Trees and Cut-sets
 Trees
 Rooted Trees
 Path Lengths in Rooted Trees
 Prefix Codes
 Binary Search Trees
 Spanning Trees and Cut-sets
 Minimum Spanning Trees
 Transport Networks
1
Basic Definitions and Properties
 tree : A connected (undirected) graph that contains no simple
cycle.
forest : A collection of disjoint trees.
terminal node : A vertex of degree 1 in a tree.
 properties of trees:
 There is a unique path between every two vertices in a tree.
 The number of vertices is one more than the number of
edges in a tree.
 A tree with two or more vertices has at least two leaves.
 equivalent definitions of trees
 A graph in which there is a unique path between every pair
of vertices is a tree.
 A connected graph with e = v - 1 is a tree.
 A graph with e = v - 1 that has no circuit(cycle) is a tree.
2
Rooted Trees
 directed tree: A directed graph is said to be a directed tree if it
becomes a tree when the directions of the edges are ignored.
 rooted tree: A directed tree is called a rooted tree if there exists
exactly one vertex whose incoming degree is 0 and the
incoming degrees of all other vertices are 1.
 internal node or branch node : A vertex of degree large than 1
in a tree.
son node, father node, and brother node
 descendant c of node a: There is a directed path from a to c.
Also, a is called to be an ancestor of c.
 the subtree with a as a root:
The subgraph T’ = (V’, E’) of T such that V’ contains a and all
of its descendants, and E’ contains the edges in all directed
paths emanating from a.
 ordered tree: A rooted tree with the edges incident from each
branch node labeled with integers 1, 2, …, i, …
 tree isomorphism
 m-ary tree: An ordered tree in which every branch node has at
most m sons.
regular tree: Every branch node of an m-ary tree has exactly
m sons.
 binary tree, left subtree, and right subtree
3
Path Lengths in Rooted Trees
 path length : The number of edges in the path from the root to
the vertex.
height of a tree : The maximum of the path lengths in the tree.
 single elimination tennis tournament
the number of games played = one less than the number of
players in the tournament
 In a regular binary tree, let I denote the sum of the path lengths
of all the branch nodes and E denote the sum of the path lengths
of the leaves in a rooted tree.
E = I + number of edges in the binary tree = I + 2i,
where i is the number of branch nodes.
4
Prefix Codes
 entropy code : One might wish to represent more frequently
used letters with shorter sequences and less frequently used
codes with longer sequences so that the overall length of the
string will be reduced.
 How one at the receiving end can unambiguously divides a long
string of 0s and 1s into sequences of letters?
Prefix code
 prefix code : A set of sequences is said to be a prefix code if no
sequence in the set is a prefix of another sequence in the set.
 binary tree of the weights w 1 ,w 2 ,...,w t :
Given a set of weights w 1 ,w 2 ,...,w t , w 1  w 2  ...  w t , a
binary tree that has t leaves with the weightsw 1 ,w 2 ,...,w t
assigned to the leaves.
 the weight of a binary tree for the weights w 1 ,w 2 ,...,w t :
W T 
t
 
= w i l w i
i 1
 
, where l w i
is the path length of
the leaf to which the weight w i is assigned .
 A binary tree T for the weights w 1 ,w 2 ,...,w t is said to be an
optimal tree if W(T) is minimum.
5
 Huffman code algorithm : A procedure for constructing a
optimal tree for a given set of weights.
 Observation : We can obtain an optimal tree T for the
weights w 1 ,w 2 ,...,w t , from an optimal tree T’ for the
weights w 1  w 2 ,w 3 ,...,w t .
 Proof. There is an optimal tree for the weights
w 1 ,w 2 ,...,w t in which the leaves to which w1 and w2 are
assigned are brothers.
Let a be a branch node of largest path length, and the weights
assigned to the sons of a are wx and wy. We have l(w1) =
l(w2) = l(wx) = l(wy). [If l(w1) < l(wx) then a contradiction
will occur.]

 Let T denote an optimal tree for w 1 ,w 2 ,...,w t in which the
leaves to which w1 and w2 are assigned are brothers.

Let
T

W( T
denote coalescing tree of

T
.

