Isomorphism and Homeomorphism of graphs


If two graphs G and H contain the same number of vertices connected in the same way, they are called isomorphic graphs (denoted by G ≅ H).

It is easier to check non-isomorphism than isomorphism. If any of these following conditions occurs, then two graphs are non-isomorphic −

  • The number of connected components are different
  • Vertex-set cardinalities are different
  • Edge-set cardinalities are different
  • Degree sequences are different


The following graphs are isomorphic −


A homomorphism from a graph G to a graph H is a mapping (May not be a bijective mapping) h: G → H such that − (x, y) ∈ E(G) → (h(x), h(y)) ∈ E(H). It maps adjacent vertices of graph G to the adjacent vertices of the graph H.

Properties of Homomorphisms

  • A homomorphism is an isomorphism if it is a bijective mapping.

  • Homomorphism always preserves edges and connectedness of a graph.

  • The compositions of homomorphisms are also homomorphisms.

  • To find out if there exists any homomorphic graph of another graph is a NPcomplete problem.

Updated on: 23-Aug-2019

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