A maximum independent set (mis) is a maximum cardinality subset of V such that there is no edge between any two vertices of it.

In particular, for any two members of a minimum connected dominating set (mcds), there are at most seven independent vertices in maximum independent set, that is |mis| [less than or equal to] 3.

Note that the cardinality of any maximal independent set is smaller than or equal to the maximum independent set, so the result holds.

We show that the maximum nonoverlapping dense blocks problem is NP-complete by a reduction from the maximum independent set problem on cubic planar graphs.

In Section 2, we define the problem formally and investigate its relation to the maximum independent set problem.

In this sense, the MNS problem is related to the maximum independent set (MIS) problem, which is defined as finding a maximum cardinality subset of vertices I of a graph G such that no two vertices in I are adjacent.

A maximum independent set on G (A, m, n) gives a maximum number of nonoverlapping blocks in A.

The graphs in this class are exactly the intersection graphs of the m x n blocks in a matrix, thus finding a maximum independent set of a graph in this class is equivalent to solving the MNDB problem of the corresponding dense matrix blocks.

Thus, a maximum cardinality subset of nonoverlapping blocks in matrix A corresponds to a maximum independent set in G [member of] [XT.

The NP-completeness proof in the next section uses a reduction from the maximum independent set (MIS) problem on cubic planar graphs and adopts orthogonal drawings.

The size of the maximum independent set in G is equal to the size of the maximum independent set in G'.

If vertex v is in a maximum independent set I, then none of its neighbors are in I.