Kuratowski's theorem

<p>In <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, Kuratowski's theorem is a mathematical <a href="/facts/Forbidden_graph_characterization/50Kex6JU">forbidden graph characterization</a> of <a href="/facts/Planar_graph/plREagr9">planar graphs</a>, named after <a href="/facts/Kazimierz_Kuratowski/8xoKvo5b">Kazimierz Kuratowski</a>.  It states that a finite graph is planar if and only if it does not contain a <a href="/facts/Glossary_of_graph_theory/W0MqKkyE">subgraph</a> that is a <a href="/facts/Subdivision_(graph_theory)/WbFo1mNW">subdivision</a> of 
  
    
      
        
          K
          
            5
          
        
      
    
    {\displaystyle K_{5}}
  
 (the <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> on five <a href="/facts/Vertex_(graph_theory)/WhlWmL3o">vertices</a>) or of 
  
    
      
        
          K
          
            3
            ,
            3
          
        
      
    
    {\displaystyle K_{3,3}}
  
 (a <a href="/facts/Complete_bipartite_graph/C1RYIdpX">complete bipartite graph</a> on six vertices, three of which connect to each of the other three, also known as the <a href="/facts/Utility_graph/CQQqp6fA">utility graph</a>).
</p>

Kuratowski's theorem open-in-new

Kuratowski's theorem