8,222 research outputs found
On recognition algorithms and structure of graphs with restricted induced cycles
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 for every
.
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 . 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 for any fixed integer . In
Chapter 2, we provide a polynomial-time algorithm to test whether a graph
contains an even hole of length at least for any fixed integer .Comment: PhD Thesis, May 2021, Princeton University, Advisor: Paul Seymou
On the structure of (pan, even hole)-free graphs
A hole is a chordless cycle with at least four vertices. A pan is a graph
which consists of a hole and a single vertex with precisely one neighbor on the
hole. An even hole is a hole with an even number of vertices. We prove that a
(pan, even hole)-free graph can be decomposed by clique cutsets into
essentially unit circular-arc graphs. This structure theorem is the basis of
our -time certifying algorithm for recognizing (pan, even hole)-free
graphs and for our -time algorithm to optimally color them.
Using this structure theorem, we show that the tree-width of a (pan, even
hole)-free graph is at most 1.5 times the clique number minus 1, and thus the
chromatic number is at most 1.5 times the clique number.Comment: Accepted to appear in the Journal of Graph Theor
Perfect Graphs
This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
Chebyshev and Conjugate Gradient Filters for Graph Image Denoising
In 3D image/video acquisition, different views are often captured with
varying noise levels across the views. In this paper, we propose a graph-based
image enhancement technique that uses a higher quality view to enhance a
degraded view. A depth map is utilized as auxiliary information to match the
perspectives of the two views. Our method performs graph-based filtering of the
noisy image by directly computing a projection of the image to be filtered onto
a lower dimensional Krylov subspace of the graph Laplacian. We discuss two
graph spectral denoising methods: first using Chebyshev polynomials, and second
using iterations of the conjugate gradient algorithm. Our framework generalizes
previously known polynomial graph filters, and we demonstrate through numerical
simulations that our proposed technique produces subjectively cleaner images
with about 1-3 dB improvement in PSNR over existing polynomial graph filters.Comment: 6 pages, 6 figures, accepted to 2014 IEEE International Conference on
Multimedia and Expo Workshops (ICMEW
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