Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/tree/recognition.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """ |
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2 Recognition Tests |
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3 ================= |
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4 |
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5 A *forest* is an acyclic, undirected graph, and a *tree* is a connected forest. |
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6 Depending on the subfield, there are various conventions for generalizing these |
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7 definitions to directed graphs. |
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8 |
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9 In one convention, directed variants of forest and tree are defined in an |
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10 identical manner, except that the direction of the edges is ignored. In effect, |
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11 each directed edge is treated as a single undirected edge. Then, additional |
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12 restrictions are imposed to define *branchings* and *arborescences*. |
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13 |
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14 In another convention, directed variants of forest and tree correspond to |
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15 the previous convention's branchings and arborescences, respectively. Then two |
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16 new terms, *polyforest* and *polytree*, are defined to correspond to the other |
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17 convention's forest and tree. |
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18 |
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19 Summarizing:: |
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20 |
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21 +-----------------------------+ |
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22 | Convention A | Convention B | |
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23 +=============================+ |
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24 | forest | polyforest | |
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25 | tree | polytree | |
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26 | branching | forest | |
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27 | arborescence | tree | |
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28 +-----------------------------+ |
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29 |
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30 Each convention has its reasons. The first convention emphasizes definitional |
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31 similarity in that directed forests and trees are only concerned with |
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32 acyclicity and do not have an in-degree constraint, just as their undirected |
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33 counterparts do not. The second convention emphasizes functional similarity |
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34 in the sense that the directed analog of a spanning tree is a spanning |
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35 arborescence. That is, take any spanning tree and choose one node as the root. |
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36 Then every edge is assigned a direction such there is a directed path from the |
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37 root to every other node. The result is a spanning arborescence. |
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38 |
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39 NetworkX follows convention "A". Explicitly, these are: |
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40 |
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41 undirected forest |
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42 An undirected graph with no undirected cycles. |
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43 |
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44 undirected tree |
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45 A connected, undirected forest. |
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46 |
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47 directed forest |
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48 A directed graph with no undirected cycles. Equivalently, the underlying |
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49 graph structure (which ignores edge orientations) is an undirected forest. |
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50 In convention B, this is known as a polyforest. |
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51 |
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52 directed tree |
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53 A weakly connected, directed forest. Equivalently, the underlying graph |
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54 structure (which ignores edge orientations) is an undirected tree. In |
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55 convention B, this is known as a polytree. |
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56 |
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57 branching |
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58 A directed forest with each node having, at most, one parent. So the maximum |
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59 in-degree is equal to 1. In convention B, this is known as a forest. |
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60 |
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61 arborescence |
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62 A directed tree with each node having, at most, one parent. So the maximum |
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63 in-degree is equal to 1. In convention B, this is known as a tree. |
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64 |
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65 For trees and arborescences, the adjective "spanning" may be added to designate |
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66 that the graph, when considered as a forest/branching, consists of a single |
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67 tree/arborescence that includes all nodes in the graph. It is true, by |
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68 definition, that every tree/arborescence is spanning with respect to the nodes |
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69 that define the tree/arborescence and so, it might seem redundant to introduce |
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70 the notion of "spanning". However, the nodes may represent a subset of |
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71 nodes from a larger graph, and it is in this context that the term "spanning" |
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72 becomes a useful notion. |
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73 |
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74 """ |
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75 |
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76 import networkx as nx |
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77 |
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78 |
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79 __all__ = ["is_arborescence", "is_branching", "is_forest", "is_tree"] |
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80 |
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81 |
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82 @nx.utils.not_implemented_for("undirected") |
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83 def is_arborescence(G): |
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84 """ |
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85 Returns True if `G` is an arborescence. |
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86 |
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87 An arborescence is a directed tree with maximum in-degree equal to 1. |
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88 |
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89 Parameters |
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90 ---------- |
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91 G : graph |
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92 The graph to test. |
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93 |
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94 Returns |
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95 ------- |
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96 b : bool |
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97 A boolean that is True if `G` is an arborescence. |
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98 |
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99 Notes |
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100 ----- |
