 
Summary: COMPLEXITY: Exercise No. 4
due in two weeks
1. a. (Test 99)Is the following problem NPcomplete?
Instance: A graph G = (V; E), a positive integer K.
Question: Does G contain an independent set of size K and a clique of size K?
b. Is the problem NPcomplete when the K = 100?
c. Is the problem NPcomplete when the K = jV j=2?
2. Show that INDEPENDENT SET remains NPcomplete even if the input graph has no clique
of size 3. (Test 95)
3. Are the following problems NPcomplete or polynomial? (prove)
CONNECTED DOMINATING SET:
Instance: A graph G = (V; E), positive integer K.
Question: Does G contain dominating set S with at most K vertices such that the subgraph
of G induced by S (i.e., the graph G S = (S; E \ S S)) is connected ?
MINIMUM LEAF SPANNING TREE:
Instance: A graph G = (V; E), a positive integer K.
Question: Is there a spanning tree for G in which the number of leaves is at most K ?
MAXIMUM LEAF SPANNING TREE: (Test 95)
Instance: A graph G, a positive integer K.
Question: Is there a spanning tree for G such that the number of leaves in the tree is at least
