Applications of the Strong Splitter Theorem: decomposition results

2014/08/11 by S. R. Kingan, Kingan, S. R.
Computer Science · Engineering · #Advanced Graph Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1408.2377

openalex publication_date 2014/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no E₄-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid E₄ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of P₉, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-r spikes Z_r, the rank 3 and 4 projective geometries F₇ and PG(3,2), respectively, the 17-element internally 4-connected matroid R₁₇, and one 12-element rank-6 matroid. All the other 3-connected members have P₉ or P₉^* as 3-decomposers. As immediate corollaries we get decomposition results for EX[P₉^*] and EX[P₉] as well as the internally 4-connected members of these classes.

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