$2$-Path Geodetic Vertex Covering of a Graph

Abstract

A vertex cover \( S \subseteq V(G) \) is called a \textit{$2$-path geodetic vertex cover} of \( G \) if for every \( v \in V(G) \setminus S \), there exist vertices \( u, w \in S \) such that $d_G(u,w) = 2$ and \( v \in I_G(u, w) \). The \textit{$2$-path geodetic vertex covering number} of \( G \), denoted \( \beta_{2pg}(G) \), 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 $2\leq a\leq b,$ there exist a connected graph $G$ such that $\beta(G)=a$ and $\beta_{2pg}=b.$ 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.