) = W( T  ) + w1 + w2
Let T’ be an optimal tree for w 1  w 2 ,w 3 ,...,w t . Let T be the
tree obtained from T’ be replacing the leaf to which w1 + w2
is assigned by a subtree of w1 and w2.
W(T) = W(T’) + w1 + w2


If W(T) > W( T ), then W(T’) > W( T  ), contradicts to the
assumption.
Q.E.D.
 The problem of constructing an optimal tree for t weights can
be reduced to that of constructing one from t - 1 weights.
6
 Binary Search Trees
 search tree for the keys K1 , K 2 ,..., K n : A binary tree with n
branch nodes and n+1 leaves, the branch nodes are labeled
K 1 , K 2 ,..., K n , and the leaves are labeled K 0 , K1 , K 2 ,..., K n ,
such that, for the branch node with the label K i , its left subtree
contains only vertices with labels K j , j  i , and its right subtree
contains only vertices with only K j , j  i .
 The maximum number of comparisons the search procedure
carried out in the worst case = the height of the corresponding
search tree
 For a given set of n keys, a search tree whose height is
log(n+1) will correspond to the best possible search
procedure.
 Since there always exists a binary tree of height log(n+1)
for any n, the problem is solved.
 Search frequencies
key
K1
frequency u1
K2
u2
…
…
Kn
un
key
<K1 >K1, <K2 … >Kn
frequency w0
w1
…
wn
The total number of comparisons will equal
n
n
j 1
j 0
 u j [ l ( K j ) 1] w j l  ( K j ) ,
where l(Kj) is the path length of the branch node that is labeled
Kj and l’(Kj) is the path length of the leaf that is labeled Kj in the
search tree.
 D. Knuth gave an optimal algorithm for the above problem.
7
Spanning Trees and Cut-sets
 spanning tree of a connected graph : A spanning tree of a
connected graph is a spanning subgraph of the graph which is a
tree.
 branch (or tree edge) : An edge of the graph that is in the tree.
chord/link (or non-tree edge) : An edge of the graph that is not
in the tree.
The set of the chords of a tree is referred to as the complement
of the tree.
 A connected graph always contains a spanning tree.
 cut-set : A (minimal) set of edges in a graph such that the
removal of the set will increase the number of connected
components in the remaining subgraph, whereas the removal of
any proper subset of it will not.
8
 The concepts of spanning trees, circuits, and cut-sets are closely
related.
 fundamental system of circuits : For a given spanning tree,
a unique circuit can be obtained by adding to the spanning
tree each of the chords. The set of e - v + 1 circuits
obtained in this way is called the fundamental system of
circuits relative to the spanning tree.
 A circuit in the fundamental system is called a
fundamental circuit.
 Since a fundamental circuit contains exactly one chord of
the spanning tree, it is referred to as the fundamental
circuit corresponding to the chord.
 For every branch in a spanning tree there is a
corresponding cut-set.
 For a given spanning tree, the set of the v - 1 cut-sets
corresponding to the v - 1 branches of the spanning tree is
called the fundamental system of cut-sets relative to the
spanning tree.
 A cut-set in the fundamental system of cut-sets is called a
fundamental cut-set.
 Since a fundamental cut-set contains exactly one tree
branch, it is referred to as the fundamental cut-set
corresponding to the branch.
9

