  This is my PhD thesis which was defended in May 2021.
  We call an induced cycle of length at least four a hole. The parity of a hole
is the parity of its length. Forbidding holes of certain types in a graph has
deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and
Thomas famously proved that a graph is perfect if and only if it does not
contain an odd hole or a complement of an odd hole. In 2002, Conforti,
Cornu\'{e}jols, Kapoor and Vu\v{s}kov\'{i}c provided a structural description
of the class of even-hole-free graphs. In Chapter 3, we provide a structural
description of all graphs that contain only holes of length $\ell$ for every
$\ell \geq 7$.
  Analysis of how holes interact with graph structure has yielded detection
algorithms for holes of various lengths and parities. In 1991, Bienstock showed
it is NP-Hard to test whether a graph G has an even (or odd) hole containing a
specified vertex $v \in V(G)$. In 2002, Conforti, Cornu\'{e}jols, Kapoor and
Vu\v{s}kov\'{i}c gave a polynomial-time algorithm to recognize even-hole-free
graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi and
Seymour provided a simpler and slightly faster algorithm to test whether a
graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour and Spirkl
provided a polynomial-time algorithm to test whether a graph contains an odd
hole. Later that year, Chudnovsky, Scott and Seymour strengthened this result
by providing a polynomial-time algorithm to test whether a graph contains an
odd hole of length at least $\ell$ for any fixed integer $\ell \geq 5$. In
Chapter 2, we provide a polynomial-time algorithm to test whether a graph
contains an even hole of length at least $\ell$ for any fixed integer $\ell
\geq 4$.
