Components - Family of functions¶

The problem definition¶

Connected components

A connected component of an undirected graph is a set of vertices that are all reachable from each other.

Notice: This problem defines on an undirected graph.

Given the following query:

pgr_connectedComponentsV(\(sql\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

• \(source = \bigcup source_i\),
• \(target = \bigcup target_i\),

The graphs are defined as follows:

The weighted undirected graph, \(G(V,E)\), is definied by:

• the set of vertices \(V\)
  • \(V = source \cup target\)
• the set of edges \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \text{ } \\ \cup \{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Given:

• \(G(V,E)\)

Then:

\(\boldsymbol{\pi} = \{(component_i, n\_seq_i, node_i)\}\)

where:

• \(component_i = \min \{node_j | node_j \in component_i\}\)
• \(n\_seq_i\) is a sequential value starting from 1 in a component.
• \(node_i \in component_i\)
• The returned values are ordered:
  • component ascending
  • node ascending

Example:

• The first component is composed of nodes 0, 1 and 4.
• The second component is composed of node 3.
• The third component is composed of nodes 2 and 5.

Strongly connected components

A strongly connected component of a directed graph is a set of vertices that are all reachable from each other.

Notice: This problem defines on a directed graph.

Given the following query:

pgr_strongComponentsV(\(sql\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

• \(source = \bigcup source_i\),
• \(target = \bigcup target_i\),

The graphs are defined as follows:

The weighted directed graph, \(G_d(V,E)\), is definied by:

• the set of vertices \(V\)
  • \(V = source \cup target \cup {start_{vid}} \cup {end_{vid}}\)
• the set of edges \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Given:

• \(G(V,E)\)

Then:

\(\boldsymbol{\pi} = \{(component_i, n\_seq_i, node_i)\}\)

where:

• \(component_i = \min {node_j | node_j \in component_i}\)
• \(n\_seq_i\) is a sequential value starting from 1 in a component.
• \(node_i \in component_i\)
• The returned values are ordered:
  • component ascending
  • node ascending

Example:

• The first component is composed of nodes 1, 2 and 4.
• The second component is composed of node 0.
• The third component is composed of node 3.
• The fourth component is composed of node 5.
• The fifth component is composed of node 6.

Biconnected components

The biconnected components of an undirected graph are the maximal subsets of vertices such that the removal of a vertex from particular component will not disconnect the component. Unlike connected components, vertices may belong to multiple biconnected components. Vertices can be present in multiple biconnected components, but each edge can only be contained in a single biconnected component. So, the output only has edge version.

Notice: This problem defines on an undirected graph.

Given the following query:

pgr_biconnectedComponents(\(sql\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

• \(source = \bigcup source_i\),
• \(target = \bigcup target_i\),

The graphs are defined as follows:

The weighted undirected graph, \(G(V,E)\), is definied by:

• the set of vertices \(V\)
  • \(V = source \cup target\)
• the set of edges \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \text{ } \\ \cup \{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Given:

• \(G(V,E)\)

Then:

\(\boldsymbol{\pi} = \{(component_i, n\_seq_i, node_i)\}\)

where:

• \(component_i = \min {node_j | node_j \in component_i}\)
• \(n\_seq_i\) is a sequential value starting from 1 in a component.
• \(edge_i \in component_i\)
• The returned values are ordered:
  • component ascending
  • edge ascending

Example:

• The first component is composed of edges 1 - 2, 0 - 1 and 0 - 2.
• The second component is composed of edges 2 - 4, 2 - 3 and 3 - 4.
• The third component is composed of edge 5 - 6.
• The fourth component is composed of edge 6 - 7.
• The fifth component is composed of edges 8 - 9, 9 - 10 and 8 - 10.

Articulation Points

Those vertices that belong to more than one biconnected component are called articulation points or, equivalently, cut vertices. Articulation points are vertices whose removal would increase the number of connected components in the graph.

Notice: This problem defines on an undirected graph.

Given the following query:

pgr_articulationPoints(\(sql\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

• \(source = \bigcup source_i\),
• \(target = \bigcup target_i\),

The graphs are defined as follows:

The weighted undirected graph, \(G(V,E)\), is definied by:

• the set of vertices \(V\)
  • \(V = source \cup target\)
• the set of edges \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \text{ } \\ \cup \{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Given:

• \(G(V,E)\)

Then:

\(\boldsymbol{\pi} = \{node_i\}\)

where:

• \(node_i\) is an articulation point.
• The returned values are ordered:
  • node ascending

Example:

• Articulation points are nodes 2 and 6.

Bridges

A bridge is an edge of an undirected graph whose deletion increases its number of connected components.

Notice: This problem defines on an undirected graph.

Given the following query:

pgr_bridges(\(sql\))

where \(sql = \{(id_i, source_i, target_i, cost_i, reverse\_cost_i)\}\)

and

• \(source = \bigcup source_i\),
• \(target = \bigcup target_i\),

The graphs are defined as follows:

The weighted undirected graph, \(G(V,E)\), is definied by:

• the set of vertices \(V\)
  • \(V = source \cup target\)
• the set of edges \(E\)
  • \(E = \begin{cases} \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \quad \text{ if } reverse\_cost = \varnothing \\ \text{ } \text{ } & \text{ } \\ \text{ } \{(source_i, target_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, cost_i) \text{ when } cost >=0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \text{ } \\ \cup \{(source_i, target_i, reverse\_cost_i) \text{ when } reverse\_cost_i >=0)\} & \quad \text{ if } reverse\_cost \neq \varnothing \\ \end{cases}\)

Given:

• \(G(V,E)\)

Then:

\(\boldsymbol{\pi} = \{edge_i\}\)

where:

• \(edge_i\) is an edge.
• The returned values are ordered:
  • edge ascending

Example:

• Bridges are edges 5 <--> 6 and 6 <--> 7.

See Also¶

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