Theorem 6.1 A circuit and the complement of any spanning
tree must have at least one edge in common.
Proof :
If there is a circuit that has no common edge with the
complement of a spanning tree, the circuit is contained in the
spanning tree.
This is impossible as a tree cannot contain a circuit. Q.E.D.
 Theorem 6.2 A cut-set and any spanning tree must have at
least one edge in common.
Proof :
If there is a cut-set that has no common edge with a spanning
tree, the removal of the cut-set will leave the spanning tree intact,
contradicts to the definition of a cut-set.
Q.E.D.
 Theorem 6.3 Every circuit has an even number of edges in
common with every cut-set.
Proof :
Corresponding to a cut-set, there is a partition of the vertices of
the graph into two subsets.
Therefore, a path connecting two vertices in one subset menu
traverse the edges in the cut-set an even number of items, since a
circuit is a path from some vertex to itself.
Q.E.D.
10
 Theorem 6.4 For a given spanning tree , let
D  e1 , e 2 , e 3 ,..., e k  be a fundamental cut-set in which e 1 is
a branch and e 2 , e 3 ,..., e k are chords of the spanning tree.
Then, e 1 is contained in the fundamental circuits
corresponding to e i for i  2 , 3 ,..., k . Moreover, e 1 is not
contained in any other fundamental circuits.
Proof:
 Let C be the fundamental circuit corresponding to the chord
e 2 . Note that e 2 is in both C and D.
 Since C and D have an even number of edges in common
and e1 is the only other edge that can possibly be in both C
and D, e 1 must be contained in C.
 A similar argument can be applied to the fundamental
circuits corresponding to the edges e 3 , e 4 ,..., e k .
 On the other hand, let C  be the fundamental circuit
corresponding to any chord not in D. C  cannot contain e 1 ,
because otherwise , C  and D will have e 1 as the only
common edge.
Q.E.D.
 Theorem 6.5 For a given spanning tree, let
C  e1 , e 2 , e 3 ,..., e k  be a fundamental circuit in which e 1 is
a chord and e 2 , e 3 ,..., e k are branched of the spanning tree.
Then, e 1 is contained in the fundamental cut-sets corresponding
to e i for i  2 , 3 ,..., k . Moreover, e1 is not contained in any
other fundamental cut-sets.
11
Minimum Spanning Trees
 The weight of a spanning tree is defined to be the sum of the
weights of the branched of the tree.
minimum spanning tree : A minimum spanning tree (MST) is
one with minimum weight.
 Kruskal’s algorithm
 Construct a subgraph of the weighted graph in step-by-step
manner, examining the edges one at a time in increasing
ordering of weights.
 An edge will be added in the partially constructed
subgraph if its inclusion does not yield a circuit, and will
be discarded otherwise.
 The construction terminates when all the edges have been
examined.
 The constructed subgraph is connected and contains no
circuit. It has minimum weight.
12
 Prim’s algorithm
 Among all edges incident with a vertex, the edge with the
smallest weight must be in an MST. [Notice that we
assume that the weights of edges are distinct here.]
 {v1,v2} = the smallest weight edge connecting with v1,
T = the spanning tree not containing {v1,v2}
U = T  {v1,v2}, the fundamental circuit corresponding to
the chord {v1,v2}
U = {v1,v2}  {v1,vx, …, v2}
Removing the edge {v1,vx} yields a spanning tree whose
weight is smaller than that of T.
 Let G’ be the graph obtained from G by coalescing the
vertices v1 and v2 with which the edge e is incident in G,
and T’ be an MST of G’. Let T denote a subgraph of G
consisting of all the edges in T’ together with the edge e, T
is an MST of G.
13

W(T) = W(T’) + w(e)
If T were not a MST of G,  MST
such that

T ,
which contains e

W( T ) < W(T).

Let
T
be the tree obtained from

v2 and removing e.
T

T
by coalescing v1 and
is a MST of G’,

W( T  ) < W(T’)
contradicts to the assumption that T’ is an MST of G’.
Q.E.D.
 Let {v1,v2} be the edge of the smallest weight. The edge
{v1,v2} must be included in an MST, we can coalesce v1
and v2 to obtain G’, and then try to determine the MST of
G’. The step is repeated until G’ becomes a single vertex.
14
Transport Networks
 A weighted directed graph is said to be a transport network if
the following conditions are satisfied:
1. It is connected and contains no loops.
2. There is one and only one vertex in the graph that has no
incoming edge.
3. There is one and only one vertex in the graph that has no
outgoing edge.
4. The weight of each edge is a nonnegative real number.
 source : the vertex that has no incoming edge.
 sink : the vertex that has no outgoing edge.
 capacity : the weight of an edge.
 flow,  : is an assignment of a nonnegative number (i, j) to
each edge (i, j) such that
 (i, j)  w(i, j) for each edge (i, j). [The amount of material to
be shipped through a route cannot exceed the capacity. ]
 all i   i , j   all k   j , k  for each vertex j except
the source a and the sink z . [The amount of material flowing
into a vertex must equal the amount of material flowing out of the
vertex.]
 the value of the flow, v  all i  a , i   all k   k , z 
 saturated of an edge (i, j) : if (i, j) = w(i, j)
unsaturated of an edge (i, j) : if (i,j) < w(i, j)
 maximum flow : A flow that achieves the largest possible value.
15
 cut : A cut-set of the undirected graph, obtained from the
transport network by ignoring the direction of the edges, that
separate the source from the sink .
 The notation  P ,?P  is used to denote a cut that divides the
vertices into two subsets P and P , where the subset P
contains the source and the subset P contains the sink.
 capacity of a cut, w  P ,?P  : the sum of the capacities of
those edges incident from the vertices in P to the vertices
in P ; that is ,
w P ,?P   w i , j  .


