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7.3 Graphs

Basic Notions  Throughout this document, we have used graphs as visual illustrations of binary relations. However, we can also consider graphs themselves as mathematical objects that are represented by binary relations. The theory of graphs has many applications in various scientific areas; they can be used to model railroad plans as well as electrical circuits, molecules as well as brain structures.

Definition 123 (Directed Graph) A  directed graph (gerichteter Graph) is a pair <V, E> of a set V of  vertices (Knoten) or  nodes and a set of E of  edges (Kanten) or  arcs where E is a binary relation on S:
G is directed graph : <=>
   exists V, E:
      G = <V, E> /\ 
      E subset V x V.

An element <x, y> in E expresses the fact that node x is connected to node y where x is called the  initial node of the corresponding edge and y is called its  terminal node. Both nodes are then  adjazent.

Please note that in above definition of a graph there may be at most one edge between any pair of nodes. A graph with potentially more than one edge between two nodes is called a  multigraph (Multigraph) which may be used to describes e.g. multiple roads between two cities. Furthermore, the edges in a graph may be labelled with real values; such a graph is called a  weighted graph (gewichteter Graph) and may describe lengths of roads or capacities of water pipes. However, we will not consider the formalization of such graphs in this document.


The matrix representation of a graph used in above example can be formalized as follows.

Definition 124 (Adjacency Matrix) Let G = <V, E> be a directed graph with |V| = n. The  adjacency matrix (Adjazenzmatrix, Nachbarschaftsmatrix) of G is the boolean n * n matrix M where M(x, y) = true if and only if <x, y> in E:
adjacency(G) :=
   let V = G0, E = G1:
      such M in V x V -> {true, false}:
         (forall x in V, y in V: M(x, y) = true <=> <x, y> in E).

The adjacency matrix is the typical implementation of a graph in a computer; most graph algorithms operate with this representation.

Example  Let R and S be binary relations on Nn for some n in N. The composition of R and S is
R o S = {<a, c>: a in Nn /\  c in Nn /\  (exists b: <a, b> in R /\  <b, c> in S)}.
For the corresponding adjacency matrix, we thus have
forall i in Nn, j in Nn:
   adjacency(<Nn, R o S>)i, j = true <=>
      exists k in Nn: Ai, k = true /\  Bk, j = true
      where A = adjacency(<Nn, R>), B = adjacency(<Nn, S>).
Written as a Java method, the composition of two adjacency matrices A and B giving a result matrix C resembles matrix multiplication:
void compose(int n, boolean[][] A, boolean[][] B, boolean[][] C)
  for (int i=0; i<n; i++)
    for (int j=0; j<n; j++)
      C[i][j] = false;
      for (int k=0; k<n; k++)
        C[i][j] = C[i][j] || (A[i][k] && B[k][j]);
  return C;

In some graphs, the direction of the edges does not matter.

Definition 125 (Undirected Graph) An  undirected graph (ungerichteter Graph) is a directed graph whose edge relation is symmetric:
G is undirected graph : <=>
   exists V, E:
      G = <V, E>,
      E subset V x V,
      E is symmetric on V.

According to this definition, an undirected graph is a special directed graph; therefore all the following definitions and results for directed graphs apply to undirected graphs as well. Because of symmetry, the only information carried by an undirected graph is whether two nodes x and y are connected or not; in the visual representation it suffices to draw a single undirected edge between two connected nodes.


An important notion is the number of edges that enter or leave a node.

Definition 126 (Degree) In a directed graph, the  indegree (eingehender Grad) of x is the number of edges whose terminal node is x:
indegG(x) := |{y in V: <y, x> in E}|
   where V=G0, E=G1.
The  outdegree (ausgehender Grad) of x is the number of edges whose initial node is x:
outdegG(x) := |{y in V: <x, y> in E}|
   where V=G0, E=G1.
The  total degree (Grad) of x is the sum of its indegree and its outdegree:
degG(x) := indegG(x) + outdegG(x).


Applied to undirected graphs, above definition makes the indegree of a node equal its outdegree and the total degree be even; since in undirected graphs the only interesting information is the total degree, this number is then usually divided by two in correspondence with the visual representation.

Even if two graphs are different, they may have the same structure.

Definition 127 (Graph Isomorphism) Two graphs are isomorphic (isomorph) if there exists a bijection between the nodes of the two graphs which preserves the edge structure:
G and G' are isomorphic : <=>
   G is directed graph /\  G' is directed graph /\ 
   exists f: f: V ->iso(E, E') V'
      where V = G0, E = G1, V' = G'0, E' = G'1.


The edges of a node yield paths that can be used to wander among nodes.

