... 4-cycle containing both v1 and v2or containing both v3 and v4.Proof. Suppose, to the contrary, that there is a 4-cycle containing both v1 and v2orcontaining both v3 and v4. By symmetry, ... Hvcontains an edge {v1,v2,v5} containing v1 and v2 ,and an edge {v3,v4,v5} containing v3 and v4. Since the edges N(vi), 1 ≤ i ≤ 4, are deletedfrom HGwhen constructing H, ... is a matching if no two edges in M are incidentto the same vertex. The matching number of G is the maximum cardinality of amatching of G.AsetS of vertices in G is a total dominating set of...