a graph with no cycles
acyclic graph
adjacency matrix of a for a digraph G = (V,A), its adjacency matrix is the n x n matrix A = (aij)
where aij = 1 if there is an edge from vertex i to vertex j and aij = 0
digraph
otherwise
adjacency matrix of a for a graph G = (V,E), its adjacency matrix is the n x n matrix A = (aij)
graph
where aij = 1 if there is an edge between vertex i and vertex j and aij = 0
otherwise
algorithm
an algorithm to solve a problem, in an informal sense, is a finite sequence
of instructions to obtain an output for a given input of the problem
alphabet
an alphabet is a finite set L the elements of which are symbols or letters
average-case
complexity of a
problem
let h(x) be the probability that an instance x of the problem is taken as a
candidate for input. Then the average-case complexity is the sum of all
terms h(x) · wA(x) where x is an instance of size n
binary search tree
assign keys to the vertices of a binary tree T such that the key of a vertex is
(1) larger than the keys of the vertices in its left subtree and (2) smaller
than the keys of the vertices in its right subtree. A binary tree with keys
defined by this rule is called a binary search tree
binary tree
a rooted tree is a binary tree if the outdegree of each intermediate vertex is
at most two
a digraph G = (V,A) where V can be partitioned into two sets V' and V''
such that each arc in A is an arc from a vertex in V' to a vertex in V''
a graph G = (V,E) where V can be partitioned into two sets V' and V'' such
that each edge is an edge between a vertex in V' and a vertex in V''
an edge in a connected graph such if this edge but not its vertices are
removed from the graph, this graph becomes disconnected
bipartite digraph
bipartite graph
bridge
bridge
a disconnecting set consisting of exactly one edge
chords of a tree
if T = (V,E') is a spanning tree in G = (V,E), the edges in G not in E' are
called the cords of T
chromatic number of
a graph
circuit
class co-NP problem
the smallest number k such that the graph is k-colourable
a closed path in which all the edges are distinct
a decision problem belongs to class co-NP if its complementary problem is
in class NP
class NP problem
a decision problem belongs to class NP if there is a polynomial algorithm
to verify the „yes“ output of that problem. The acronym NP stands for
„nondeterministic-polynomial“.
class P problem
a decision problem belongs to class P if there is a polynomial algorithm to
solve every instance of the problem
clique
closed directed path
closed path
coloured graph
a complete subgraph in a graph G
a directed path from a vertex to itself
a path between a vertex and itself
a graph is said to be coloured if each of its vertices is assigned a colour
such that no two adjacent vertices have the same colour
complete bipartite
graph
complete digraph
complete graph
component
connected graph
cutset of a graph
a simple bipartite graph G = (V',V'',E) with an edge between every vertex
in V' and every vertex in V''. Such a graph is then denoted by Kp,q if there
are p vertices in V' and q vertices in V''.
a digraph whose underlying graph is complete
a simple graph with n vertices with an edge between every pair of vertices.
Such graph is the denoted by Kn.
a component C of a graph G = (V,E) is the subgraph induced by any class
of the equivalence E defined as follows: for two vertices v and w in V we
have v, wE if there is a path between v and w
a graph is connected if there is a path between every pair of vertices in it
a disconnecting set D of a graph G is called a cutset of G if no proper
subset of D is a disconnecting subset of G
cycle
decision problem
a circuit in which all the vertices are distinct
a problem p as a subset of L* x L* is called a decision problem if whenever
(z,y) is in p, then y is the empty string
degree of a vertex
depth first search
spanning tree of a
graph
the number of edges incident on that vertex
the acyclic subgraph consisting of the n vertices of the graph and the n-1
edges resulting from a depth first search algorithm
descendant of a vertex vertex v is a descendant of vertex u if there is a directed path from u to v
a structure G = (V,A) where V is a finite set of vertices and A is a finite set
digraph
of arcs such that each arc a in A is associated with an ordered pair of
vertices v and w. We write e = (v,w) and say that (1) a is an arc from v to
w, (2) vertex v is adjacent to vertex w, (3) vertex w is adjacent from vertex
v, (4) arc a is incident from v and (5) arc e is incident to w. Two vertices
are adjacent if there is an arc from one to the other.
a closed directed path with all its arcs distinct
directed circuit
a closed directed path with all its vertices distinct
directed cycle
directed Hamiltonian a closed directed Hamiltonian cycle
cycle
directed Hamiltonian a directed path in a digraph from a vertex to another vertex in which each
vertex occurs exactly once
path
a digraph with a real number assigned to each arc
directed network
a directed path from a vertex v to a vertex w in a digraph G = (V,A) is an
directed path
alternating finite sequence of vertices and arcs of the form
v
,
a
,
v
,
a
,
v
,
a
,

