Tag: Sang-ilOum

Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb F$, the list contains only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general.

We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most~$k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

This is joint work with Mamadou M. Kanté, Eun Jung Kim, and O-joung Kwon.

Bouchet introduced isotropic systems in 1983 unifying some combinatorial features of binary matroids and 4-regular graphs. The concept of isotropic system is a useful tool to study vertex-minors of graphs and yet it is not well  known. I will give an introduction to isotropic systems.

For a vertex v of a graph G, the local complementation at v is an operation to obtain a new graph denoted by G*v from G such that two distinct vertices x, y are adjacent in G*v if and only if both x, y are neighbors of v and x, y are non-adjacent, or at least one of x, y is not a neighbor of v and x, y are adjacent. A graph H is a vertex-minor of a graph G if H is obtained from G by a sequence of local complementation and vertex deletions. Interestingly vertex-minors have been used in the study of measurement-based quantum computing on graph states.

Motivated by the big success of the graph minor structure theory developed deeply by Robertson and Seymour since 1980s, we propose a similar theory for vertex-minors. This talk will illustrate similarities between graph minors and graph vertex-minors and give a survey of known theorems and open problems on vertex-minors of graphs.