Georgia Institute of Technology College of Sciences

A matroid $M$ is a pair $(E, \mathcal{I})$ where $E$ is a finite set, called the {\em ground set} of $M$, and $\mathcal{I}$ is a non-empty collection of subsets of $E$, called {\em independent sets} of $M$, such that (1) a subset of an independent set is independent; and (2) if $I$ and $J$ are independent sets with $|I| < |J|$, then exists $x \in J \backslash I$ such that $I \cup \{x\}$ is independent. 

A graph $G$ gives rise to a matroid $M(G)$ where the ground set is $E(G)$ and a subset of $E(G)$ is independent if it spans a forest. Another example is a matroid that comes from a matrix over a field $F$: the ground set $E$ is the set of all columns and a subset of $E$ is independent if it is linearly independent over $F$. 

Tutte's Wheel and Whirl Theorem and Seymour's Splitter Theorem are two well-known inductive tools for proving results for 3-connected graphs and matroids. In this talk, we will give a survey on induction theorems for various versions of 4-connected matroids and graphs.