
# using dfs for finding eulerian path traversal






def 	UpperCamelCase   (							lowerCAmelCase__			,      lowerCAmelCase__			,      lowerCAmelCase__			,      lowerCAmelCase__=None		):
					'''simple docstring'''
					lowercase						     =						(path or []) + [u]
					for v in graph[u]:
										if visited_edge[u][v] is False:
															lowercase     ,						lowercase						     =						True, True
															lowercase						     =						dfs(lowerCAmelCase__			,      lowerCAmelCase__			,      lowerCAmelCase__			,      lowerCAmelCase__		)
					return path



def 	UpperCamelCase   (							lowerCAmelCase__			,      lowerCAmelCase__		):
					'''simple docstring'''
					lowercase						     =						0
					lowercase						     =						-1
					for i in range(lowerCAmelCase__		):
										if i not in graph.keys():
															continue
										if len(graph[i]		) % 2 == 1:
															odd_degree_nodes += 1
															lowercase						     =						i
					if odd_degree_nodes == 0:
										return 1, odd_node
					if odd_degree_nodes == 2:
										return 2, odd_node
					return 3, odd_node



def 	UpperCamelCase   (							lowerCAmelCase__			,      lowerCAmelCase__		):
					'''simple docstring'''
					lowercase						     =						[[False for _ in range(max_node + 1		)] for _ in range(max_node + 1		)]
					lowercase     ,						lowercase						     =						check_circuit_or_path(lowerCAmelCase__			,      lowerCAmelCase__		)
					if check == 3:
										print('''graph is not Eulerian'''		)
										print('''no path'''		)
										return
					lowercase						     =						1
					if check == 2:
										lowercase						     =						odd_node
										print('''graph has a Euler path'''		)
					if check == 1:
										print('''graph has a Euler cycle'''		)
					lowercase						     =						dfs(lowerCAmelCase__			,      lowerCAmelCase__			,      lowerCAmelCase__		)
					print(lowerCAmelCase__		)



def 	UpperCamelCase   (							):
					'''simple docstring'''
					lowercase						     =						{1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
					lowercase						     =						{1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
					lowercase						     =						{1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
					lowercase						     =						{1: [2, 3], 2: [1, 3], 3: [1, 2]}
					lowercase						     =						{
					    1: [],
					    2: []
					    # all degree is zero
					}
					lowercase						     =						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 decimal import Decimal

from numpy import array



def 	UpperCamelCase   (							lowerCAmelCase__		):
					'''simple docstring'''
					lowercase						     =						Decimal

					# Check if the provided matrix has 2 rows and 2 columns
					# since this implementation only works for 2x2 matrices
					if len(lowerCAmelCase__		) == 2 and len(matrix[0]		) == 2 and len(matrix[1]		) == 2:
										# Calculate the determinant of the matrix
										lowercase						     =						float(
										    d(matrix[0][0]		) * d(matrix[1][1]		) - d(matrix[1][0]		) * d(matrix[0][1]		)		)
										if determinant == 0:
															raise ValueError('''This matrix has no inverse.'''		)

										# Creates a copy of the matrix with swapped positions of the elements
										lowercase						     =						[[0.0, 0.0], [0.0, 0.0]]
										lowercase     ,						lowercase						     =						matrix[1][1], matrix[0][0]
										lowercase     ,						lowercase						     =						-matrix[1][0], -matrix[0][1]

										# Calculate the inverse of the matrix
										return [
										    [(float(d(lowerCAmelCase__		)		) / determinant) or 0.0 for n in row] for row in swapped_matrix
										]
					elif (
					    len(lowerCAmelCase__		) == 3
					    and len(matrix[0]		) == 3
					    and len(matrix[1]		) == 3
					    and len(matrix[2]		) == 3
					):
										# Calculate the determinant of the matrix using Sarrus rule
										lowercase						     =						float(
										    (
										        (d(matrix[0][0]		) * d(matrix[1][1]		) * d(matrix[2][2]		))
										        + (d(matrix[0][1]		) * d(matrix[1][2]		) * d(matrix[2][0]		))
										        + (d(matrix[0][2]		) * d(matrix[1][0]		) * d(matrix[2][1]		))
										    )
										    - (
										        (d(matrix[0][2]		) * d(matrix[1][1]		) * d(matrix[2][0]		))
										        + (d(matrix[0][1]		) * d(matrix[1][0]		) * d(matrix[2][2]		))
										        + (d(matrix[0][0]		) * d(matrix[1][2]		) * d(matrix[2][1]		))
										    )		)
										if determinant == 0:
															raise ValueError('''This matrix has no inverse.'''		)

										# Creating cofactor matrix
										lowercase						     =						[
										    [d(0.0		), d(0.0		), d(0.0		)],
										    [d(0.0		), d(0.0		), d(0.0		)],
										    [d(0.0		), d(0.0		), d(0.0		)],
										]
										lowercase						     =						(d(matrix[1][1]		) * d(matrix[2][2]		)) - (
										    d(matrix[1][2]		) * d(matrix[2][1]		)
										)
										lowercase						     =						-(
										    (d(matrix[1][0]		) * d(matrix[2][2]		)) - (d(matrix[1][2]		) * d(matrix[2][0]		))
										)
										lowercase						     =						(d(matrix[1][0]		) * d(matrix[2][1]		)) - (
										    d(matrix[1][1]		) * d(matrix[2][0]		)
										)
										lowercase						     =						-(
										    (d(matrix[0][1]		) * d(matrix[2][2]		)) - (d(matrix[0][2]		) * d(matrix[2][1]		))
										)
										lowercase						     =						(d(matrix[0][0]		) * d(matrix[2][2]		)) - (
										    d(matrix[0][2]		) * d(matrix[2][0]		)
										)
										lowercase						     =						-(
										    (d(matrix[0][0]		) * d(matrix[2][1]		)) - (d(matrix[0][1]		) * d(matrix[2][0]		))
										)
										lowercase						     =						(d(matrix[0][1]		) * d(matrix[1][2]		)) - (
										    d(matrix[0][2]		) * d(matrix[1][1]		)
										)
										lowercase						     =						-(
										    (d(matrix[0][0]		) * d(matrix[1][2]		)) - (d(matrix[0][2]		) * d(matrix[1][0]		))
										)
										lowercase						     =						(d(matrix[0][0]		) * d(matrix[1][1]		)) - (
										    d(matrix[0][1]		) * d(matrix[1][0]		)
										)

										# Transpose the cofactor matrix (Adjoint matrix)
										lowercase						     =						array(lowerCAmelCase__		)
										for i in range(3		):
															for j in range(3		):
																				lowercase						     =						cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
										lowercase						     =						array(lowerCAmelCase__		)
										for i in range(3		):
															for j in range(3		):
																				inverse_matrix[i][j] /= d(lowerCAmelCase__		)

        # Calculate the inverse of the matrix
										return [[float(d(lowerCAmelCase__		)		) or 0.0 for n in row] for row in inverse_matrix]
					raise ValueError('''Please provide a matrix of size 2x2 or 3x3.'''		)




