ez WT 
ieQ 
for some Q C {1,...,k},Q # 0. The cycles of the 
basis will be called basic cycles. 
The number k above is constant for all cycle basis 
and is called the cyclomatic number of G. If the graph 
has p connected components the cyclomatic number 
equals #(E) — #(V) + p. It is denoted by v(G). 
2.4 Hypergraphs 
If V is à a finite non empty set and F, E C P(V), a 
family of non empty subsets of V such that 
U E; — V, 
E;cE 
then the couple H, H — (V, E), is called a hypergraph. 
The elements of V are referred to as the vertices of the 
hypergraph. The edges or hyperedges are the elements 
of E. 
2.5 Matroids 
In 1935, H.Whitney introduced the matroid concept 
in order to investigate linear independence in an ab- 
stract way. Let À = {a1,...,an} be a finite set and 
F C P(A). (A, F) is defined to be a matroid on A if, 
and only if, 
L.'FeF = FA, 
fail € T,for4 —1,...,n, 
Fecf,FZzOGPCF:-Ircf, 
ow N 
and for each S C A, the members of F that are 
maximal in S have the same cardinality. 
The elements of A are called the elements of the ma- 
troid (A, F) and the elements of F are called the in- 
dependent sets of the matroid. Maximal independent 
sets are called matroid bases and minimal dependent 
sets are called circuits. 
It is very easy to show [3][Chapter 2] that, if 
C = {H € E(G) : H is a cycle}, 
and 
F = {I € P(C) : I is an independent set of cycles}, 
then (C, F) is à matroid on C whose bases contain 
v(G) elements. 
If a weight w(c) for each element c of C is defined 
—for instance, w(c) could be taken as /(c); the length 
616 
of c in G— then the weight w(B) of a basis B, B € F, 
is defined as 
w(B) — 3 ^ w(o). 
ceB 
For an elementary introduction to the subject 
see [2]. 
2.6 NP-completeness 
The class NP contains the problems solvable by non- 
deterministic polynomial algorithms. The subclass 
NP-complete of NP contains the hardest problems of 
NP in the sense that a solution for a NP-complete 
problem is a solution to any other problem in NP 
through a transformation by a polynomial time algo- 
rithm. 
A nondeterministic algorithm is an algorithm that 
at each step has several choices for the next step. A 
polynomial or polynomial time algorithm is an algo- 
rithm that gives a result after a number of steps which 
is bounded by a polynomial function. 
3 NETWORK DISCRETE MODELS 
In [4] the practical convenience that network discrete 
models be available is discussed in connection with 
the many algorithms of a discrete nature involved in 
the software; the conclusion is that comprehensive 
discrete network models and their corresponding dis- 
crete software modules are missing concepts in our 
systems. 
A widely long since accepted abstraction is the 
correspondence between a network NE and a graph 
G(NE): to each unknown parameter group? a graph 
vertex is associated; there is an edge connecting 
two vertices if their corresponding parameters are in- 
volved in a same observation (see [6]). 
It is also known how an undirected graph G(N) 
describes the zero-nonzero sparsity pattern of a block 
symmetric matrix and that G(NE) — G(N) if N is 
the normal equations matrix of NE. 
This, however, represents only a part of the prob- 
lem. 
To improve the situation consider the net- 
work adjustment [sparse] design matrix A = 
(Qij)1<i<m,1<j<n and the hypergraph H such that 
#(V(H)) = n, and a numbering p of the vertices 
  
3A parameter group stands for a set of parameters related 
in the obvious way. In a photogrammetric block, the three 
coordinates of a point constitute a parameter group, also the 
six orientation elements of an image; the set of selfcalibration 
parameters constitutes as well another parameter group. 
is 
foi 
or 
th 
for 
ple 
lig 
po. 
CT 
tro 
th: 
dir 
the 
Opt 
pli 
so, 
els