* VERTEX COVER: find a

minimum vertex cover (of the edges) of a graph G with c vertices and m edges;

Fortunately, a maximal match can be constructed by a simple greedy algorithm and its vertices are a vertex cover with cardinality C [less than or equal to] 2*(VC) where VC is the cardinality of a

minimum vertex cover [19].

Let [V*.sub.c] be a minimum vertex cover of G, and let P* be a minimum monochromatic clique partition of H.

On the other hand, we can obtain a monochromatic clique partition P of H from a minimum vertex cover [V*.sub.c] of G, as follows.

We show this by an approximation factor preserving reduction from

MINIMUM VERTEX COVER.

Let S = {[v.sub.1], [v.sub.2], ..., [v[alpha].sub.0]} be a

minimum vertex cover of G.

Finding a maximum independent set is equivalent to finding a maximum clique or a

minimum vertex cover of a graph, and all three problems are NP-hard [Garey and Johnson 1979].

Given a graph G, let us consider a

minimum vertex cover [C.sup.*] ([absolute value of [C.sup.*]] = [Tau](G)) and the corresponding maximum independent set [S.sup.*].

Consequently, computing the minimum connected tropical subgraph in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] determines the

minimum Vertex Cover of G.

Note that Lemma 2 does not imply that finding a

minimum vertex cover is in P if a minimum twin-cover is provided--in fact, a

minimum vertex cover may not intersect with a minimum twin-cover at all (e.g., in a complete graph with one missing edge).

Note that Lemma 2 does not imply that finding a

minimum vertex cover is in P if a minimum twin-cover is pro[v.sub.i]ded - in fact, a

minimum vertex cover may not intersect with a minimum twin-cover at all (e.g., in a complete graph with one missing edge).

Note that the maximum independent set and

minimum vertex cover problems are complementary.