Definition 128 (Path) A  path (Pfad) in a graph is a sequence of nodes connected by edges:
p is path in G : <=>
   (exists n in N>0: p: Nn -> V /\ 
      forall i in Nn-1: <pi, pi+1> in E)
   where V=G0, E=G1.
The length (Länge) of a path is the number of edges it contains:
length(p) := such n in N: p: Nn+1 -> V.
A path from x to y has initial node x and terminal node y : <=>
p is path from x to y : <=>
   p0 = x /\  pn = y
      where n = length(p).
A path is  simple (einfach) if it does not contain any edge twice:
p is simple : <=>
   forall i in Nn, j in Nn: <pi, pi+1> = <pj, pj+1> => i = j
      where n = length(p).
A path is  elementary (elementar) if it does not contain any node twice:
p is elementary : <=>
   forall i in Nn, j in Nn: pi = pj => i = j
      where n = length(p).
A path is a  cycle (Kreis) or  circuit if it terminates in its initial node:
p is cycle : <=> exists x: p is path from x to x.

Reachability  We will now investigate how to determine whether two nodes are directly or indirectly connected in a graph.

Definition 129 (Reachable) A node y is  reachable (erreichbar) from a node x in a graph G if there is a path in G from x to y:
y is reachable from x in G : <=>
   exists p: p is path in G /\  p is path from x to y.

The predicate "is reachable" is, for fixed graph G=<V, E>, a binary relation on V, as well as the edge relation E is. Actually we can derive from E the reachability relation by a general construction that we are going to elaborate.

Definition 130 (Reflexive and Transitive Closure) Let R be a binary relation on S. The  reflexive closure (reflexiver Abschluß) of R on S is the smallest relation that contains R and is reflexive on S:
reflexiveS(R) :=
   such R' subset S x S:
      R subset R' /\  R' is reflexive on S /\ 
      forall R": (R subset R" /\  R" is reflexive on S) => R' subset R".
The  transitive closure (transitiver Abschluß) of R on S is the smallest relation that contains R and is transitive on S:
transitiveS(R) :=
   such R' subset S x S:
      R subset R' /\  R' is transitive on S /\ 
      forall R": (R subset R" /\  R" is transitive on S) => R' subset R".

We then can derive the following result.

Proposition 131 (Reachability is Closure of Edge Relation) We define the reachability relation
RG := {<x, y> in G0 x G0: y is reachable from x in G}.
Then, for any directed graph <V, E>, R<V, E> is the reflexive and transitive closure of E on V:
forall V, E: <V, E> is directed graph =>
   R<V, E> = reflexiveV(transitiveV(E)).

Example  Take the graph <N_5, E> with
E = {<0, 1>, <1, 2>, <1, 3>, <2, 3>, <3, 4>}
which can be depicted as:
Since <0, 1> in E and <1, 3> in E, <0, 3> is in the transitive closure of E. Since also <3, 4> is in E, also <0, 4> is in the transitive closure of E, i.e., node 4 is reachable from 0. In total, we have:
transitiveN5(E) =
   {<0, 1>, <0, 2>, <0, 3>, <0, 4>, <1, 2>, <1, 3>, <1, 4>, <2, 3>, <2, 4>}.
In addition, every node is reachable from itself:
reflexiveN5(E) = {<0, 0>, <1, 1>, <2, 2>, <3, 3>, <4, 4>}.

Above definitions of closures are too inefficient for a practical algorithm that decides the reachability of a node; a direct construction of the reflexive closure is provided by the following result which is easy to establish.

Proposition 132 (Reflexive Closure) The reflexive closure of R on S is the union of R and the identity relation on S:
forall S, R: R subset S x S =>
   reflexiveS(R) = R union (S x S).

An efficient computation of the transitive closure is based on the following operation that composes a relation R an arbitrary number of times.

Definition 131 (Exponentiation of Relations) 
R0S := {<x, x>: x in S}
Ri+1S := RiS o R.

RS0 is the identity relation and RS1 is R. Consequently, e.g.

RS3 = R o R o R.

The transitive closure is then characterized by the following result which we state without proof.

Proposition 133 (Transitive Closure) 
forall S, R: R subset S x S =>
   transitiveS(R) = union {R</sup>iS: i in N>0}

In other words, the transitive closure of R is the union of all members of the sequence

[R, R union (R o R), R union (R o R) union (R o R o R), ...]
that starts with R and computes each subsequent member by adding to its predecessor the transitive consequences of R. Since every member is contained in its successor, the transitive closure is the "limit" of this sequence. This approach is constructive because one can show that the sequence converges, i.e., from a certain point on no more elements are added; the final sequence member represents the transitive closure.

Proposition 134 (Transitive Closure) Let R be a binary relation on S where S has n elements. Then union_1 <= i <= n RiS is the transitive closure of R:
forall S, R: R subset S x S =>
   transitiveS(R) = union_1 <= i <= n RiS
   where n = |S|.

