An algorithm is used to obtain simple proofs of these two known relations in the theory of matched graphs; A graph with a unique 1-factor contains a matched bridge; an n-connected graph with a 1-factor has at least n totally covered vertices, for n≥2. The proof of the second result is extended to show some totally covered vertex lies within a distance of 2 from at least n-1 others, for n≥3.
Gabow, Harold N., "Algorithmic Proofs of Two Relations Between Connectivity and the 1-Factors of a Graph ; CU-CS-094-76" (1976). Computer Science Technical Reports. 92.