,
va
,r
,
v
112
23
3
r

1

1
r
where v1 = v, vr = w and ai is an arc
1

i

r
from vi to vi+1 for
directed tree
disconnected graph
disconnecting set of a
graph
a digraph is called a directed tree if its underlying graph is a tree
a graph that is not connected
A subset D of edges of a graph G = (V,E) is called a disconnecting set of G
if the deletion of the edges in D from G makes G into a disconnected graph
empty string
the string of length 0 denoted by Φ
empty word
the word of length 0 denoted by Φ
Eulerian circuit
Eulerian graph
Eulerian path
even vertex
exponential algorithm
a closed Eulerian path
a graph that has an Eulerian circuit
a path in a graph in which every edge of the graph exactly once
a vertex with an even degree
an algorithm whose complexity violates all polynomial bounds
exterior face of a
plane graph
f = O(g)
face of a plane graph
a face whose region is unbounded
let f and g be two functions from the set of natural numbers to the set of
non-negative real numbers. If there is a positive constant c and a natural
number n0 such that f(n) ≤ c · g(n) for all n ≥ n0, we write f = O(g)
a two-dimensional region defined by edges in a plane graph, the various
vertices and edges then define the boundaries of these faces
fixed-length character codes in which each character in a set is assigned a unique fixed-length
binary number
codes
a graph with no cycles
forest
a regular binary tree is full if all its leaves are at the same level
full regular binary
tree
function of higher
order
let c = lim [f(n)/g(n)] as n goes to plus infinity. We say that g is of higher
order than f if c = 0
function of lower
order
let c = lim [f(n)/g(n)] as n goes to plus infinity. We say that f is of lower
order than g if c = 0
fundamental cutset of
a spanning tree
if T is any spanning tree in G = (V,E), the deletion of any edge in T makes
T disconnected by creating two subtrees with vertex sets W and W' such
that D(W,W') is a cutset of G [this is a corollary of theorem 7.1.1]. Thus,
corresponding to each edge e of a spanning tree T, there is a unique cutset
DT(e) called the fundamental cutset of T with respect to the edge e.
fundamental cycle
If e is a chord in a graph G joining the vertices u and v, the edges of the
unique path in a spanning tree T between u and v together with the edge e
form a unique cycle in G which is called the fundamental cycle of G
relative to T with respect to the chord e and is denoted by CT(e)
graph
a structure G = (V,E) consisting of a finite set V of vertices (nodes) and a
finite set of edges such that each edge e is associated with a pair of vertices
v and w. We write e = {v,w}and say that (1) e is an edge between v and w,
(2) e is an edge incident on both v and w, (3) e joins the vertices v and w .
In this case both v and w are adjacent vertices and they are incident on e.
given a computer program with its input, will it ever halt?
halting problem
Hamiltonian cycle
Hamiltonian graph
Hamiltonian path
Hamiltonianconnected graph
height of a tree
Hilbert's tenth
problem
a closed Hamiltonian path
a graph that has a Hamiltonian cycle
a path between two vertices in a graph in which each vertex occurs exactly
once
a graph in which there is a Hamiltonian path between every pair of vertices
the highest level occurring in the tree
given a polynomial equation in more variables with integer coefficients,
does it have any integer solutions?
homeomorphic graphs two graphs are said to be homeomorphic or identical to each other within
vertices of degree 2) if they both can be obtained from the same graph G
by introducing new vertices of degree 2 on its edges
incidence matrix of a for a digraph G = (V,A) where V = {1,2,…,n}and A = {a1,a2,…,am}, its
incidence matrix is the n x m matrix B = (bik) where rows represent the set
digraph
of all vertices and columns the set of all arcs, bik = –1 if ak is incident from
i, bik = 1 if ak is incident to i, and bik = 0 otherwise
incidence matrix of a
graph
indegree of a vertex
induced subgraph
input of a problem
instance of a problem
for a graph G = (V,E) where V = {1,2,…,n}and E = {e1,e2,…,em}, its
incidence matrix is the n x m matrix B = (bik) where the rows represent the
set of vertices and the columns the set of all edges and bik = 1 ek if is
