Math 6a  Solution Set 4
Problem # 1
Liu 4.10
A set of vertices in an undirected graph is said to be a
dominating set
if every vertex not
in the set is adjacent to one or more vertices in the set.
A
minimal dominating set
is a
dominating set such that no proper subset of it is also a dominating set.
(a) For the graph in Fig. 4P.2 find two minimal dominating sets of different sizes.
(b) Let the vertices of a graph represent cities and the edges represent communication links
between cities. Give a physical interpretation of a dominating set in this case.
(c) Let the 64 squares of a chessboard be represented by 64 vertices. Let there be an edge
between two vertices if their corresponding squares are in the same row, same column, or
same (backward or forward) diagonal.
It is known that five queens can be placed on the
chessboard so that they will dominate all 64 squares. Moreover, five is the minimum number
of queens that is needed. Restate this result in graph theoretic terms.
Solution.
(a) First we label the top vertex of the graph
a,
the four vertices below it, from left to
right,
b, c, d, e,
the vertex immediately below them
f
and the two lowest vertices, from left
to right,
g
and
h.
Then, it is readily seen that the sets
{
b, e, f
}
and
{
a, f
}
are both minimal
dominating sets of different sizes.
(b) A dominating set in this case is a collection of cities such that any city is in direct
communication with at least one of the cities in the collection.
(c) Note that a queen placed on square
a
dominates square
b
precisely when
a
is adjacent to
b.
Thus, the placement of five queens such that they dominate the entire board is equivalent
to a dominating set of size 5 in the graph. The result also shows that any such dominating
set will be minimal, and in fact
five
is the minimum size of any minimal dominating set.
Problem # 2
Liu 4.11
A set of vertices in an undirected graph is said to be an
independent set
if no two vertices
in it are adjacent. A
maximal independent set
is an independent set which will no longer be
one when any vertex is added to the set.
(a) For the graph in Fig. 4P.2 find two maximal independent sets of different sizes.
(b) How can the problem of placing eight queens on a chessboard so that no one captures
another be stated in graph theoretic terms?
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Solution.
(a) Note that the sets
{
b, e, f
}
and
{
a, f
}
found in the previous problem are also independent.
Thus, as they are dominating sets, any other vertex in the graph is adjacent to one in either
set, so it follows that they are maximally independent, again of different sizes.
(b) Two queens are in a position where neither captures the other if and only if the squares
they are on are not adjacent.
Thus, the problem of finding eight queens such than none
captures the other is equivalent to finding eight squares, none of which are adjacent, which
is the same as finding an independent set of size eight.
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 Fall '07
 Wilson
 Math, Vertex, Order theory, Partially ordered set, minimal dominating sets

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