Lecture Notes Discrete Optimization - Applied Mathematics

are the same is called a loop. A graph that has neither parallel edges nor loops is said to
be simple. Note that in a simple graph every edge e = (u,v) ∈ E is uniquely identified
by its endpoints u and v. Unless stated otherwise, we assume that undirected graphs are
simple. We denote by n and m the number of nodes and edges of G, respectively. A
complete graph is a graph that contains an edge for every (unordered) pair of nodes. That
is, a complete graph has m=n(n−1)/2 edges.
A subgraph H of G is a graph such that V(H)⊆ V and E(H)⊆E and each e∈E(H) has
the same endpoints in H as in G. Given a subset V ′ ⊆ V of nodes and a subset E ′ ⊆ E of
edges of G, the subgraph H of G induced by V ′ and E ′ is defined as the (unique) subgraph
H of G with V(H) = V ′ and E(H) = E ′ . Given a subset E ′ ⊆ E, G\E ′ refers to the
subgraph H of G that we obtain if we delete all edges in E ′ from G, i.e., V(H)= V and
E(H)=E\ E ′ . Similarly, given a subset V ′ ⊆ V, G\V ′ refers to the subgraph of G that
we obtain if we delete all nodes in V ′ and its incident edges from G, i.e., V(H)= V \V ′
and E(H) = E\{(u,v)∈E | u∈ V ′ }. A subgraph H of G is said to be spanning if it
contains all nodes of G, i.e., V(H)= V.
A path P in an undirected graph G is a sequence P = 〈v 1 ,...,v k 〉 of nodes such that
e i = (v i ,v i+1 ) (1 ≤ i < k) is an edge of G. We say that P is a path from v 1 to v k , or a
v 1 ,v k -path. P is simple if all v i (1≤i≤k) are distinct. Note that if there is a v 1 ,v k -path
in G, then there is a simple one. Unless stated otherwise, the length of P refers to the
number of edges of P. A path C=〈v 1 ,...,v k = v 1 〉 that starts and ends in the same node
is called a cycle. C is simple if all nodes v 1 ,...,v k−1 are distinct. A graph is said to be
acyclic if it does not contain a cycle.
A connected component C ⊆ V of an undirected graph G is a maximal subset of nodes
such that for every two nodes u,v∈ C there is a u,v-path in G. A graph G is said to be
connected if for every two nodes u,v∈ V there is a u,v-path in G. A connected subgraph
T of G that does not contain a cycle is called a tree of G. A spanning tree T of G is a tree
of G that contains all nodes of G. A subgraph F of G is a forest if it consists of a (disjoint)
union of trees.
A directed graph G = (V,E) is defined analogously with the only difference that edges
are directed. That is, every edge e is associated with an ordered pair(u,v)∈ V ×V. Here
u is called the source (or tail) of e and v is called the target (or head) of e. Note that,
as opposed to the undirected case, edge (u,v) is different from edge (v,u) in the directed
case. All concepts introduced above extend in the obvious way to directed graphs.
1.5 Sets, etc.
Let S be a set and e /∈ S. We will write S+e as a short for S∪{e}. Similarly, for e∈S we
write S−e as a short for S\{e}.
The symmetric difference of two sets S and T is defined as S△T =(S\T)∪(T \ S).
We use N, Z, Q and R to refer to the set of natural, integer, rational and real numbers,
respectively. We use Q + and R + to refer to the nonnegative rational and real numbers,
respectively.
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