





# using dfs for finding eulerian path traversal





def 		_SCREAMING_SNAKE_CASE     ( lowerCAmelCase__							,lowerCAmelCase__							,lowerCAmelCase__							,lowerCAmelCase__=None						):
		lowerCamelCase_	:     Union[str, Any] 						=      (path or []) + [u]
		for v in graph[u]:
				if visited_edge[u][v] is False:
						lowerCamelCase_						,      lowerCamelCase_	:     Tuple 						=      True, True
						lowerCamelCase_	:     Any 						=      dfs(__lowercase							,__lowercase							,__lowercase							,__lowercase						)
		return path


def 		_SCREAMING_SNAKE_CASE     ( lowerCAmelCase__							,lowerCAmelCase__						):
		lowerCamelCase_	:     Tuple 						=      0
		lowerCamelCase_	:     List[Any] 						=      -1
		for i in range(__lowercase						):
				if i not in graph.keys():
						continue
				if len(graph[i]						) % 2 == 1:
						odd_degree_nodes += 1
						lowerCamelCase_	:     Tuple 						=      i
		if odd_degree_nodes == 0:
				return 1, odd_node
		if odd_degree_nodes == 2:
				return 2, odd_node
		return 3, odd_node


def 		_SCREAMING_SNAKE_CASE     ( lowerCAmelCase__							,lowerCAmelCase__						):
		lowerCamelCase_	:     Dict 						=      [[False for _ in range(max_node + 1						)] for _ in range(max_node + 1						)]
		lowerCamelCase_						,      lowerCamelCase_	:     str 						=      check_circuit_or_path(__lowercase							,__lowercase						)
		if check == 3:
				print('graph is not Eulerian'						)
				print('no path'						)
				return
		lowerCamelCase_	:     str 						=      1
		if check == 2:
				lowerCamelCase_	:     Optional[Any] 						=      odd_node
				print('graph has a Euler path'						)
		if check == 1:
				print('graph has a Euler cycle'						)
		lowerCamelCase_	:     Optional[int] 						=      dfs(__lowercase							,__lowercase							,__lowercase						)
		print(__lowercase						)


def 		_SCREAMING_SNAKE_CASE     ( ):
		lowerCamelCase_	:     Tuple 						=      {1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
		lowerCamelCase_	:     Tuple 						=      {1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
		lowerCamelCase_	:     Any 						=      {1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
		lowerCamelCase_	:     Optional[int] 						=      {1: [2, 3], 2: [1, 3], 3: [1, 2]}
		lowerCamelCase_	:     List[Any] 						=      {
		    1: [],
		    2: []
		    # all degree is zero
		}
		lowerCamelCase_	:     Union[str, Any] 						=      10
		check_euler(__lowercase							,__lowercase						)
		check_euler(__lowercase							,__lowercase						)
		check_euler(__lowercase							,__lowercase						)
		check_euler(__lowercase							,__lowercase						)
		check_euler(__lowercase							,__lowercase						)


if __name__ == "__main__":
		main()








from __future__ import annotations

from typing import Generic, TypeVar

__a   :							str						= TypeVar("T")


class    __lowercase   (						Generic[T]	):







     '''simple docstring'''




     def __init__(					self							:      Any			,						UpperCamelCase_							:      T					):


         """simple docstring"""



         __A       =						data
         __A       =						self
         __A       =						0


class    __lowercase   (						Generic[T]	):







     '''simple docstring'''




     def __init__(					self							:      Optional[Any]					):


         """simple docstring"""



         __A       =						{}




     def 		lowerCAmelCase_  (					self							:      Union[str, Any]			,						UpperCamelCase_							:      T					):


         """simple docstring"""



         __A       =						DisjointSetTreeNode(UpperCamelCase_					)




     def 		lowerCAmelCase_  (					self							:      List[Any]			,						UpperCamelCase_							:      T					):


         """simple docstring"""



         __A       =						self.map[data]
         if elem_ref != elem_ref.parent:
             __A       =						self.find_set(elem_ref.parent.data					)
         return elem_ref.parent




     def 		lowerCAmelCase_  (					self							:      int			,						UpperCamelCase_							:      DisjointSetTreeNode[T]			,						UpperCamelCase_							:      DisjointSetTreeNode[T]					):


         """simple docstring"""



         if nodea.rank > nodea.rank:
             __A       =						nodea
         else:
             __A       =						nodea
             if nodea.rank == nodea.rank:
                 nodea.rank += 1





     def 		lowerCAmelCase_  (					self							:      Tuple			,						UpperCamelCase_							:      T			,						UpperCamelCase_							:      T					):


         """simple docstring"""



         self.link(self.find_set(UpperCamelCase_					)			,						self.find_set(UpperCamelCase_					)					)


class    __lowercase   (						Generic[T]	):







     '''simple docstring'''




     def __init__(					self							:      Optional[int]					):


         """simple docstring"""



         __A       =						{}




     def 		lowerCAmelCase_  (					self							:      int			,						UpperCamelCase_							:      T					):


         """simple docstring"""



         if node not in self.connections:
             __A       =						{}




     def 		lowerCAmelCase_  (					self							:      Dict			,						UpperCamelCase_							:      T			,						UpperCamelCase_							:      T			,						UpperCamelCase_							:      int					):


         """simple docstring"""



         self.add_node(UpperCamelCase_					)
         self.add_node(UpperCamelCase_					)
         __A       =						weight
         __A       =						weight





     def 		lowerCAmelCase_  (					self							:      Dict					):


         """simple docstring"""



         __A       =						[]
         __A       =						set()
         for start in self.connections:
             for end in self.connections[start]:
                 if (start, end) not in seen:
                     seen.add((end, start)					)
                     edges.append((start, end, self.connections[start][end])					)
         edges.sort(key=lambda UpperCamelCase_					: x[2]					)

         # creating the disjoint set
         __A       =						DisjointSetTree[T]()
         for node in self.connections:
             disjoint_set.make_set(UpperCamelCase_					)

         # MST generation
         __A       =						0
         __A       =						0
         __A       =						GraphUndirectedWeighted[T]()
         while num_edges < len(self.connections					) - 1:
             __A     ,						__A     ,						__A       =						edges[index]
             index += 1
             __A       =						disjoint_set.find_set(UpperCamelCase_					)
             __A       =						disjoint_set.find_set(UpperCamelCase_					)
             if parent_u != parent_v:
                 num_edges += 1
                 graph.add_edge(UpperCamelCase_			,						UpperCamelCase_			,						UpperCamelCase_					)
                 disjoint_set.union(UpperCamelCase_			,						UpperCamelCase_					)
         return graph




