Question 12: Recall that in the vertex cover problem, we are trying t…

Recall that in the vertex cover problem, we are trying to find a smallest subset $S$ of vertices in a graph $G$ such that $G \setminus S$ is an independent set; in other words, for every edges $(u, v)$ in the graph $G$, we have that $\{u, v\} \cap S$ is non-empty.

Consider the algorithm that repeats the following steps as long as the graph has at least one edge available: select a vertex of maximum degree (say $w$), and include $w$ in the solution; thereafter, delete $w$ and all edges incident to $w$.

Based on the above data, answer the given subquestions.

Recall that the size of an optimal vertex cover is at most twice the size of a maximum matching of a graph G. Are there examples of graphs where the size of the optimal vertex cover is exactly equal to twice the size of its maximum matching? For example, this is not true for a path on three vertices: the size of both the maximum matching and minimum vertex cover is 1.