On the edge-to-vertex geodetic number of a graph

Abstract

Let G D .V; E/ be a connected graph with at least three vertices. For vertices u and v in G; the distance d.u; v/ is the length of a shortest u v path in G: A u v path of length d.u; v/ is called a u v geodesic. For subsets A and B of V; the distance d.A; B/; is defined as d.A; B/ D mi n fd.x; y/ W x 2 A; y 2 Bg. A u v path of length d.A; B/ is called an A B geodesic joining the sets A; B Â V; where u 2 A and v 2 B: A vertex x is said to lie on an A B geodesic if x is a vertex of an A B geodesic. A set S Â E is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S: The edge-to-vertex geodetic number g ev .G/ of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-to-vertex geodetic set of cardinality g ev .G/ is an edge-to-vertex geodetic basis of G: Any edge-to-vertex geodetic basis is also called a g ev-set of G: It is shown that if G is a connected graph of size q and diameter d; then g ev .G/ Ä q d C 2: It is proved that, for a tree T with q 2; g ev .T / D q d C 2 if and only if T is a caterpillar. For positive integers r; d and l 2 with r Ä d Ä 2r; there exists a connected graph G with rad G D r; d iam G D d and g ev .G/ D l: Also graphs G for which g ev .G/ D q; q 1 or q 2 are characterized.

The Edge-to-Vertex Geodetic Number of a Graph

“…There are interesting applications of these concepts to the problem of designing the route for a shuttle and communication network design. We further extend these concepts to the edge set of G and present several interesting results in [10].…”

“…7 g are the only g ev -sets of G so that every g ev -set contains the edge v 1 v 2 : Hence the edge v 1 v 2 is the unique edge-to-vertex geodetic edge of G: The following theorems from [10] are used in the sequel. Theorem 1.…”

The edge-to-edge geodetic domination number of a graph

“…An u-v path of length d(A, B) is called an A-B geodesic joining the sets A and B, where u ∈ A and v ∈ B. A vertex x is said to lie on an A-B geodesic if x is a vertex of an A-B geodesic [21]. A set S ⊆ E(G) is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The edge-to-vertex geodetic number g ev (G) of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-tovertex geodetic set of cardinality g ev (G) is called an g ev -set of G. The edge-to-vertex geodetic number of a graph was studied in [21,24,25,27].…”

“…A vertex x is said to lie on an A-B geodesic if x is a vertex of an A-B geodesic [21]. A set S ⊆ E(G) is called an edge-to-vertex geodetic set if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The edge-to-vertex geodetic number g ev (G) of G is the minimum cardinality of its edge-to-vertex geodetic sets and any edge-tovertex geodetic set of cardinality g ev (G) is called an g ev -set of G. The edge-to-vertex geodetic number of a graph was studied in [21,24,25,27]. A set S ⊆ E(G) is called an edge-to-edge geodetic set of G if every edge of G is an element of S or lies on a geodesic joining a pair of edges of S. The edge-to-edge geodetic number g ee (G) of G is the minimum cardinality of its edge-to-edge geodetic sets and any edge-to-edge geodetic set of cardinality g ee (G) is said to be a g ee -set of G. This concept was studied in [1].…”

“…Then the problem is equivalent to finding the size of a minimum independent edge dominating set of G. The edge-to-vertex geodetic concepts have many applications in location theory and convexity theory. There are interesting applications of these concepts to the problem of designing the route for a shuttle and communication network design [21]. In the case of designing the route for a shuttle, although all the vertices are covered by the shuttle when considering edgeto-vertex geodetic sets, some of the edges may be left out.…”