Graph Theory (1) ------ Graph Representation

A graph G = (V, E) is composed of vertex set V and edge set E. Each edge is a vertex pair (v, w), where v, w, and V. If the vertex is ordered, the graph is a directed graph.

A path in the figure is a vertex sequence w1, w2, w3,..., wk, making (wi, wi + 1) ε E, 1 <= I <= k. The path length is the number of edges in the path.

If a path exists from each vertex to other vertex in an undirected graph, the path is connected. Directed Graphs of this nature are strongly connected. Remove the direction from the arc of the directed graph.

If the graph is connected, the directed graph is weakly connected.

Simple two-dimensional array representation of an image

Represent with an adjacent matrix. For each edge (u, v), set A [u] [v] = 1, otherwise it is 0. if the edge has a permission, set the array element to the permission. The required space is bytes (V2 ).

If the graph is dense (with many edges), the adjacent matrix is an appropriate representation. If it is sparse, the better solution is to use the adjacent table.

Graph joining table Representation

For each vertex, use a table to store all adjacent vertices. At this time, the space requirement is O (E + V ).

class Vertex(object):    def __init__(self,key):        self.id=key        self.adj={}    def addNeighbor(self,nbr,weight=0):        self.adj[nbr]=weight    def getNeighbors(self):        return self.adj.keys()    def getId(self):        return self.id    def getWeight(self,key):        return self.adj[key]class Graph(object):    def __init__(self):        self.vertexlist={}        self.size=0    def addVertex(self,key):        vertex=Vertex(key)        self.vertexlist[key]=vertex        self.size+=1        return vertex    def getVertex(self,key):        return self.vertexlist.get(key)    def __contains__(self,key):        if key in self.vertexlist:            return True        else:            return False    def addEdge(self,f,t,weight=0):        if f not in self.vertexlist:            self.addVertex(f)        if t not in self.vertexlist:            self.addVertex(t)        self.vertexlist[f].addNeighbor(self.vertexlist[t],weight)    def getVertices(self):        return self.vertexlist.keys()    def __iter__(self):        return iter(self.vertexlist.values())