



# using dfs for finding eulerian path traversal













def 			lowerCAmelCase__	(					lowerCamelCase_	:	Dict ,lowerCamelCase_	:	List[str] ,lowerCamelCase_	:	int ,lowerCamelCase_	:	List[Any]=None):





				'''simple docstring'''

				lowerCAmelCase__  :       Union[str, Any]				=  (path or []) + [u]
				for v in graph[u]:
								if visited_edge[u][v] is False:
												lowerCAmelCase__  :       Any				=  True, True
												lowerCAmelCase__  :       List[Any]				=  dfs(lowerCAmelCase_ ,lowerCAmelCase_ ,lowerCAmelCase_ ,lowerCAmelCase_)
				return path






def 			lowerCAmelCase__	(					lowerCamelCase_	:	Tuple ,lowerCamelCase_	:	List[Any]):





				'''simple docstring'''

				lowerCAmelCase__  :       Union[str, Any]				=  0
				lowerCAmelCase__  :       Optional[int]				=  -1
				for i in range(lowerCAmelCase_):
								if i not in graph.keys():
												continue
								if len(graph[i]) % 2 == 1:
												odd_degree_nodes += 1
												lowerCAmelCase__  :       Any				=  i
				if odd_degree_nodes == 0:
								return 1, odd_node
				if odd_degree_nodes == 2:
								return 2, odd_node
				return 3, odd_node






def 			lowerCAmelCase__	(					lowerCamelCase_	:	Union[str, Any] ,lowerCamelCase_	:	Optional[Any]):





				'''simple docstring'''

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






def 			lowerCAmelCase__	(					):





				'''simple docstring'''

				lowerCAmelCase__  :       Union[str, Any]				=  {1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
				lowerCAmelCase__  :       str				=  {1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
				lowerCAmelCase__  :       List[Any]				=  {1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
				lowerCAmelCase__  :       str				=  {1: [2, 3], 2: [1, 3], 3: [1, 2]}
				lowerCAmelCase__  :       List[Any]				=  {
				    1: [],
				    2: []
				    # all degree is zero
				}
				lowerCAmelCase__  :       Dict				=  10
				check_euler(lowerCAmelCase_ ,lowerCAmelCase_)
				check_euler(lowerCAmelCase_ ,lowerCAmelCase_)
				check_euler(lowerCAmelCase_ ,lowerCAmelCase_)
				check_euler(lowerCAmelCase_ ,lowerCAmelCase_)
				check_euler(lowerCAmelCase_ ,lowerCAmelCase_)


if __name__ == "__main__":
			main()






from __future__ import annotations

from collections.abc import Iterable, Iterator
from dataclasses import dataclass

__snake_case       :	Optional[int]            =(3, 9, -1_1, 0, 7, 5, 1, -1)
__snake_case       :	str            =(4, 6, 2, 0, 8, 1_0, 3, -2)



@dataclass
class  lowerCamelCase__    :




		'''simple docstring'''



		snake_case_			=42
		snake_case_			=42



class  lowerCamelCase__    :




		'''simple docstring'''



		def __init__(self	,__lowerCamelCase				)  ->    None:

						"""simple docstring"""






						lowerCAmelCase__  :       Node | None				=  None
						for i in sorted(__lowerCamelCase	,reverse=__lowerCamelCase				):
										lowerCAmelCase__  :       Dict				=  Node(__lowerCamelCase	,self.head				)



		def __iter__(self				)  ->    Iterator[int]:

						"""simple docstring"""






						lowerCAmelCase__  :       Optional[Any]				=  self.head
						while node:
										yield node.data
										lowerCAmelCase__  :       Dict				=  node.next_node



		def __len__(self				)  ->    int:

						"""simple docstring"""






						return sum(1 for _ in self				)



		def __str__(self				)  ->    str:

						"""simple docstring"""






						return " -> ".join([str(__lowerCamelCase				) for node in self]				)






def 			lowerCAmelCase__	(					lowerCamelCase_	:	SortedLinkedList ,lowerCamelCase_	:	SortedLinkedList):





				'''simple docstring'''

				return SortedLinkedList(list(lowerCamelCase_) + list(lowerCamelCase_))


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							__snake_case       :	Any            =SortedLinkedList
							print(merge_lists(SSL(test_data_odd), SSL(test_data_even)))