Since the reflexive and transitive closure of the edge relation describes the reachability relation, we thus have a constructive method of computing the reachability relation of a graph <V, E>; we start with the identity relation and add n times the transitive consequences of R:

reachability(V, E):
   n = |V|
   R0 = {<x, x>: x in V}
   for (i=0; i < n; i++)
      Ri+1 = Ri union (Ri o E)
   return Rn

For an actual implementation, we usually turn to the representation of the graph as a n * n matrix M. Since in this representation the composition of two relations resembles matrix multiplication the computation of reachability is basically a sequence of matrix multiplications (see this example.

Trees  Trees are a special kind of graphs that are of particular relevance in computer science: they provide a way to represent hierarchical structures such as a family genealogy, an organizational chart, or a file system.

Definition 132 (Tree) A  tree (Baum) is a directed graph such that there is exactly one node, the  root (Wurzel), that has indegree zero, every other node has indegree one, and every node can be reached from the root.
T is tree : <=>
   T is directed graph /\ 
   exists r in V: indeg(r) = 0 /\ 
      forall x in V-{r}:
         indeg(x) = 1 /\ 
         x is reachable from r in T
   where V = T0.
root(T) := (such r in V: indeg(r) = 0) where V = T0.

Trees are depicted with the root node at the top and all arcs directed downwards (such that arrowheads need not be given)5.


We state without proof the following important property of trees.

Proposition 135 (Trees and Cycles) A tree has only cycles of length 0:
forall T: T is tree =>
  ~(exists p: p is path in T /\  length(p) > 0 /\  p is cycle).

The following notions describe relations among tree nodes.

Definition 133 (Node Relations) Let T be a tree. A node y is called a  child (Kind) of x if there is an edge from x to y in T:
y is child of x in T: <=>
   <x, y> in E where E = T1.
x is then called the  parent (Elternteil) of y:
parentT(y) := such x in V: <x, y> in E
   where V = T0, E = T1.
A node x is a  leaf (Blatt), if it does not have children:
x is leaf in T : <=> x in V /\  ~exists y: y is child of x in T
    where V = T0.
A node x is an  ancestor (Vorfahre) of y if there is a path from x to y in T:
x is ancestor of y in T : <=>
   exists p: p is path in T /\  p is path from x to y.
y is then called a  descendant (Nachkomme) of x:
y is descendant of x in T : <=> x is ancestor of y in T.

Please note that in a tree every node (apart from the root) has a parent and that this parent is unique. Every node is a descendant of the root; the only descendant of a leaf node is the node itself.

Example  In the tree
node a is the parent of b and an ancestor of leaf c.

In a tree, all paths from the root are unique.

Proposition 136 (Unique Root Paths) Let T be a tree with root r. Then the path from r to every node of T is unique:
forall T: T is tree =>
   forall x in V:
      exists p: p is path in T /\  p is path from r to x /\ 
         forall p': (p' is path in T /\  p' is path from r to x) => p' = p
   where V = T0, r = root(T).

Because of this result, the following notions are uniquely defined.

Definition 134 (Level and Height) The  level (Stufe) of a node x in a tree is the length of the path from the root of the tree to x:
levelT(x) := length(p)
   where p = such p: p is path in T /\  p is path from root(T) to x.
The  height (Höhe) of a tree is the maximum level of its nodes:
height(T) := max {levelT(x): x in V} where V = T0.

The definition implies that the root node has level 0 and the level of a node is one plus the level of its parent.

Example  In the tree
a has level 1, b has level 2, and c has level 3. The height of the tree is 3.

We conclude this section by a result which relates the degrees of nodes to the heights of trees.

Proposition 137 (Height of Binary Trees) Let T be a tree of height h where every node has an outdegree of at most 2. Then the number of nodes in the tree is less than 2h+1:
forall T: (T is tree /\  forall x in V: outdeg(x) <= 2) => |V| < 2h+1
   where V = T0, h = height(T).

Proof  Let T be a tree where every node has an outdegree of at most 2. We proceed by complete induction on the height of T.
  1. Assume the height is h = 0. Then |V| = 1 < 2 = 2h+1.
  2. Assume the height is h > 0. Consequently the root of T has a child that is the root of a tree of height h-1 and possibly a second child that is the root of a tree of height less than or equal h-1. By the induction hypothesis, we thus have
    |V| <= 1+(2h-1)+(2h-1) = 2h+1-1 < 2h+1.

It is easy to generalize above result to trees whose outdegrees are bounded by an arbutrary limit k (how?).

Author: Wolfgang Schreiner
Last Modification: October 4, 1999

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