The Edge Geodesic Transversal Problem

For a given connected graph $G(V,E)$, the geodesic transversal problem finds a set S of vertices of G with minimum cardinality such that every maximal geodesic of G contains at least one vertex of S. The cardinality of such a smallest set S is called the gt-number of G and is denoted as $gt(G)$. In this paper, we introduce the Edge geodesic transversal problem as the edge-based extension of the geodesic transversal problem. The problem of determining a set S⊆ E(G) with minimum number of elements such that every maximal geodesic P in G traverses at least one edge of S is called the edge geodesic transversal problem, and the cardinality of such a set is called the edge geodesic transversal number or the gtₑ-number of G, denoted as gtₑ(G). Further, we investigate the effect of an edge cut of a graph on the edge geodesic transversal problem and prove a novel result called the edge cut transversal theorem which is used in the computation of edge geodesic transversal number of hypercube, stacked prism and ladder graph.