Let \(G(V,E)\) be a connected undirected graph. The following algorithm \(\mathtt{getBICONNECTEDCOMPONENTS(G)}\) obtains correctly all of its cutvertices and biconnected components in the time complexity \(\mathcal O(|V|+|E|)\).

Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph

``` FOR v\in V visited[v]:=NIL; // unmark all vertices in `\(G\)` NEXT B=\emptyset; // initiate empty set of all biconnected components S=\text{empty stack}; // initiate empty stack enum:=-1; FOR v\in V IF visited[v]=NIL THEN CALL \{\mathtt{calculateLowpoint(G,v,B,S,enum)}\}; // print all cutvertices and store biconnected components in `\(B\)`; ENDIF NEXT RETURN B; // return all found biconnected components // Function `\(\mathtt{calculateLowpoint(G,v,B,S,m)}\)` calculates the lowpoint of `\(v\)`. // The lowpoint is lowest depth of all descendants of v in the depth-first-search tree FUNCTION \mathtt{calculateLowpoint(G,v,B,S,enum)} enum:=enum+1; visited[v]:=enum; numberOfDescendants[v]:=0; // set the number of descendants to 0 lowpoint:=enum; S.push(v); // put `\(v\)` on a stack FOR n\in f(v)// for all (neighbors) `\(n\)` of `\(v\)` IF visited[n] = NIL THEN // if the neighbor `\(n\)` has not yet been visited numberOfDescendants[v]:=numberOfDescendants[v]+1; // increase the number of descendants lowpointneighbor:=\mathtt{calculateLowpoint(G,n,B,S,enum)}; // depth-first search for the next neighbor `\(n\)` IF lowpointneighbor < lowpoint THEN lowpoint:=lowpointneighbor; ENDIF IF visited[v]==0 THEN ;// special case for root of DFS ;// root is an cutvertex iif there are two or more descendants IF numberOfDescendants[v]\ge 2 THEN outputComponent(G,B,S,v); // a new cutvertex `\(v\)` and a new biconnected component found ENDIF ELSE IF (lowpointneighbor \ge visited[v] THEN outputComponent(G,B,S,v); // a new cutvertex `\(v\)` and a new biconnected component found ENDIF ELSE IF visited[n] < lowpoint THEN lowpoint=visited[n]; ENDIF ENDIF NEXT RETURN lowpoint; ENDFUNCTION // function `\(\mathtt{outputComponent(G,B,S,v)}\)` prints the cutvertex `\(v\)` // and returns a subgraph of `\(G\)`, // which is its biconnected component FUNCTION outputComponent(G,B,S,v) PRINT v; // new cutvertex found V:=\emptyset; // initiate an empty set of vertices REPEAT x:=S.pull(); // pull vertex from stack V:=V\cup \{x\}; // add vertex to set `\(V\)` UNTIL x=v; // until the vertex `\(v\)` is reached B:=B\cup \{G[V]\}; // output the subgraph of `\(G\)` induced by `\(V\)` (=biconnected component of `\(G\)`) S.push(v); // put cutvertex `\(v\)` on a stack as part of the next biconnected component ENDFUNCTION

Proofs: 1

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  1. Sedgewick, Robert: "Algorithmen in C++", Addison-Wesley, 1992, 1st Edition