Merging of TOSCA Cloud Topology Templates - IAAS

2 Fundamentals
2 Fundamentals
This chapter provides the fundamentals necessary for this thesis. In Section 2.1 the definitions
from Graph Theory used throughout this work are provided. Section 2.2 gives a brief
introduction to Cloud Computing and Section 2.3 presents the syntax of the TOSCA specification
in detail. Finally, Section 2.4 defines the fundamentals and the approach for designing
extendable frameworks.
2.1 Graph Theory
One of the underlying theoretical principles of this thesis is graph theory. This section introduces
the terms and definitions used throughout the rest of the thesis.
Informally a graph or general graph is a set of nodes, also called vertices, and a set of edges.
There exist many slightly different formal definitions and notations of graphs. The formal
definitions in this master thesis are based on [47], [8] and [48].
Definition 2.1 (Undirected graph): An undirected graph G = (V, E) consists of a set V of vertices
and a set E of edges and every e ∈ E of G connects two, not necessarily distinct, vertices of
V. For every graph G, the following holds true: V ∩ E = ∅. A graph G is called simple if it has
no edge ending at the same vertex (loop) or parallel edges.
Note: It is assumed that the sets and of a graph are finite.
Definition 2.2 (Incidence and adjacency): Given a graph G = (V, E) and an edge e ∈ E, one
can write e = {u, v} whereas u, v ∈ V are vertices that are called the ends of e. If v is an end of
e, we say v is incident to e. Two vertices that are the ends of an edge e, are called adjacent to
each other. The same holds true for two edges that are incident with a common vertex.
Definition 2.3 (Subgraph): Given a graph G = (V, E), a graph H = (W, F) is a subgraph of G,
if W ⊆ V and F ⊆ E. We say G contains H or H is contained in G and write G ⊇ H or H ⊆ G
respectively.
Definition 2.4 (Path and cycle): Given a graph G = (V, E) a path is a linear sequence of vertices
that are adjacent if they are consecutive in the sequence and nonadjacent otherwise. No
vertex is repeated in the sequence. A cycle is a closed sequence where u = u and u , u ∈ V,
i.e. the starting and ending vertex are identical.
Definition 2.5 (Connected graph and components): A graph G = (V, E) is connected if there
exists a path from every u ∈ V to every v ∈ V. Otherwise G is called disconnected. A maximum
connected subgraph is called a component.
Definition 2.6 (Directed graph): A directed graph, or short digraph, = (, ) consists of a
set of vertices and a set of directed edges (arcs) such that to every ∈ a unique ordered
pair (, ) ∈ has been assigned. The vertex u is called tail and the vertex v is called head of
the arc = (, ). A loop is an arc = (, ) where head and tail are the same vertices. Two
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