std::size_t, alignof(_Tp)>
      |                                                         ^
/usr/include/c++/13/type_traits:1357:37: error: ‘size_t’ is not a member of ‘std’; did you mean ‘size_t’?
 1357 |     : public integral_constant<std::size_t, 0> { };
      |                                     ^~~~~~
/usr/lib/gcc/x86_64-linux-gnu/13/include/stddef.h:214:23: note: ‘size_t’ declared here
  214 | typedef __SIZE_TYPE__ size_t;
      |                       ^~~~~~
/usr/include/c++/13/type_traits:1357:46: error: template argument 1 is invalid
 1357 |     : public integral_constant<std::size_t, 0> { };
      |                                              ^
/usr/include/c++/13/type_traits:1359:26: error: ‘std::size_t’ has not been declared
 1359 |   template<typename _Tp, std::size_t _Size>
      |                          ^~~
/usr/include/c++/13/type_traits:1360:21: error: ‘_Size’ was not declared in this scope
 1360 |     struct rank<_Tp[_Size]>
      |                     ^~~~~
/usr/include/c++/13/type_traits:1360:27: error: template argument 1 is invalid
 1360 |     struct rank<_Tp[_Size]>
      |                           ^
/usr/include/c++/13/type_traits:1361:37: error: ‘size_t’ is not a member of ‘std’; did you mean ‘size_t’?
 1361 |...