""" A Python Class A simple Python graph class, demonstrating the essential facts and functionalities of graphs. """ class Graph(object): def __init__(self, graph_dict={}): """ initializes a graph object """ self.__graph_dict = graph_dict def vertices(self): """ returns the vertices of a graph """ return list(self.__graph_dict.keys()) def edges(self): """ returns the edges of a graph """ return self.__generate_edges() def add_vertex(self, vertex): """ If the vertex "vertex" is not in self.__graph_dict, a key "vertex" with an empty list as a value is added to the dictionary. Otherwise nothing has to be done. """ if vertex not in self.__graph_dict: self.__graph_dict[vertex] = [] def add_edge(self, edge): """ assumes that edge is of type set, tuple or list; between two vertices can be multiple edges! """ edge = set(edge) vertex1 = edge.pop() if edge: # not a loop vertex2 = edge.pop() else: # a loop vertex2 = vertex1 if vertex1 in self.__graph_dict: self.__graph_dict[vertex1].append(vertex2) else: self.__graph_dict[vertex1] = [vertex2] def __generate_edges(self): """ A static method generating the edges of the graph "graph". Edges are represented as sets with one (a loop back to the vertex) or two vertices """ edges = [] for vertex in self.__graph_dict: for neighbour in self.__graph_dict[vertex]: if {neighbour, vertex} not in edges: edges.append({vertex, neighbour}) return edges def __str__(self): res = "vertices: " for k in self.__graph_dict: res += str(k) + " " res += "\nedges: " for edge in self.__generate_edges(): res += str(edge) + " " return res def find_isolated_vertices(self): """ returns a list of isolated vertices. """ graph = self.__graph_dict isolated = [] for vertex in graph: print(isolated, vertex) if not graph[vertex]: isolated += [vertex] return isolated def find_path(self, start_vertex, end_vertex, path=[]): """ find a path from start_vertex to end_vertex in graph """ graph = self.__graph_dict path = path + [start_vertex] if start_vertex == end_vertex: return path if start_vertex not in graph: return None for vertex in graph[start_vertex]: if vertex not in path: extended_path = self.find_path(vertex, end_vertex, path) if extended_path: return extended_path return None def find_all_paths(self, start_vertex, end_vertex, path=[]): """ find all paths from start_vertex to end_vertex in graph """ graph = self.__graph_dict path = path + [start_vertex] if start_vertex == end_vertex: return [path] if start_vertex not in graph: return [] paths = [] for vertex in graph[start_vertex]: if vertex not in path: extended_paths = self.find_all_paths(vertex, end_vertex, path) for p in extended_paths: paths.append(p) return paths def is_connected(self, vertices_encountered = set(), start_vertex=None): """ determines if the graph is connected """ gdict = self.__graph_dict vertices = gdict.keys() if not start_vertex: # chosse a vertex from graph as a starting point start_vertex = vertices[0] vertices_encountered.add(start_vertex) if len(vertices_encountered) != len(vertices): for vertex in gdict[start_vertex]: if vertex not in vertices_encountered: if self.is_connected(vertices_encountered, vertex): return True else: return True return False def vertex_degree(self, vertex): """ The degree of a vertex is the number of edges connecting it, i.e. the number of adjacent vertices. Loops are counted double, i.e. every occurence of vertex in the list of adjacent vertices. """ adj_vertices = self.__graph_dict[vertex] degree = len(adj_vertices) + adj_vertices.count(vertex) return degree def degree_sequence(self): """ calculates the degree sequence """ seq = [] for vertex in self.__graph_dict: seq.append(self.vertex_degree(vertex)) seq.sort(reverse=True) return tuple(seq) @staticmethod def is_degree_sequence(sequence): """ Method returns True, if the sequence "sequence" is a degree sequence, i.e. a non-increasing sequence. Otherwise False is returned. """ # check if the sequence sequence is non-increasing: return all( x>=y for x, y in zip(sequence, sequence[1:])) def delta(self): """ the minimum degree of the vertices """ min = 100000000 for vertex in self.__graph_dict: vertex_degree = self.vertex_degree(vertex) if vertex_degree < min: min = vertex_degree return min def Delta(self): """ the maximum degree of the vertices """ max = 0 for vertex in self.__graph_dict: vertex_degree = self.vertex_degree(vertex) if vertex_degree > max: max = vertex_degree return max def density(self): """ method to calculate the density of a graph """ g = self.__graph_dict V = len(g.keys()) E = len(self.edges()) return 2.0 * E / (V *(V - 1)) def diameter(self): """ calculates the diameter of the graph """ v = self.vertices() pairs = [ (v[i],v[j]) for i in range(len(v)-1) for j in range(i+1, len(v))] smallest_paths = [] for (s,e) in pairs: paths = self.find_all_paths(s,e) smallest = sorted(paths, key=len)[0] smallest_paths.append(smallest) smallest_paths.sort(key=len) # longest path is at the end of list, # i.e. diameter corresponds to the length of this path diameter = len(smallest_paths[-1]) return diameter @staticmethod def erdoes_gallai(dsequence): """ Checks if the condition of the Erdoes-Gallai inequality is fullfilled """ if sum(dsequence) % 2: # sum of sequence is odd return False if Graph.is_degree_sequence(dsequence): for k in range(1,len(dsequence) + 1): left = sum(dsequence[:k]) right = k * (k-1) + sum([min(x,k) for x in dsequence[k:]]) if left > right: return False else: # sequence is increasing return False return True if __name__ == "__main__": g = { "a" : ["d"], "b" : ["c"], "c" : ["b", "c", "d", "e"], "d" : ["a", "c"], "e" : ["c"], "f" : [] } graph = Graph(g) print(graph) for node in graph.vertices(): print(graph.vertex_degree(node)) print("List of isolated vertices:") print(graph.find_isolated_vertices()) print("""A path from "a" to "e":""") print(graph.find_path("a", "e")) print("""All pathes from "a" to "e":""") print(graph.find_all_paths("a", "e")) print("The maximum degree of the graph is:") print(graph.Delta()) print("The minimum degree of the graph is:") print(graph.delta()) print("Edges:") print(graph.edges()) print("Degree Sequence: ") ds = graph.degree_sequence() print(ds) fullfilling = [ [2, 2, 2, 2, 1, 1], [3, 3, 3, 3, 3, 3], [3, 3, 2, 1, 1] ] non_fullfilling = [ [4, 3, 2, 2, 2, 1, 1], [6, 6, 5, 4, 4, 2, 1], [3, 3, 3, 1] ] for sequence in fullfilling + non_fullfilling : print(sequence, Graph.erdoes_gallai(sequence)) print("Add vertex 'z':") graph.add_vertex("z") print(graph) print("Add edge ('x','y'): ") graph.add_edge(('x', 'y')) print(graph) print("Add edge ('a','d'): ") graph.add_edge(('a', 'd')) print(graph)