We will use the following notation for We will use the following notation for independence and covering problems independence and covering problems

(G) : maximum size of independent set maximum size of independent set

(G) : maximum size of matching maximum size of matching

(G) : minimum size of vertex cover minimum size of vertex cover

(G) : minimum size of edge cover minimum size

please prove that:

(1)In a graph G, In a graph G, SV(G) is an independent set if and is an independent set if and only if only if S is a vertex cover, and hence is a vertex cover, and hence (G) + (G) = n(G)

(2)If G has no isolated vertices, then If G has no isolated vertices, then (G) + (G) = n(G)

(3)If G is a bipartite graph with no isolated vertices, If G is a bipartite graph with no isolated vertices, then (G) = (G) z (max independent set = min edge cover) (max independent set = min edge cover)

(Hint for part ii: use a maximum matching, then construct an cover by adding edges to cover unsaturated vertices arbitrarily. The size gives a bound on '(G) in terms of n and '(G). Then consider a minimum edge cover and show that it is a disjoint union of stars (complete bipartite graphs K1,t). Take an edge from each of these stars to form a matching. This gives a bound on '(G) in terms of n and '(G).