Editorial for IOI '12 P2 - Parachute Rings

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A suboptimal solution covering all but the last subtask considers the following conditions:

if there is a vertex $V$ ~V~ of degree $\ge 4$ ~\ge 4~, no other vertex can be critical (because removing $V$ ~V~ still leaves one or more vertices of degree $\ge 3$ ~\ge 3~); so if there is more than one vertex of degree $\ge 4$ ~\ge 4~, there are no critical vertices;
if there is a vertex $V$ ~V~ of degree $3$ ~3~, each critical vertex is either $V$ ~V~ or one of its neighbors;
if there is a cycle, all critical vertices lie on the cycle;
if the graph is linear (a set of disjoint paths), all of its vertices are critical.

These checks can be easily extended to the dynamic case of the last subtask: the only nontrivial check is keeping track of cycle formation, which can be dealt with using suitable data structures (union-find d.s., etc.).

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