Contractible Elements in Graphs and Matroids

Authors:
Haidong Wu
Affiliations:
Department of Mathematics, The University of Mississippi, University, MS 38677, USA (e-mail: hwu@olemiss.edu
Venue:
Combinatorics, Probability and Computing
Year:
2003

Citing 4
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Contractible edges in 3-connected graphs

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Covering contractible edges in 3-connected graphs: II. characterizing those with covers of size three

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On contractible and vertically contractible elements in 3-connected matroids and graphs

Discrete Mathematics

Contractible edges in a k-connected graph

CJCDGCGT'05 Proceedings of the 7th China-Japan conference on Discrete geometry, combinatorics and graph theory
Every DFS Tree of a 3-Connected Graph Contains a Contractible Edge

Journal of Graph Theory

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Abstract

Let G be a simple 3-connected graph with at least fivevertices. Tutte [13] showed that G has at least onecontractible edge. Thomassen [11] gave a simple proof of this factand showed that contractible edges have many applications. In thispaper, we show that there are at most $\frac{|V(G)|}{5}$ verticesthat are not incident to contractible edges in a 3-connected graphG. This bound is best-possible. We also show that if avertex v is not incident to any contractible edge inG, then v has at least four neighbours having degreethree, and each such neighbour is incident to exactly twocontractible edges. We give short proofs of several results oncontractible edges in 3-connected graphs as well. We also study thecontractible elements for k-connected matroids. We partiallysolve an open problem for regular matroids.