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101 In another convention, an arborescence is known as a *tree*. |
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102 |
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103 See Also |
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104 -------- |
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105 is_tree |
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106 |
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107 """ |
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108 return is_tree(G) and max(d for n, d in G.in_degree()) <= 1 |
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109 |
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110 |
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111 @nx.utils.not_implemented_for("undirected") |
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112 def is_branching(G): |
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113 """ |
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114 Returns True if `G` is a branching. |
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115 |
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116 A branching is a directed forest with maximum in-degree equal to 1. |
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117 |
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118 Parameters |
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119 ---------- |
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120 G : directed graph |
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121 The directed graph to test. |
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122 |
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123 Returns |
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124 ------- |
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125 b : bool |
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126 A boolean that is True if `G` is a branching. |
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127 |
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128 Notes |
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129 ----- |
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130 In another convention, a branching is also known as a *forest*. |
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131 |
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132 See Also |
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133 -------- |
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134 is_forest |
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135 |
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136 """ |
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137 return is_forest(G) and max(d for n, d in G.in_degree()) <= 1 |
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138 |
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139 |
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140 def is_forest(G): |
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141 """ |
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142 Returns True if `G` is a forest. |
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143 |
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144 A forest is a graph with no undirected cycles. |
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145 |
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146 For directed graphs, `G` is a forest if the underlying graph is a forest. |
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147 The underlying graph is obtained by treating each directed edge as a single |
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148 undirected edge in a multigraph. |
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149 |
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150 Parameters |
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151 ---------- |
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152 G : graph |
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153 The graph to test. |
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154 |
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155 Returns |
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156 ------- |
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157 b : bool |
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158 A boolean that is True if `G` is a forest. |
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159 |
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160 Notes |
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161 ----- |
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162 In another convention, a directed forest is known as a *polyforest* and |
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163 then *forest* corresponds to a *branching*. |
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164 |
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165 See Also |
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166 -------- |
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167 is_branching |
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168 |
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169 """ |
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170 if len(G) == 0: |
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171 raise nx.exception.NetworkXPointlessConcept("G has no nodes.") |
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172 |
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173 if G.is_directed(): |
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174 components = (G.subgraph(c) for c in nx.weakly_connected_components(G)) |
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175 else: |
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176 components = (G.subgraph(c) for c in nx.connected_components(G)) |
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177 |
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178 return all(len(c) - 1 == c.number_of_edges() for c in components) |
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179 |
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180 |
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181 def is_tree(G): |
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182 """ |
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183 Returns True if `G` is a tree. |
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184 |
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185 A tree is a connected graph with no undirected cycles. |
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186 |
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187 For directed graphs, `G` is a tree if the underlying graph is a tree. The |
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188 underlying graph is obtained by treating each directed edge as a single |
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189 undirected edge in a multigraph. |
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190 |
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191 Parameters |
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192 ---------- |
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193 G : graph |
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194 The graph to test. |
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195 |
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196 Returns |
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197 ------- |
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198 b : bool |
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199 A boolean that is True if `G` is a tree. |
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200 |
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201 Notes |
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202 ----- |
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203 In another convention, a directed tree is known as a *polytree* and then |
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204 *tree* corresponds to an *arborescence*. |
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205 |
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206 See Also |
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207 -------- |
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208 is_arborescence |
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209 |
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210 """ |
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211 if len(G) == 0: |
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212 raise nx.exception.NetworkXPointlessConcept("G has no nodes.") |
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213 |
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214 if G.is_directed(): |
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215 is_connected = nx.is_weakly_connected |
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216 else: |
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217 is_connected = nx.is_connected |
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218 |
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219 # A connected graph with no cycles has n-1 edges. |
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220 return len(G) - 1 == G.number_of_edges() and is_connected(G) |