2018/07/30 by Nelson, Peter, Nomoto, Kazuhiro · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.11543
A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets E of points in PG(n-1,2) for which |E ∩ P| is not a basis of P for any plane P, or as the subsets X of \mathbbF2n containing no linearly independent triple x,y,z for which x+y,y+z,x+z,x+y+z ∉ X. We prove a decomposition theorem that exactly determines the structure of all claw-free matroids. The theorem states that claw-free matroids either belong to one of three particular basic classes of claw-free matroids, or can be constructed from these basic classes using a certain 'join' operation.