DIRECCION TALLER DE SERVICIOS PÚBLICOS
INFRAESTRUCTURA JAQUE, COMPLEJO PORTAL DE LA YUNGAS Y POLIDEPORTIVOS
Let V denote the set of virtual links4 in G
m. The goal is to prove the number of
linearly independent paths in Y is ||Gm|| − |V|, and these linearly independent paths
are sufficient to identify all tree links in the original graph G.
For all redundant paths generated by mapping from C to Y , they can be divided into 3 categories, i.e., 1) trivial topology, 2) linearly dependent paths, and 3) du- plicate paths. Note in Fig. E.8–Fig. E.14, for all paths (or path segments) colored as blue/green/red, they represent virtual links if they are outside G; and real simple paths otherwise.
1) Trivial topology (a graph with no nodes or links). As shown in Fig. E.8,
combining the red and green paths w.r.t. m1 (Gm involves rm+1m1m+2r), cycle
rm+1m1m+2r is formed. However, the corresponding path of rm+1m1m+2r after re-
moving the virtual links and the resulting isolated nodes is empty. Therefore, this trivial path has no contribution for identifying real links in G.
2)Linearly dependent paths. The following 3 cases can generate redundant lin-
early dependent paths.
(2.i). Consider the three cycles constructed w.r.t. m2 (shown in Fig.E.9). The
4G
G
m2
r
mi mj
e1 e2
Figure E.9: Path construction w.r.t. m2.
G m1 r m2 mi e1 ma e2 m+1 m 2 +
Figure E.10: Path construction w.r.t. ma.
paths associated with blue + red cycle and blue + green cycle are mie1m2 and
m2e2mj, respectively. However, the path associated red+green cycle, mie1m2e2mj,
is the sum of previously constructed paths mie1m2 and m2e2mj. Therefore, redun-
dant path mie1m2e2mj is linearly dependent with the other two.
(2.ii). Now consider the case shown in Fig.E.10. For Gm, there exists one and
only one monitor ma with link m+2ma colored as blue in the m+2 → ma direction,
according to the rules for blue tree construction. Now consider ma. According to
the s-t numbering rule and the processing of ear decomposition, we have f(m+ 2) <
f (ma) and g(m+2) < g(ma); therefore, m+2ma is colored5 as green in the other
direction. Then m2e1ma and mie2ma are two paths obtained from green + blue
cycle and green + red cycle w.r.t. ma, respectively. However, the path generated by
the blue + red cycle is the sum of m2e1maand mie2ma, thus redundant in Y .
5Note m
a might have more than one neighbor w with f(w) < f(ma) and g(w) < g(ma), in which case we can still choose m+
G m1 r m2 mb e1 mj e2 m1+ m+2
Figure E.11: Path construction w.r.t. mb.
G m1 r m2 ma e1 e 2 m1+ m2 +
Figure E.12: Duplicate paths w.r.t. m1 and m+2.
(2.iii). Analogously, there exists one and only one monitor mb with link m+1mb
colored as blue in the m+
1 → mb direction (see Fig. E.11). Based on the non-
separating ear decomposition, we have f(m+
1) > f (mb) and g(m+1) < g(mb); there-
fore, m+
1mb is colored6as red in the other direction. Then mbe1m2 and mbe2mj are
two paths obtained from red + blue cycle and red + green cycle w.r.t. mb, respec-
tively. However, the path generated by the blue + green cycle is the sum of mbe1m2
and mbe2mj, thus not providing new information for link identifications in G.
3) Duplicate paths. Each of the following 3 cases can generate the same path
twice.
(3.i). We have considered combining the red and green paths w.r.t. m1. Now
consider the paths associated with the blue + red cycle and blue + green cycle. As Fig.E.12 displays, the paths associated with these two cycles are exactly the same,
6Note m
bmight have more than one neighbor u with f(u) > f(ma) and g(u) < g(ma), in which case we can still choose m+
G m1 r m2 mb e1 m+1 m 2 +
Figure E.13: Duplicate paths w.r.t. m+ 1. G m1 r m2 mb e1 ma e2 mc mr1 mrN mgN mg1 1 2 m+1 m 2 +
Figure E.14: Other virtual links connecting to m+
1 or m+2.
i.e., path m1e1m2 is generated twice.