i P , j P

 Theorem 6.6
The value of any flow in a given transport network is less than
or equal to the capacity of any cut in the network.
Proof :
Let  be a flow and  P ,?P  be a cut.
For the source a,
  a , i     j , a     a , i   v .
alli

allj

alli

For a vertex p  a in P,
 p , i    j , p   0
alli

allj

16
We have



p
,
i


j
,
p




 


p P  alli
allj

   p , i     j , p 
v 
p P ; alli


p P ; allj
p P ; i P
 p , i  

p P ; i P
 p , i 


     j , p      j , p 
p P ; j P
 p P ; j P

Note that
  p , i     j , p  .
p P ; i P
Thus , (*) becomes
v  
p P ; j P
p P ; i P
But, since

p P ; j P
v 
(*)
 p , i  

p P ; j P
 j , p 
  j , p   0, we have
   p , i   w  p , i   w  P , P  .
p P ; i P
(**)
Q.E.D.
p P ; j P
 Eq. (**): For any cut  P ,?P  , the value of a flow equal the sum
of flows in the edges from the vertices in P to the vertices in P
minus the sum of flows in the edges from the vertices in P to
the vertices in P.
17
Constructing a Maximum Flow in a Transport
Network
 Iterated improvement: For any cut  P ,?P  , the values of a flow
in a transport network equal the sum of flows in the edges from
the vertices in P to the vertices in P minus the sum of flows in
the edges from the vertices in P to the vertices in P.
 Whenever we can construct a flow  the value of which is equal
to the capacity of some cut, we can be certain that  is a
maximum flow.
 Labeling procedure:
1. The source a is labeled (-, ). It means that (out from
nowhere) the source can supply an infinite amount of
material to the other vertices.
2. A vertex b that is adjacent from a is labeled (a+, b),
where b is equal to w(a, b) - (a, b), if w(a, b) > (a, b);
it is not labeled
if w(a, b) = (a, b).
3. [forward] Those vertices that are adjacent to or from the
labeled vertices are scanned. A vertex q that is adjacent
from b is labeled (b+, q), where q is equal to the min{b,
w(b, q) - (b, q)},
if w(b, q) >
(b, q).
It is not labeled
if w(b, q) = (b, q).
The total incoming flow into q from b can be increased by
q.
18
4. [backward] A vertex q is labeled (b-, q), where q is
equal to the min{b, (b, q)},
if
(b, q) > 0.
It is not labeled
if (b, q) = 0.
The total incoming flow from q to b can be decreased by
q.
5. The vertex q might be adjacent to or from more than one
labeled vertex.
 If we repeat the procedure of labeling the vertices that are
adjacent to or from the labeled vertices, on of the following two
cases shall arise:
1. Sink z is labeled, say, (y+, z). Increase the flow in (y, z) by z.
Notice that y must be labeled either (q+, y) or, (q-, y) with
y  z from some q.
If y is labeled (q+, y), increase the flow in (q, y) by z.
If y is labeled (q-, y), decrease the flow in (y, q) by z. So the
increment from y to z is compensated.
2. Sink z is not labeled. Denote all the labeled vertices by P and
all the unlabeled vertices by P . The fact that sink z is not
labeled means that the flow in each of the edges incident from
the vertices in P to the vertices in P is equal to the capacity
of that edge, and that the flow in each of the edges incident
from the vertices in P to the vertices in P is equal to zero.
We have thus obtained a flow, the value of which is equal to
the capacity of the cut  P ,?P  . By (**), we find a maximum
flow.
19
Intersection Graphs
 hereditary property
 intersection graph : Let F be a family of nonempty set. The
intersection graph of F is obtained by representing each set in F
by a vertex and connecting two vertices by an edge if and only
if their corresponding sets intersect.
 Marczewski, 1945 : When F is allowed to be an arbitrary
family of sets, the class of graphs obtained as intersection
graphs is simply all undirected graphs
 The intersection graph is called the interval graph if
F is a family of intervals on a linearly ordered set.
 unit interval graph
 proper interval graph
 Roberts, 1969 : The classes of unit interval graph
and proper interval graph coincide.
 circular-arc graph
proper circular-arc graph : not only is no arc properly
contained in another but also no pair of arcs together cover the
entire circle.
 permutation graph : A permutation diagram consists of n points
on two parallel lines and n straight line segments matching the
points. The intersection graph of the line segments is called a
permutation graph.
 triangulated graph property : Every simple cycle of length
strictly greater than 3 posses a chord.
20
triangulated graph (chordal graph)
 G is a triangulated graph if and only if G is the intersection
graph of a family of subtrees of a tree.
 transitive orientation property : Each edge can be assigned a
one-way direction in such a way that the resulting oriented
graph (V,F) satisfies the following condition:
abF and bcF imply acF (a,b,c  V).
comparability graph
 Three typical graphs
Property C:
G is a comparability graph
G is a comparability graph
Property C :
Property T:
G is a triangulated graph
Property T :
G is a triangulated graph
Interval graph
 T+ C
permutation graph
 C+ C
split graph
 T+ T
21