Abstract
For a connected graph G = (V, E) of order n ≥ 2, a set \({S\subseteq V}\) is a 2-edge geodetic set of G if each edge \({e\in E - E(S)}\) lies on a u-v geodesic with d(u, v) = 2 for some vertices u and v in S. The minimum cardinality of a 2-edge geodetic set in G is the 2-edge geodetic number of G, denoted by eg2(G). It is proved that for any connected graph G, β1(G) ≤ eg2(G), where β1(G) is the matching number of G. It is shown that every pair a, b of integers with 2 ≤ a ≤ b is realizable as the matching number and 2-edge geodetic number, respectively, of some connected graph. We determine bounds for the 2-edge geodetic number of Cartesian product of graphs. Also we determine the 2-edge geodetic number of certain classes of Cartesian product graphs. The 2-edge geodetic number of join of two graphs is obtained in terms of the 2-edge geodetic number of the factor graphs. Open image in new window
Highlights
By a graph G = (V, E), we mean a finite undirected connected graph without loops or multiple edges
The radius rad G and diameter diam G are defined by rad G = min{e(v) : v ∈ V } and diam G = max{e(v) : v ∈ V }, respectively
A set S of vertices is an edge geodetic set of a graph G if each edge of G lies on a geodesic between two vertices from S, and the minimum cardinality of an edge geodetic set is the edge geodetic number eg(G)
Summary
A set S ⊆ V is called a k-geodetic set of G if each vertex in V − S lies on a k-geodesic of vertices in S. The Cartesian product of two graphs G and H , denoted by G H , has the vertex set V (G) × V (H ), where two distinct vertices (x1, y1) and (x2, y2) are adjacent if and only if either x1 = x2 and y1 y2 ∈ E(H ), or y1 = y2 and x1x2 ∈ E(G). Theorem 1.1 [11] Each extreme vertex of a connected graph G belongs to every edge geodetic set of G. Theorem 1.2 [12] For an integer k ≥ 1, each k-edge geodetic set of a connected graph G with at least two vertices contains every k-extreme vertex of G. The projections πG(S) and πH (S) are edge geodetic sets of G and H , respectively
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
