5. Graph Definitions

Undirected (Ungerichteter) Graphs

An undirected graph $G=(V,E)$ is made-up of a set of vertices $V\ne \varnothing$ ($V$ is the vertex set, de: Knotenmenge) and a set of edges $E$ (edge set, de: Kantenmenge).

The standard notation for $|V|=n$ and for $|E|=m$.

Every element $e\in E$ is an unordered pair $e=\{u,v\}$ (where $u\ne v$).

• The vertices (Knoten) $u,v$ are called endpoints (Endpunkte) of $e$.
• We call $u$ adjacent (adjazent oder benachbart) to $v$ and $e$ incident (inzident oder anliegend).
• The degree (Knotengrad) $\text{deg}(v)$ of a vertex $v$ is the number of edges that are incident to $v$, i.e. $\text{deg}(v)=|\{e\in E|v\text{ Endpoint of }e\}|$.
• If a vertex has $\text{degree}(v)=0$, it is called an isolated vertex (isolierter Knoten).

Vocabulary:

• A sequence of vertices $(v_0,v_1,\dots ,v_k)$ (with $v_i\in V$ for all $i$) is a walk (german “Weg”) if $\{v_i,v_{i+1}\}$ is an edge for each $0\le i\le k-1$. We say that $v_0$ and $v_k$ are the endpoints (german “Startknoten” and “Endknoten”) of the walk. The length of the walk $(v_0,v_1,\dots ,v_k)$ is $k$.
  • A walk of length $l$ connects $l+1$ vertices!
• A sequence of vertices $(v_0,v_1,\dots ,v_k)$ is a closed walk (german “Zyklus”) if it is a walk, $k\ge 2$ and $v_0=v_k$.
• A sequence of vertices $(v_0,v_1,\dots ,v_k)$ is a path (german “Pfad”) if it is a walk and all vertices are distinct (i.e., $v_i\ne v_j$ for $0\le i<j\le k$).
• A sequence of vertices $(v_0,v_1,\dots ,v_k)$ is a cycle (german “Kreis”) if it is a closed walk, $k\ge 3$ and all vertices (except $v_0$ and $v_k$) are distinct.
• An Eulerian walk (german “Eulerweg”) is a walk that contains every edge exactly once.
• A closed Eulerian walk (german “Eulerzyklus”) is a closed walk that contains every edge exactly once.
• A Hamiltonian path (german “Hamiltonpfad”) is a path that contains every vertex.
• A Hamiltonian cycle (german “Hamiltonkreis”) is a cycle that contains every vertex.
• For $u,v\in V$, we say $u$ reaches $v$ (or $v$ is reachable from $u$; german ”$u$ erreicht $v$”) if there exists a walk with endpoints $u$ and $v$, or equivalently, there exists a path with endpoints $u$ and $v$.
  • This reachability relation is an equivalence class.
• A connected component of $G$ (german “Zusammenhangskomponente”) is an equivalence class of the (equivalence) relation defined as follows: Two vertices $u,v\in V$ are equivalent if $u$ reaches $v$.
• A graph $G$ is connected (german “zusammenhängend”) if for every two vertices $u,v\in V$, $u$ reaches $v$, or equivalently, if there is only one connected component.
• A graph is acyclic (german “azyklisch”) if it does not have any cycles.
• A graph $G$ is a tree (german “Baum”) if it is connected and has no cycles.
• A graph $G$ is complete when it’s set of edges is $\{\{u,v\}|u,v\in V,u\ne v\}$.
• A graph $G$ is transitive when for any two edges $\{u,v\}$ and $\{v,w\}$ in $E$, the edge $\{u,w\}$ is also in $E$.
• A graph $G$ is bipartite if it’s possible to partition the vertices into two sets $V_1$ and $V_2$ that are disjoint and cover the graph. Any edge $\{u,v\}$ has to have one endpoint in $V_1$ and the other in $V_2$.

Self loops (Schleifen) are not allowed $⟹(v,v)\notin E$ and $(u,v)$ can only appear once in the set of edges (no multigraphs, Multigraphen).

Directed (Gerichtete) Graphs

In a directed graph, an edge $e\in E$ is an ordered pair of vertices $(u,v)$ where $u\ne v$

• $v$ is the direct successor (Nachfolger) of $u$
• $u$ is the direct predecessor (Vorgänger) of $v$ We also call $u$ the start vertex (Startknoten) and $v$ the end vertex (Endknoten) of $e$.

We now differentiate between the in-degree ${\text{deg}}_{\text{in}}(v)$ (Eingangsgrad) and the out-degree ${\text{deg}}_{\text{out}}(v)$ (Ausgangsgrad) of a vertex v:

• the in-degree is the number of edges that have $v$ as the end vertex
• The out-degree is the number of edges that have $v$ as the start vertex

A vertex with ${\text{deg}}_{\text{in}}(v)=0$ is called a source (Quelle) and a vertex with ${\text{deg}}_{\text{out}}(v)=0$ is called a sink (Senke).

Vocabulary

• All walks, closed walks, paths and cycles are now called directed (gerichteter).
• For $u,v\in V$ we say that $u$ reaches $v$ (erreicht) if there is a directed path with endpoints $u$ and $v$.
• A graph is called a directed acyclic graph or DAG (gerichteter azyklischer Graph) if there is no directed cycles.
• Two vertices are strongly connected if there exists both a path from $u$ to $v$ and $v$ to $u$.

Same as for undirected graphs, self loops $(u,u)\in E$ are not allowed and there cannot be multiple $(u,v)\in E$. We can however have $(u,v)\wedge (v,u)\in E$.

Cut Vertex and Edges

A vertex $v$ in a connected graph is called a cut vertex (or articulation point) if the subgraph obtained by removing $v$ and all it’s incident edges is disconnected.

An edge $e$ in a connected graph is called a cut edge (or bridge) if the subgraph obtained by removing $e$ (but keeping all vertices) is disconnected.

Bipartite Graph

A graph $G$ is bipartite if it’s possible to partition the vertices into two sets $V_1$ and $V_2$ that are disjoint and cover the graph. Any edge $\{u,v\}$ has to have one endpoint in $V_1$ and the other in $V_2$.

Bipartite Graph

A graph is bipartite if and only if it does not contain any odd cycles.

Representations

Adjacency Matrix

$A_{uv}=\{\begin{array}{ll}1& \text{if }(u,v)\in E\\ 0& \text{otherwise}\end{array}$ The matrix representing an undirected graph is always symmetric.

Adjacency List

$\text{Adj}[u]=\text{List of all successors of }u$. Represented as a list of lists in memory.

Runtimes

Op/Run	Adj. Matrix	Adj. List
$(u,v)\in E$	$O(1)$	$O(1+{\mathrm{deg}}_{\text{out}}(u))$
List successors	$O(n)$	$O(1+{\mathrm{deg}}_{\text{out}}(u))$