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

defined a directed graph to be a set of pairs over a base set (of nodes). These objects we have seen
in the beginning of this course and called them relations. So directed graphs are special relations.
We will now introduce some nomenclature based on this intuition.
Directed Graphs
Idea: Directed Graphs are nothing else than relations
Definition 365 Let G = 〈V, E〉 be a directed graph, then we call a node v ∈ V
initial, iff there is no w ∈ V such that 〈w, v〉 ∈ E. (no predecessor)
terminal, iff there is no w ∈ V such that 〈v, w〉 ∈ E. (no successor)
In a graph G, node v is also called a source (sink) of G, iff it is initial (terminal) in G.
Example 366 The node 2 is initial, and the nodes 1 and 6 are terminal in
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c○: Michael Kohlhase 217
For mathematically defined objects it is always very important to know when two representations
are equal. We have already seen this for sets, where {a, b} and {b, a, b} represent the same set:
the set with the elements a and b. In the case of graphs, the condition is a little more involved:
we have to find a bijection of nodes that respects the edges.
Graph Isomorphisms
Definition 367 A graph isomorphism between two graphs G = 〈V, E〉 and G ′ = 〈V ′ , E ′ 〉
is a bijective function ψ : V → V ′ with
directed graphs undirected graphs
〈a, b〉 ∈ E ⇔ 〈ψ(a), ψ(b)〉 ∈ E ′ {a, b} ∈ E ⇔ {ψ(a), ψ(b)} ∈ E ′
Definition 368 Two graphs G and G ′ are equivalent iff there is a graph-isomorphism ψ
between G and G ′ .
Example 369 G1 and G2 are equivalent as there exists a graph isomorphism ψ :=
{a ↦→ 5, b ↦→ 6, c ↦→ 2, d ↦→ 4, e ↦→ 1, f ↦→ 3} between them.
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c○: Michael Kohlhase 218
Note that we have only marked the circular nodes in the diagrams with the names of the elements
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