The Geodetic vertex covering number of a graph

Abstract

A subset $S$ of vertices in a connected graph $G$ of order at least two is called a geodetic vertex cover if $S$ is both a geodetic set and a vertex covering set. The minimum cardinality of a geodetic vertex cover is the geodetic vertex covering number of $G$ denoted by $g_\alpha(G)$. Any geodetic vertex cover of cardinality $g_\alpha(G)$ is a $g_\alpha$ - set of $G$. Some general properties satisfied by geodetic vertex covering number of a graph are studied. The geodetic vertex covering number of several classes of graphs are determined. Some bounds for $g_\alpha(G)$ are obtained and the graphs attaining these bounds are characterized. A few realization results are given for the parameter $g_\alpha(G)$.