32,852 research outputs found
Star Decompositions of Bipartite Graphs
In Chapter 1, we will introduce the definitions and the notations used throughout this thesis. We will also survey some prior research pertaining to graph decompositions, with special emphasis on star-decompositions and decompositions of bipartite graphs. Here we will also introduce some basic algorithms and lemmas that are used in this thesis.
In Chapter 2, we will focus primarily on decomposition of complete bipartite graphs. We will also cover the necessary and sufficient conditions for the decomposition of complete bipartite graphs minus a 1-factor, also known as crown graphs and show that all complete bipartite graphs and crown graphs have a decomposition into stars when certain necessary conditions for the decomposition are met. This is an extension of the results given in "On claw-decomposition of complete graphs and complete bigraphs" by Yamamoto, et. al. We will propose a construction for the decomposition of the graphs.
In Chapter 3, we focus on the decomposition of complete equipartite tripartite graphs. This result is similar to the results of "On Claw-decomposition of complete multipartite graphs" by Ushio and Yamamoto. Our proof is again by construction and we propose how it might extend to equipartite multipartite graphs. We will also discuss the 3-star decomposition of complete tripartite graphs.
In Chapter 4 , we will discuss the star decomposition of 4-regular bipartite graphs, with particular emphasis on the decomposition of 4-regular bipartite graphs into 3-stars. We will propose methods to extend our strategies to model the problem as an optimization problem. We will also look into the probabilistic method discussed in "Tree decomposition of Graphs" by Yuster and how we might modify the results of this paper to star decompositions of bipartite graphs.
In Chapter 5, we summarize the findings in this thesis, and discuss the future work and research in star decompositions of bipartite and multipartite graphs
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Homogeneous sets, clique-separators, critical graphs, and optimal -binding functions
Given a set of graphs, let be the optimal -binding function of
the class of -free graphs, that is,
In this paper, we combine the
two decomposition methods by homogeneous sets and clique-separators in order to
determine optimal -binding functions for subclasses of -free graphs
and of -free graphs. In particular, we prove the following
for each :
(i)
(ii) $\
f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),\
f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin
\mathcal{O}(\omega),\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.G\chi(G)>\chi(G-u)u\in V(G)(P_5,banner)$-free graphs
Towards obtaining a 3-Decomposition from a perfect Matching
A decomposition of a graph is a set of subgraphs whose edges partition those
of . The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011
states that every connected cubic graph can be decomposed into a spanning tree,
a 2-regular subgraph, and a matching. It has been settled for special classes
of graphs, one of the first results being for Hamiltonian graphs. In the past
two years several new results have been obtained, adding the classes of plane,
claw-free, and 3-connected tree-width 3 graphs to the list.
In this paper, we regard a natural extension of Hamiltonian graphs: removing
a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely,
removing a perfect matching from a cubic graph leaves a disjoint union
of cycles. Contracting these cycles yields a new graph . The graph is
star-like if is a star for some perfect matching , making Hamiltonian
graphs star-like. We extend the technique used to prove that Hamiltonian graphs
satisfy the 3-decomposition conjecture to show that 3-connected star-like
graphs satisfy it as well.Comment: 21 pages, 7 figure
Fork-decomposition of strong product of graphs
Decomposition of arbitrary graphs into subgraphs of small size is assuming importance in the literature. There are several studies on the isomorphic decomposition of graphs into paths, cycles, trees, stars, sunlet etc. Fork is a tree obtained by subdividing any edge of a star of size three exactly once. In this paper, we investigate the necessary and sufficient for the fork-decomposition of Strong product of graphs
Fast Recognition of Partial Star Products and Quasi Cartesian Products
This paper is concerned with the fast computation of a relation on the
edge set of connected graphs that plays a decisive role in the recognition of
approximate Cartesian products, the weak reconstruction of Cartesian products,
and the recognition of Cartesian graph bundles with a triangle free basis.
A special case of is the relation , whose convex closure
yields the product relation that induces the prime factor
decomposition of connected graphs with respect to the Cartesian product. For
the construction of so-called Partial Star Products are of particular
interest. Several special data structures are used that allow to compute
Partial Star Products in constant time. These computations are tuned to the
recognition of approximate graph products, but also lead to a linear time
algorithm for the computation of for graphs with maximum bounded
degree.
Furthermore, we define \emph{quasi Cartesian products} as graphs with
non-trivial . We provide several examples, and show that quasi
Cartesian products can be recognized in linear time for graphs with bounded
maximum degree. Finally, we note that quasi products can be recognized in
sublinear time with a parallelized algorithm
Разноразмерные древесные разложения полных графов
In this paper we explore the problem on decomposition of complete graphs into trees with various amounts of chain-, star-, comet-, and double-star-like vertices. The results of stadies are presented in the form of tables
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