I still have problem with the notion of a minor versus a topological minor.

  1. Prove that $K_7$ has $K_4$ as a topological minor.

  2. Let $v_1,...,v_6$ a labeling of $K_6$ and let $G$ be the graph obtained from $K_6$ by removing the edges $\{v_1v_2,v_1v_3,v_4v_5, v_4v_6\}$. Prove that $G$ contain $K_5$ as a minor but not as a topological minor.

My Answers:

  1. Be deleting $3$ vertices of $K_7$ we obtain $K_4$ and thus $K_4$ is a topological minor of $K_7$.

  2. By contracting the edge $v_1v_4$ we obtain $K_5$. Therefore $K_5$ is a minor of $G$. Since the vertices $v_2$, $v_3$, $v_4$ and $v_5$ each have degree $4$ and vertices $v_1$ and $v_4$ each have degree $3$, it's impossible for $K_5$ to be a topological minor of $G$ because if it were, at least five vertices need to have degree $4$.

Are my arguments correct?

  • $\begingroup$ Are my arguments corrects ? $\endgroup$
    – idm
    Jun 8, 2015 at 12:36

1 Answer 1


For completeness I'll include the definitions of graph minor and topological graph minor, but yes, your arguments are sound.

We say that a graph $H$ is a minor of $G$ (sometimes denoted $H \prec G$) if you can form $H$ by deleting vertices, deleting edges, or contracting edges in $G$.

We say that $H$ is a topological minor of $G$ if there exists a subgraph of $G$ (the result of deleting edges and vertices in $G$) that is isomorphic to a subdivision of $H$.


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