On recognition algorithms and structure of graphs with restricted induced cycles
CoRR(2023)
Abstract
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 $\ell
\geq 4$.
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