Total edge geodetic number of a graph

Abstract

An edge geodetic set $S \subseteq V$ is said to be a total edge geodetic set if the subgraph induced by S has no isolated vertices. The minimum cardinality of a total edge geodetic set of G is the total edge geodetic number and is denoted by $g_{1t}(G)$. A total edge geodetic set of cardinality $g_{1t}(G)$ is called $g_{1t}(G)$-set. In this paper, total edge geodetic number $[g_{1t}(G)]$ of graph is introduced. We concentrate on the study of how $g_{1t}(G)$ changes under various operations on $G$. Finally, we study $g_{1t}(G)$ of some special graph.