Articulation Points, Bridges, Bi-

connectivity“Critical nodes, critical edges” and connectivity

Ivaylo Kenov

Telerik Corporationhttp:/telerikacademy.

com

Telerik Academy Student

Table of Contents

1. Connectivity2. Articulation points3. Bridges4. Bi-connectivity

K-connectivity5. Algorithm6. Additional information

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ConnectivityConnecting the chain

Connectivity (1) Connected component of undirected graph – a subgraph in which any two nodes are connected to each other by paths.

Connectivity (2) A simple way to find number of connected components - loop through all nodes and start a DFS or BFS traversing from any unvisited node.

Each time you start a new traversing - you find a new connected component!

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Connectivity (3)

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foreach node from graph G{ if node is unvisited

{ DFS(node); counterOfComponents++;}

}

Algorithm:

*Note: Do not forget to mark each node in the DFS as visited!

Connectivity (4) Connected graph - basically a graph with one connected component

In every connected graph a path exists between any two nodes

Easy algorithm for checking whether a graph is connected - if the previous code return one connected component - graph is connected!

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Articulation pointWhat is it?

Articulation point (or node)

Usually used in unordered connected graphs (containing one connected component ).

If removed (with all its edges) divides the graph into 2 or more connected components.

Bridge"Articulation edge"

Bridge The same as articulation points but this time we remove edges.

If removed divides the graph into 2 or more connected components.

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Bi-connectivityGraph with no articulation points

Bi-connectivity (1) An unordered connected graph is called bi-connected when if we remove 1 of its nodes (no matter which one), the graph will remain connected (with 1 connected component).

Has no articulation points.

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Bi-connectivity (2) k-vertex-connected graph is a graph in which if we remove any k-1 number of nodes, it will remain connected.w

k-edge-connected graph is a graph in which if we remove any k-1 number of edges, it will remain connected.

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AlgorithmFind articulation points

Algorithm (1) One easy (but not efficient) way to determine articulation point (and bi-connectivity) is to remove node by node from the graph and check whether it is still connected.

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Algorithm (2) A node K is an articulation point if there are two other nodes I and J that each path between I and J goes through K.

This statement does not work for the root of the DFS traversal tree - it has no parent nodes.

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Algorithm (3) So how to check the root? Easy! If it has two or more child elements - it is articulation point!

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Algorithm (4) We use two arrays:

Prenumerator[] - we number the nodes in the graph with DFS instead of just mark them as visited

Lowest[] - the least numbered node that can be obtained by a back edge to one of its ancestors or the least node that can be obtained by a back edge from any one of its descendants.

The check - if the DFS number of the node is <= to the Lowest of any of its children

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Algorithm (5)

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function DFS(i) { prenumerator[i] = ++DFScounter;

for (j = 0; j < numberOfNodes; j++) if (A[i][j] != 0 && prenum[j] == 0) { A[i][j] = 2; /* building DFS tree of the graph */ DFS(j); } }

The altered DFS to numerate nodes:

Algorithm (6)

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/* traversing the tree in postorder */ function postOrder(i) { unsigned j;

for (j = 0; j < numberOfNodes; j++) if (2 == A[i][j]) postOrder(j);

lowest[i] = prenumerator[i];

for (j = 0; j < numberOfNodes; j++) if (1 == A[i][j]) lowest[i] = min(lowest[i], prenumerator[j]); for (j = 0; j < numberOfNodes; j++) if (2 == A[i][j]) lowest[i] = min(lowest[i], lowest[j]); }

DFS tree traversal with post-order:

Algorithm (7)

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function findArticPoints() { articulationPoints[], count; for (i = 0; i < numberOfNodes; i++) { prenum[i] = 0; lowest[i] = 0; artPoints[i] = 0; } DFScounter = 0; DFS(0); for (i = 0; i < numberOfNodes; i++) if (0 == prenumerator[i]) { “Graph is not connected!"; return; } postOrder(0); count = 0; for (i = 0; i < numberOfNodes; i++) //checking root if (2 == A[0][i]) count++; if (count > 1) artPoints[0] = 1;

Continues…

Articulation points algorithm:

Algorithm (8)

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Continues…/* checking the rest of the nodes*/ for (i = 1; i < numberOfNodes; i++) { for (j = 0; j < numberOfNodes; j++) if (2 == A[i][j] && lowest[j] >= prenum[i]) break; if (j < n) artPoints[i] = 1; }

Print the result from arcPoints[];}

Articulation points algorithm:

*Note: If you have a graph with huge number of nodes, you should use Stack instead of recursion in the algorithm!

AlgorithmLive demo

Additional information Bi-connectivity: http://www.seas.gwu.edu/~ayoussef/cs212/graphsearch.html#biconnectivity

Algorithm for articulation points: http://nbangla.blogspot.com/2012/12/cpp-articulation-points-detection-algorithm.html

Algorithm for bridges: http://www.aspfree.com/c/a/code-examples/articulation-edges-and-vertexes/

“Programming = ++Algorithms” by P. Nakov;

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