Splitting and contractible edges in 4-connected graphs

  • Authors:

  • Akira Saito

  • Affiliations:

  • Department of Applied Mathematics, Nihon University, Sakurajosui 3-25-40 Setagaya-Ku, Tokyo 156-8550, Japan

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

Let G be a graph and let x be a vertex of degree four with NG(x) = {a, b, c, d}. Then the operation of deleting x and adding the edges ab and cd is called splitting at x. An edge e of a graph G is said to be k-contractible if contraction of e yields a k-connected graph. Splitting has been studied as a reduction method to preserve edge-connectivity. In this paper, we consider splitting and 4-contractible edges as tools for reduction of 4-connected graphs. We prove that for a 4-connected graph G of order at least six, there exists either a 4-contractible edge or a vertex eligible for splitting preserving 4-connectedness near every vertex in G.