Revisiting the vertex cover of graphs

A set U of vertices of graph G is a vertex cover of G if every edge in G is incident with a vertex in U. The minimum cardinality of such set is the vertex covering number of G and is denoted by (G). This paper characterizes bipartite graphs in terms of vertex cover. It provides the vertex covering number of (i) the complement of a nonempty bipartite graph, (ii) the powers of paths and cycles, and (iii) one supergraph of planar grid.