COMPLEXITY: Exercise No. 4 due in two weeks Summary: COMPLEXITY: Exercise No. 4 due in two weeks 1. a. (Test 99)Is the following problem NP-complete? 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 NP-complete when the K = 100? c. Is the problem NP-complete when the K = jV j=2? 2. Show that INDEPENDENT SET remains NP-complete even if the input graph has no clique of size 3. (Test 95) 3. Are the following problems NP-complete 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 Collections: Computer Technologies and Information Sciences