Past exam of the computer science course of the University of Oxford / 2023 / Combinatorial optimisation / 3 / d / ii / Solution

First test for an isolated vertex, reporting infeasibility if one exists. Otherwise compute a maximum cardinality matching $M$ using a general-graph polynomial-time algorithm, then add one arbitrary incident edge for each unmatched vertex in a matching. The proof in (i) shows that the resulting edge cover has minimum cardinality $|V|-|M|$.

The final scan takes $O(|V|+|E|)$ time with adjacency lists; the maximum cardinality matching computation is polynomial. One may obtain it from a general-graph maximum-weight matching routine with every weight equal to one.^[ref]