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Every DFS Tree of a 3-Connected Graph Contains a Contractible Edge


  • Contract grant sponsors: .

  • Contract grant sponsors: Alexander von Humboldt Fellowship; Deutsche Forschungsgemeinschaft; contract grant number: GRK 1408.


Tutte proved that every 3-connected graph G on more than 4 vertices contains a contractible edge. We strengthen this result by showing that every depth-first-search tree of G contains a contractible edge. Moreover, we show that every spanning tree of G contains a contractible edge if G is 3-regular or if G does not contain two disjoint pairs of adjacent degree-3 vertices.