(3.ii). Following the similar argument, the paths associated with the blue + red cycle and blue + green cycle w.r.t. m+
2 are also the same, i.e., path m2e2ma(shown
in Fig. E.12) is generated twice when mapping from C to Y . Note ma cannot be
the same as m2 since m2 only appears in the last ear. Therefore, path m2e2ma is
non-trivial.
(3.iii). Similar to (3.i) and (3.ii), the paths associated with the blue + red cycle and blue + green cycle w.r.t. m+
1 are also the same, i.e., mbe1m2 as shown in
Fig. E.13 appears twice when mapping from C to Y . In addition, mbe1m2 is non-
trivial since m2has the highest ear level which means mb 6= m2.
4) We have discussed 7 cases in 1)–3), each case providing a redundant path
virtual links in Gm is also 7, i.e., rm+1, rm+2, m+1m1, m+2m1, m+2ma (Fig. E.10),
and m+
1mb (Fig. E.11). However, in addition to these 7 virtual links, there might
exist other virtual links in Gm. As Fig. E.14 shows, suppose there are other N1
virtual links (colored as green and no color on the other direction as m+
1mb has
been colored as blue) connecting m+
2 and mgi (i = 1, · · · , N1) and N2 virtual links
(colored as red and no color on the other direction as m+
1mbhas been colored as blue)
connecting m+
1 and mri (i = 1, · · · , N2). For the 3 paths constructed w.r.t. each
node in {mg1, · · · , mgN1, mr1, · · · , mrN2}, there exists one path which is linearly
dependent with the other two. To prove this claim, consider mg1as an example. Two
paths7m
2e1mg1and mce2mg1(see Fig.E.14) can be obtained from the green + blue
cycle and green + red cycle w.r.t. mg1. However, the path associated with the
blue + red cycle w.r.t. mg1 is the sum of m2e1mg1 and mce2mg1, thus linearly
dependent with the other two. The same argument can be applied to other nodes in {mg1, · · · , mgN1, mr1, · · · , mrN2}. Therefore, we can identify another N1 + N2
linearly dependent paths in Y regarding these N1+ N2virtual links.
In sum, 1)–4) cover all the possible cases of redundant paths when mapping from C to Y and the number of these redundant paths is 7 + N1 + N2, which is
also the number of virtual links (denoted by |V|) in Gm. Therefore, removing these
redundant paths, the cardinality of the resulting path set Y′is ||G
m||−(7+N1+N2) =
||Gm|| − |V|.
Now we can explore if Y′ is sufficient to identify all tree links in G∗
ex. Recall
that each path (for identifying tree links) constructed by STPC consists of one or two path segments, each path segment terminating at the first monitor it encounters. If we let each segment terminate at the last monitor before traversing a virtual link and combine any two segments w.r.t. real node v (v can be either monitor or non- monitor), then a new measurement path set J is formed. It is easy to show (using
7Note m
STLI) that this new path set J is sufficient to identify all tree links in G∗
ex. Recall
the mapping process from cycle set C to path set Y . Each path in Y is obtained by removing the virtual links and all resulting isolated nodes of the corresponding cycle in C. Then each path PJ in J can be obtained by applying the same operation to the
corresponding cycle CJ (CJ ⊃ PJ) in C; therefore, each path in J must be involved
as one element in Y . Accordingly, Y is sufficient to identify all tree links in G∗ ex.
Since Y′ is obtained by removing redundant paths in the procedure of mapping C
to Y , then Y′ is also sufficient to identify all tree links in G∗
ex, i.e., the measurement
matrices associated with J, Y and Y′ have the same column rank.
In sum, for graph Gm with ||Gm|| − |V| real links, the property of Y′ is that it
contains exactly ||Gm|| − |V| paths, with each path containing only tree links in Gex∗ .
Therefore, the measurement matrix associated with Y′ is a square matrix with full
rank.