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

that represent the nodes for convenience, the only thing that matters for graphs is which nodes
are connected to which. Indeed that is just what the definition of graph equivalence via the
existence of an isomorphism says: two graphs are equivalent, iff they have the same number of
nodes and the same edge connection pattern. The objects that are used to represent them are
purely coincidental, they can be changed by an isomorphism at will. Furthermore, as we have
seen in the example, the shape of the diagram is purely an artifact of the presentation; It does not
matter at all.
So the following two diagrams stand for the same graph, (it is just much more difficult to state
the graph isomorphism)
Note that directed and undirected graphs are totally different mathematical objects. It is easy
to think that an undirected edge {a, b} is the same as a pair 〈a, b〉, 〈b, a〉 of directed edges in
both directions, but a priory these two have nothing to do with each other. They are certainly
not equivalent via the graph equivalent defined above; we only have graph equivalence between
directed graphs and also between undirected graphs, but not between graphs of differing classes.
Now that we understand graphs, we can add more structure. We do this by defining a labeling
function from nodes and edges.
Labeled Graphs
Definition 370 A labeled graph G is a triple 〈V, E, f〉 where 〈V, E〉 is a graph and
f : V ∪ E → R is a partial function into a set R of labels.
Notation 371 write labels next to their vertex or edge. If the actual name of a vertex does
not matter, its label can be written into it.
Example 372 G = 〈V, E, f〉 with V = {A, B, C, D, E}, where
E = {〈A, A〉, 〈A, B〉, 〈B, C〉, 〈C, B〉, 〈B, D〉, 〈E, D〉}
f : V ∪ E → {+, −, ∅} × {1, . . . , 9} with
f(A) = 5, f(B) = 3, f(C) = 7, f(D) = 4, f(E) = 8,
f(〈A, A〉) = −0, f(〈A, B〉) = −2, f(〈B, C〉) = +4,
f(〈C, B〉) = −4, f(〈B, D〉) = +1, f(〈E, D〉) = −4
-0
-2 +1 -4
5 3 4 8
-4
7
+4
c○: Michael Kohlhase 219
Note that in this diagram, the markings in the nodes do denote something: this time the labels
given by the labeling function f, not the objects used to construct the graph. This is somewhat
confusing, but traditional.
Now we come to a very important concept for graphs. A path is intuitively a sequence of nodes
that can be traversed by following directed edges in the right direction or undirected edges.
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