We introduce and study a natural variant of matroid amalgams. For matroids M(A) and N(B) with M.([email protected]?B)=N|([email protected]?B), we define a splice of M and N to be a matroid L on [email protected]?B with L|A=M and L.B=N. We show that splices exist for each such pair of matroids M and N; furthermore, there is a freest splice of M and N, which we call the free splice. We characterize when a matroid L([email protected]?B) is the free splice of L|A and L.B. We study minors of free splices and the interaction between free splice and several other matroid operations. Although free splice is not an associative operation, we prove a weakened counterpart of associativity that holds in general and we characterize the triples for which associativity holds. We also study free splice as it relates to various classes of matroids.
[1]
Tom Brylawski,et al.
Modular constructions for combinatorial geometries
,
1975
.
[2]
Henry Crapo,et al.
The free product of matroids
,
2005,
Eur. J. Comb..
[3]
Douglas G. Kelly,et al.
The Higgs factorization of a geometric strong map
,
1978,
Discret. Math..
[4]
Marc Noy,et al.
Lattice path matroids: enumerative aspects and Tutte polynomials
,
2003,
J. Comb. Theory, Ser. A.
[5]
Henry Crapo,et al.
A unique factorization theorem for matroids
,
2005,
J. Comb. Theory, Ser. A.
[6]
R. A. Main,et al.
Non-Algebraic Matroids exist
,
1975
.
[7]
Primitive elements in the matroid-minor Hopf algebra
,
2005,
math/0511033.
[8]
Tom Brylawski,et al.
An Affine Representation for Transversal Geometries
,
1975
.