For any graph G = [V E}. let 0(0) denote the number of connected components of G. having an odd number of vertices. Then we have the following: Tutte's Theorem: A graph G = (V, E) has a perfect matching if and only if for every subset S of V we have 0(G S) 3 ISI. If G does not have a perfect matching. then a set S such that 0(G S) :> S is called a atter set. If Sis a Tutte set for Gbut no proper subset of Sis a Tutte set. then Sis called a minimal Tutte set for G. A vertex '1.' E V(G) is a ctcw center of G if and only if i; is the center of an induced K13 in G: i.e.. some three independent vertices of G which are all adjacent to c. A graph is factor critical if the removal of every vertex leaves a graph which admits a perfect matching. A connected graph G is said to be latconnected if it has more than 1:: vertices and remains connected whenever fewer than k vertices are removed. [a] Let S be a minimal Tutte set for a connected graph G of even order [number of vertices] that has no perfect matching. Let ISI = k. and 0K: S} = t. Then show that i. t 2 k + 2 and. ii. each vertex in S is adjacent to vertices in at least t kl 1 distinct odd components of G S (Hint: use minimality). [b] Use the results of item a] to show the following: If G is a connected graph of even order that does not have a perfect matching and if S is a minimal Tutte set for G. then every element of Sis a claw center. (c) Let G be a clawfree graph. Say if each one of the following statements is correct or not. and explain briefly why. i. Every clawfree graph admits a perfect matching. ii. Every clawfree graph of even order admits a perfect matching. iii. Every connected clawfree graph of even order admits a perfect matching. iv. Every clawfree graph of odd order is factorcritical. v. Every connected clawfree graph of odd order is factorcritical. vi. Every 2connected clawfree graph of odd order is factorcritical