给定一个由顶点数V和边E[ ]定义的无向图，任务是在无向图中找到最大独立顶点集。
Independent Set: An independent set in a graph is a set of vertices which are not directly connected to each other.
注意：假设至少有一种方法可以从图中的任何顶点遍历到另一个顶点，即该图具有一个连通分量。
例子：
Input: V = 3, E = { (1, 2), (2, 3) }
Output: {1, 3}
Explanation:
Since there are no edges between 1 and 3, and we cannot add 2 to this since it is a neighbour of 1, this is the Maximal Independent Set.
Input: V = 8,
E = { (1, 2), (1, 3), (2, 4), (5, 6), (6, 7), (4, 8) }
Output: {2, 3, 5, 7, 8}
方法：
这个问题是一个 NP-Hard 问题，只能在指数时间内解决（截至目前）。
请按照以下步骤解决问题：
- 遍历图的顶点并使用回溯来检查顶点是否可以包含在最大独立集中。
- 每个顶点有两种可能性，是否可以包含在最大独立集中。
- 最初开始，考虑所有顶点和边。一个接一个，选择一个顶点。从图中移除该顶点，将其从最大独立集 中排除，并递归遍历剩余的图以找到最大独立集。
- 否则，考虑最大独立集中的选定顶点并从中删除其所有邻居。继续寻找排除其邻居的最大独立集。
- 对所有顶点重复此过程并打印获得的最大独立集。
下面是上述方法的实现：
Python3
# Python Program to implement
# the above approach
# Recursive Function to find the
# Maximal Independent Vertex Set
def graphSets(graph):
# Base Case - Given Graph
# has no nodes
if(len(graph) == 0):
return []
# Base Case - Given Graph
# has 1 node
if(len(graph) == 1):
return [list(graph.keys())[0]]
# Select a vertex from the graph
vCurrent = list(graph.keys())[0]
# Case 1 - Proceed removing
# the selected vertex
# from the Maximal Set
graph2 = dict(graph)
# Delete current vertex
# from the Graph
del graph2[vCurrent]
# Recursive call - Gets
# Maximal Set,
# assuming current Vertex
# not selected
res1 = graphSets(graph2)
# Case 2 - Proceed considering
# the selected vertex as part
# of the Maximal Set
# Loop through its neighbours
for v in graph[vCurrent]:
# Delete neighbor from
# the current subgraph
if(v in graph2):
del graph2[v]
# This result set contains VFirst,
# and the result of recursive
# call assuming neighbors of vFirst
# are not selected
res2 = [vCurrent] + graphSets(graph2)
# Our final result is the one
# which is bigger, return it
if(len(res1) > len(res2)):
return res1
return res2
# Driver Code
V = 8
# Defines edges
E = [ (1, 2),
(1, 3),
(2, 4),
(5, 6),
(6, 7),
(4, 8)]
graph = dict([])
# Constructs Graph as a dictionary
# of the following format-
# graph[VertexNumber V]
# = list[Neighbors of Vertex V]
for i in range(len(E)):
v1, v2 = E[i]
if(v1 not in graph):
graph[v1] = []
if(v2 not in graph):
graph[v2] = []
graph[v1].append(v2)
graph[v2].append(v1)
# Recursive call considering
# all vertices in the maximum
# independent set
maximalIndependentSet = graphSets(graph)
# Prints the Result
for i in maximalIndependentSet:
print(i, end =" ")输出：
2 3 8 5 7
时间复杂度： O(2 N )
辅助空间： O(N)
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