Discrete Mathematics, Spring 2004
Homework 9 Sample Solutions
7.1 #36.
A vertex
v
in a tree
T
is a
center
for
T
if the eccentricity of
v
is minimal;
that is, if the maximum length of a simple path starting from
v
is less than or equal to the
maximum length of a simple path starting from any other vertex
w
. Show that a tree has
either one or two centers.
Solution
.
Let
P
be a simple path of maximal length in
T
.
(Such a path surely exists;
in fact, we have seen this construction before, when establishing equivalent characterizations
of trees.) The length of this path is either even or odd. In the first case, denote this length
by 2
m
; then we have
P
= (
v
0
, v
1
, . . . , v
m
−
1
, v
m
, v
m
+1
, . . . , v
2
m
−
1
, v
2
m
)
,
where all of the
v
i
are distinct. We claim that
v
m
is the unique center for
T
.
To see this, notice first that the eccentricity of
v
m
is actually equal to
m
(clearly it must be
at least
m
). If it were not, then there would exist a simple path (
v
m
=
w
0
, w
1
, . . . , w
k
=
w
)
with
k > m
. Since trees are acyclic, there are only two possibilities:
•
w
j
=
v
m
+
j
for all
j
≤
some nonnegative integer
j
0
, and
w
j
/
∈ {
v
0
, . . . , v
2
m
}
for all
j > j
0
, or
•
w
j
=
v
m
−
j
for all
j
≤
j
0
, and
w
j
/
∈ {
v
0
, . . . , v
2
m
}
for all
j > j
0
.
In the first case, the path
P
′
= (
v
0
, v
1
, . . . , v
m
=
w
0
, w
1
, . . . , w
k
=
w
) would be a simple
path of length greater than that of
P
, contradicting the way in which we chose
P
. In the
second case, the path
P
′′
= (
w
=
w
k
, . . . , w
1
, w
0
=
v
m
, v
m
−
1
, . . . , v
0
) would be a simple path
of length greater than that of
P
, again a contradiction.
Hence the eccentricity of
v
m
is equal to
m
.
Now consider any of the remaining vertices
u
in
T
.
If
u
is one of the
v
i
, then its eccentricity must be
> m
, for if
u
=
v
i
with
i
≤
m
then (
v
i
, v
i
+1
, . . . , v
2
m
) is a simple path of length
> m
, and if
u
=
v
i
with
i > m
then
(
v
i
, v
i
−
1
, . . . , v
0
) is a simple path of length
> m
. If
u
is not one of the
v
i