incident on i and bik = 0 otherwise
in a digraph, it is the number of arcs incident to that vertex
if, for a graph G = (V,E), W is any subset of V, the subgraph of G induced
by W is the graph H =(W,F) where f is an edge in F if f = {u,v}where f is in
E and both u and v are in W.
for any problem p we have the corresponding metamathematical problem:
given a string z in L*, find a string y in L* such that (z,y) is in p or report
that no such string exists. Here the string z is called an input of the problem
for any problem p we have the corresponding metamathematical problem:
given a string z in L*, find a string y in L* such that (z,y) is in p or report
that no such string exists. Here the string z is called an instance of the
problem
interior face of a plane a face whose region is bounded
graph
intermediate vertex
a vertex in a rooted tree that is not a terminal vertex
internal vertex
intractable problem
a vertex in a rooted tree that is not a terminal vertex
a problem for which no polynomial algorithm is known and for which is
conjectured that no such algorithm exists is called an intractable problem
k-colourable graph
a graph that can be coloured using k colours
key of a vertex
suppose that T is a binary tree. To each vertex v of this tree, a real number
k(v) is assigned. This number k(v) is called the key of the vertex
a path with k edges
the set of all strings or words from L
k-path
L*
labelled tree
a tree with n vertices is called a labelled tree if each vertex is assigned a
unique label i where i is a positive integer between 1 and n
leaf
a vertex in a rooted tree whose outdegree is 0
left descendant
if T is a binary tree and if the outdegree of a vertex is 2, then v has two
immediate descendants - a left descendant and a right descendant. (If the
outdegree of v is 1, then the unique descendant may be considered as
either the left descendant or the right descendant)
left subtree of a vertex the tree rooted at the left immediate descendant of a vertex is called the left
subtree of this vertex
the number of symbols in the string
length of a string
length of a word
the number of symbols in the word
level of a vertex
linearly equivalent
functions
the number of arcs in the path from the root to the vertex
two functions f and g are linearly equivalent if f = O(g) and
we write f >< g. The relation >< is an equivalence relation
loop
metamathematical
problem
an edge joining a vertex to itself
for any problem p we have the corresponding metamathematical problem:
given a string z in L*, find a string y in L* such that (z,y) is in p or report
that no such string exists
g = O(f)
minimal spanning tree a spanning tree T in G is a minimal spanning tree (MST) if the weight of T
does not exceed the weight of any other spanning tree in G
mixed graph
multigraph
multiple edges
network
NPC
in G = (V,E) at least one element is an arc and at least element is an edge
a graph with more than one edges joining vertices
several edges that join the same vertices
a graph with a real number assigned to each edge
the class of NP complete problems is denoted by NPC
NP-complete problem a problem in NP that is NP-hard is said to be NP-complete
NP-hard problem
a decision problem p is NP-hard if every problem in NP can be transformed
into it polynomially. In other words an NP-hard problem cannot be easier
than any problem in NP
number of basic
operations needed
if A is an algorithm to solve (every instance of) a problem and if x is an
instance of the problem, the number of basic operations needed to solve x
using A is denoted by wA(x)
odd vertex
orientation
outdegree of a vertex
output of a problem
a vertex with an odd degree
a digraph G' obtained from a graph G by changing each edge of G into an
arc
in a digraph, it is the number of arcs incident from that vertex
for any problem p we have the corresponding metamathematical problem:
given a string z in L*, find a string y in L* such that (z,y) is in p or report
that no such string exists. Here the string y is called the output of the
problem.
path between two
vertices
a path between two vertices v and w in a graph G = (V,E) is an alternating
finite sequence of vertices and edges of the form
v
,,
e
v
,
e
,
v
,
e
,

