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

Authors:
Amr Elmasry;Kurt Mehlhorn;Jens M. Schmidt
Affiliations:
COMPUTER SCIENCE DEPARTMENT, UNIVERSITY OF COPENHAGEN, DENMARK;MPI FÜR INFORMATIK, CAMPUS E1 4, 66123, SAARBRÜCKEN, GERMANY;DEPARTMENT OF COMPUTER SCIENCE, FU BERLIN, GERMANY
Venue:
Journal of Graph Theory
Year:
2013

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Abstract

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. © 2013 Wiley Periodicals, Inc. (Contract grant sponsors: Alexander von Humboldt Fellowship; Deutsche Forschungsgemeinschaft; contract grant number: GRK 1408.)