(untitled) (European Journal of Pure and Applied Mathematics, 2025)
A vertex cover \( S V(G) \) is called a 2-path geodetic vertex cover of \( G \) if for every \( v V(G) S \), there exist vertices \( u, w S \) such that d_G(u,w) = 2 and \( v I_G(u, w) \). The 2-path geodetic vertex covering number of \( G \), denoted \( β_2pg(G) \), is the minimum cardinality of a 2-path geodetic vertex covering of \( G \). In this paper, we show that given two positive integer a and b such that 2≤ a≤ b, there exist a connected graph G such that β(G)=a and β_2pg=b. As a consequence, the difference between the 2-path geodetic vertex covering number and the classical vertex covering number of a graph can be made arbitrarily large. We characterize graphs with small and large values of the 2-path closure absorbing vertex covering number. Furthermore, we provide necessary and sufficient conditions for the 2-path geodetic vertex covers in certain graph operations. The exact values of 2-path geodetic vertex cover numbers of these graphs are also determined.