Sang-il Oum (엄상일), Survey on vertex-minors

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.