,
vev
,r
,rwhere v1 = v, vr = w and ei is an edge
112233
r

1

1
between vi and vi+1 for 1  i  r
planar graph
plane graph
polynomial algorithm
a graph that can be drawn so that no two edges intersect except at a vertex
a graph drawn on a plane so that no two edges intersect except at a vertex
an algorithm to solve a problem p is called a polynomial algorithm if its
worst-case complexity fA(n) is O(nk) for some fixed positive integer k
polynomially
transformable
problems
a decision problem p is polynomially transformable to a decision problem q
if the following two conditions hold: (1) there exists a function f(x) that
will transform every instance of x of p to an instance of q such that the
answer to x is „yes“ if and only if the answer to f(x) is „yes“ and (2) there is
a polynomial algorithm to compute f(x) for every x
polynomial-time
algorithm
an algorithm to solve a problem p is called a polynomial-time algorithm if
its worst-case complexity fA(n) is O(nk) for some fixed positive integer k
prefix
a word w (in a finite alphabet) is a prefix of another word v if v = wp where
p is another word
prefix code
a character code is a prefix code if it has the prefix property
prefix property
problem
a character code is said to have the prefix property if no code for a
character is a prefix of the ode for another character
a problem p is a subset of L* x L*
provably difficult
problem
a problem is provably difficult if can be proved that any algorithm which
will solve (every instance of) the problem is an exponential algorithm
rate of growth of a
function
given a relation ><, the rate of growth of a function f is its equivalence
class which may be represented by a canonical member from that class
reachability matrix of for a digraph with n vertices, it is an n x m matrix R = (rij) where rij is 1 if
there is a directed path from i to j and 0 otherwise
a graph
regular binary tree
a rooted tree is a regular binary tree if the outdegree of each intermediate
vertex is exactly two
right descendant
if T is a binary tree and if the outdegree of a vertex is 2, then v has two
immediate descendants - a left descendant and a right descendant. (If the
outdegree of v is 1, then the unique descendant may be considered as either
the left descendant or the right descendant)
right subtree of a
vertex
the tree rooted at the right immediate descendant of a vertex is called the
right subtree of this vertex
rooted tree
a directed tree is a rooted tree if (1) there is exactly one vertex (called a
root) with indegree 0, and (2) the indegree of every vertex is 1.
a graph without loops or multiple edges
a simple path is a path with its vertices distinct
for any problem p we have the corresponding metamathematical problem:
given a string z in L*, find a string y in L* such that (z,y) is in p or report
that no such string exists. Here the string y is called the solution of the
problem.
simple graph
simple path
solution of a problem
spanning tree
string
strong component
strong component
strong orientation
strongly connected
digraph
strongly connected
pair
strongly orientable
graph
subgraph
a subgraph T of a graph with n vertices is a spanning tree in the graph if (a)
T is a tree and (b) T has n vertices
an ordered finite sequence of symbols from an alphabet L
a strong component C of a digraph G = (V,E) is the subgraph induced by
any class of the equivalence E defined as follows: for any two vertices v
and w in V that are a strongly connected pair, we have v, wE
an orientation of a graph that is strongly connected
a digraph in which every pair of vertices is strongly connected
a pair of vertices is strongly connected if each is connected to the other
a graph that has a strong orientation
a graph G' = (V',E') is a subgraph of G = (V,E) if V' is a subset of V and E'
is a subset of E.
subtree at a vertex
if T is a rooted tree, a subtree T' at vertex v is a rooted tree T' = (V',E') such
that (1) the root of T' is at v, (2) V' consists of v and all its descendants, and
(3) E' contains all the arcs of all the directed paths in T form v to all the
leaves
terminal vertex
tournament
tree
a vertex in a rooted tree whose outdegree is 0
a simple digraph in which for every pair of vertices v and w either there is
an arc from v to w or from w to v but not both
a connected forest
tree
a tree is a connected graph with no cycles
tree pruned at vertex
v
a tree is said to have been pruned at vertex v if all the descendants of
vertex v have been removed along with al the arcs of all the directed paths
from v so that v becomes a leaf of the pruned tree
undecidable problem
a problem is undecidable if there is no algorithm that will solve every
instance of the problem
underlying graph
unilaterally connected
digraph
unilaterally connected
pair
a digraph where each arc is treated as an edge
a digraph in which every pair of vertices is unilaterally connected
variable-length
character codes
vertex connected from
vertex
vertex connected to
vertex
weakly connected
graph
weight of a spanning
tree
codes in which each character in a set is assigned a unique variable-length
binary number
if there is a directed path from vertex v to vertex w then w is connected
from v
if there is a directed path from vertex v to vertex w then v is connected to w
a pair of vertices is unilaterally connected if one of them is connected to
the other
a digraph whose underlying graph is connected
if we associate a real number (called the weight) with each edge of a graph
G o that G becomes a network, the weight of a spanning tree T in G is then
the sum of the weights of all the edges in T
weighted digraph
weighted graph
well-characterized
problem
a digraph with a real number assigned to each arc
a graph with a real number assigned to each edge
a decision problem is said to be well-characterized if it is both NP and coNP
word
an ordered finite sequence of symbols from an alphabet L
worst-case complexity the worst-case complexity of the algorithm A to solve a problem p is
defined to be fA(n) where fA(n) = Max {wA(x) | x is an instance of p and size
of a problem
of x is n}