Block triangular decomposition of directed graph into strongly connected components












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$begingroup$


Let $G=(V,E)$ with $Esubseteq Vtimes V$ be a directed graph with (directed) adjacency matrix $A=A(G)$.



I remember having seen a decomposition of $G$ into subgraphs $S_i=(V_i,E_i)$ for $iinmathcal J$ some index set $mathcal J$, such that




  1. $V=bigsqcup_{iinmathcal J}V_i$


  2. $S_isubseteq G$ is a stronlgy connected subgraph (possibly trivial, i.e. a point)


  3. $E ,bigcap ,(V_itimes V_j)=emptyset$ whenever $j<i$.


This also casts $A$ into a block triangular form.



Q) What is the name of this decomposition? Or is there anything fishy with it?






graph-theory reference-request topological-graph-theory







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let $G=(V,E)$ with $Esubseteq Vtimes V$ be a directed graph with (directed) adjacency matrix $A=A(G)$.



    I remember having seen a decomposition of $G$ into subgraphs $S_i=(V_i,E_i)$ for $iinmathcal J$ some index set $mathcal J$, such that




    1. $V=bigsqcup_{iinmathcal J}V_i$


    2. $S_isubseteq G$ is a stronlgy connected subgraph (possibly trivial, i.e. a point)


    3. $E ,bigcap ,(V_itimes V_j)=emptyset$ whenever $j<i$.


    This also casts $A$ into a block triangular form.



    Q) What is the name of this decomposition? Or is there anything fishy with it?






    graph-theory reference-request topological-graph-theory







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $G=(V,E)$ with $Esubseteq Vtimes V$ be a directed graph with (directed) adjacency matrix $A=A(G)$.



      I remember having seen a decomposition of $G$ into subgraphs $S_i=(V_i,E_i)$ for $iinmathcal J$ some index set $mathcal J$, such that




      1. $V=bigsqcup_{iinmathcal J}V_i$


      2. $S_isubseteq G$ is a stronlgy connected subgraph (possibly trivial, i.e. a point)


      3. $E ,bigcap ,(V_itimes V_j)=emptyset$ whenever $j<i$.


      This also casts $A$ into a block triangular form.



      Q) What is the name of this decomposition? Or is there anything fishy with it?






      graph-theory reference-request topological-graph-theory







      share|cite|improve this question









      $endgroup$




      Let $G=(V,E)$ with $Esubseteq Vtimes V$ be a directed graph with (directed) adjacency matrix $A=A(G)$.



      I remember having seen a decomposition of $G$ into subgraphs $S_i=(V_i,E_i)$ for $iinmathcal J$ some index set $mathcal J$, such that




      1. $V=bigsqcup_{iinmathcal J}V_i$


      2. $S_isubseteq G$ is a stronlgy connected subgraph (possibly trivial, i.e. a point)


      3. $E ,bigcap ,(V_itimes V_j)=emptyset$ whenever $j<i$.


      This also casts $A$ into a block triangular form.



      Q) What is the name of this decomposition? Or is there anything fishy with it?






      graph-theory reference-request topological-graph-theory




      graph-theory reference-request topological-graph-theory






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 2 '18 at 23:21









      MarloMarlo

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