In graph theory, two graphs 
  
    
      
        G
      
    
    {\displaystyle G}
  
 and 
  
    
      
        
          G
          ′
        
      
    
    {\displaystyle G'}
  
 are homeomorphic if there is a graph isomorphism from some subdivision of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to some subdivision of 
  
    
      
        
          G
          ′
        
      
    
    {\displaystyle G'}
  
.  If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the topological sense.
 In general, a subdivision of a graph G (sometimes known as an expansion) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w } and {w,v }.
 For example, the edge e, with endpoints {u,v }:
 can be subdivided into two edges, e1 and e2, connecting to a new vertex w:
 The reverse operation, smoothing out or smoothing  a vertex w with regards to the pair of edges (e1, e2) incident on w, removes both edges containing w and replaces (e1, e2) with a new edge that connects the other endpoints of the pair. Here, it is emphasized that only degree-2 (i.e., 2-valent) vertices can be smoothed.
 For example, the simple connected graph with two edges, e1 {u,w } and e2 {w,v }: 
 has a vertex (namely w) that can be smoothed away, resulting in:
 Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.
 The barycentric subdivision subdivides each edge of the graph.  This is a special subdivision, as it always results in a bipartite graph.  This procedure can be repeated, so that the nth barycentric subdivision is the barycentric subdivision of the n−1st barycentric subdivision of the graph.  The second such subdivision is always a simple graph.
 It is evident that subdividing a graph preserves planarity.  Kuratowski's theorem states that 
 In fact, a graph homeomorphic to K5 or K3,3 is called a Kuratowski subgraph.
 A generalization, following from the Robertson–Seymour theorem, asserts that for each integer g, there is a finite obstruction set of graphs 
  
    
      
        L
        (
        g
        )
        =
        
          {
          
            G
            
              i
            
            
              (
              g
              )
            
          
          }
        
      
    
    {\displaystyle L(g)=\left\{G_{i}^{(g)}\right\}}
  
 such that a graph H is embeddable on a surface of genus g if and only if H contains no homeomorphic copy of any of the 
  
    
      
        
          G
          
            i
          
          
            (
            g
            )
            
          
        
      
    
    {\displaystyle G_{i}^{(g)\!}}
  
.  For example, 
  
    
      
        L
        (
        0
        )
        =
        
          {
          
            
              K
              
                5
              
            
            ,
            
              K
              
                3
                ,
                3
              
            
          
          }
        
      
    
    {\displaystyle L(0)=\left\{K_{5},K_{3,3}\right\}}
  
 consists of the Kuratowski subgraphs.
 In the following example, graph G and graph H are homeomorphic.
 If G′ is the graph created by subdivision of the outer edges of G and H′ is the graph created by subdivision of the inner edge of H, then G′ and H′ have a similar graph drawing:
 Therefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.