General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

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We represented the maze as a graph for clarity.
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〈a, e〉, 〈e, i〉, 〈i, j〉,
〈f, j〉, 〈f, g〉, 〈g, h〉,
〈d, h〉, 〈g, k〉, 〈a, b〉
〈m, n〉, 〈n, o〉, 〈b, c〉
〈k, o〉, 〈o, p〉, 〈l, p〉
Now, we are interested in circuits, which we will also represent as graphs.
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, a, p
Let us look at the theory of graphs first (so we know what we are doing)
c○: Michael Kohlhase 213
Graphs and trees are fundamental data structures for computer science, they will pop up in many
disguises in almost all areas of CS. We have already seen various forms of trees: formula trees,
tableaux, . . . . We will now look at their mathematical treatment, so that we are equipped to talk
and think about combinatory circuits.
We will first introduce the formal definitions of graphs (trees will turn out to be special graphs),
and then fortify our intuition using some examples.
Basic Definitions: Graphs
Definition 360 An undirected graph is a pair 〈V, E〉 such that
V is a set of vertices (or nodes) (draw as circles)
E ⊆ {{v, v ′ } | v, v ′ ∈ V ∧ (v = v ′ )} is the set of its undirected edges (draw as lines)
Definition 361 A directed graph (also called digraph) is a pair 〈V, E〉 such that
V is a set of vertices
E ⊆ V × V is the set of its directed edges
Definition 362 Given a graph G = 〈V, E〉. The in-degree indeg(v) and the out-degree
outdeg(v) of a vertex v ∈ V are defined as
indeg(v) = #({w | 〈w, v〉 ∈ E})
outdeg(v) = #({w | 〈v, w〉 ∈ E})
Note: For an undirected graph, indeg(v) = outdeg(v) for all nodes v.
c○: Michael Kohlhase 214
We will mostly concentrate on directed graphs in the following, since they are most important for
the applications we have in mind. Many of the notions can be defined for undirected graphs with
a little imagination. For instance the definitions for indeg and outdeg are the obvious variants:
indeg(v) = #({w | {w, v} ∈ E}) and outdeg(v) = #({w | {v, w} ∈ E})
In the following if we do not specify that a graph is undirected, it will be assumed to be
directed.
This is a very abstract yet elementary definition. We only need very basic concepts like sets and
ordered pairs to understand them. The main difference between directed and undirected graphs
can be visualized in the graphic representations below:
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