STRONG DOUBLY EDGE GEODETIC PROBLEM IN GRAPHS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. For a graph G(V (G),E(G), the problem to find a set S ⊆ V (G) where every edge in E(G) is covered by least two fixed geodesics between the vertices in S is called the strong doubly edge geodetic problem and the cardinality of the smallest such S is the strong doubly edge geodetic number of G. In this paper the computational complexity for strong doubly edge geodetic problem is studied and also some bounds for general graphs are derived.


INTRODUCTION
Consider a graph G(V (G), E(G)), with order |V (G)| and size |E(G)|. An (x − y) geodesic is the length of the shortest path between the vertices x and y. For a graph G, the length of the maximum geodesic is called the graph diameter, denoted as diam(G). Harary et al introduced a graph theoretical parameter in [2] called the geodetic number of a graph and it was further studied in [3]. Let I [u, v] be the set of all vertices lying on some u − v geodesic of G, and for some non empty subset S of V (G), I[S] = ∪ u,v∈S I [u, v]. The set S of vertices of G is called a geodetic set of G, if I[S] = V . A geodetic set of minimum cardinality is called minimum geodetic set of G. The cardinality of the minimum geodetic set of G is the geodetic number g(G) of G. The geodetic set decision problem is NP-complete [12]. The set S ⊆ V (G) is an edge geodetic cover of G if every edge of G is contained in the geodesic between some pair of vertices in S, and the cardinality of minimum edge geodetic cover is called the edge geodetic number of G denoted as g 1 (G) [13].
Strong geodetic problem is a variation of geodetic problem and is defined in [10] as follows.
The cardinality of the minimum strong geodetic set is the strong geodetic number of G and is denoted by sg(G). The strong geodetic problem was later studied in [4] [15].The edge version of the strong geodetic problem is defined in [11] i.e. a set S ⊆ V (G) is called a strong edge geodetic set if for any pair x, y ∈ S a shortest path P xy can be assigned such that The cardinality of the smallest strong edge geodetic set of G is called the strong edge geodetic number and is denoted as sg e (G).
In [16] another variant of geodetic problem named doubly geodetic problem is introduced and is defined as follows: For any graph G(V, E) two geodesics g p (x, y) and g q (u, v) are distinct if

MOTIVATION
Smarandache geometries and Smarandache multispaces are sigficant topics in Mathematics.
In graph theory the study of Smarandache path k− cover is initiated in [1]. Also, in [14] Smarandache edge geodetic set is defined. A set T ⊆ V (G) is a Smarandache edge geodetic set if each edge in E(G) lies on at least two geodesic between the vertices in T .
In [11] an urban road network problem is modelled into a graph where the vertices represent the bus stops or junctions and the edges represent the roads connecting them, subjected to the condition that a road is a geodesic and it is patrolled by a pair of road inspectors by placing one inspector at each end. Also, one pair of road inspectors is not assigned to more than one road segment. The strong edge geodetic problem is to identify the minimum number of road inspectors to patrol the urban road network i.e. each edge is patrolled by at least one pair of inspectors. Assume that the inspectors are working under the following condition. In order to guard and secure a road(geodesic), there should necessarily be a communication between the inspectors who are guarding that particular road(geodesic). Suppose there is a communication problem or other network related issues between a pair of inspectors then the edges in the fixed geodesic patrolled by that particular pair of inspectors are unsecured. To avoid this situation we can arrange at least two pair of inspectors for an edge. In this arrangement, even if one pair of inspector had lost the communication, then the other pair of inspectors can guard that particular edge. With this motivation the strong doubly edge geodetic set can be defined.

STRONG DOUBLY EDGE GEODETIC NUMBER
A set S ⊆ V (G) is called a strong doubly edge geodetic set of if each edge e ∈ E(G) lies on at least two distinct fixed geodesics between vertices of S. The strong doubly edge geodetic number sdg e (G) is the minimum cardinality of a strong doubly edge geodetic set.
For the graph given in Figure 1: the set {b, f , g} forms a minimum strong edge geodetic set whereas the {a, b, c, f , g} is a strong doubly edge geodetic set. Thus sg e (G) = 3 and sdg e (G) =

5.
A graph G which has a strong doubly edge geodetic set is called a strong doubly edge geodetic graph. The complete graph K n and the graph (K n − e) are not strong doubly edge geodetic graphs.
A graph may have strong doubly geodetic set but not strong doubly edge geodetic set. For example, complete graph K n is a strong doubly geodetic graph but not a strong doubly edge geodetic graph.
Throughout this paper, we assume G to be a strong doubly edge geodetic graph.

COMPUTATIONAL COMPLEXITY
The proof for the NP-completeness of the strong doubly edge geodetic problem for general graphs can be reduced from the vertex cover problem which is already proved to be NPcomplete.
Theorem 1. Strong doubly edge geodetic problem is NP-complete for general graphs.
Proof. The graphḠ(V ,Ē) is constructed from a given graph G(V, E) as follows : The vertex set V = {a, b} ∪V ∪V ∪V where V induces a clique inḠ and V is an independent set of order forms a set of simplicial vertices and they are the elements of any strong doubly edge geodetic set inḠ. Let T be a vertex cover set of G. We will prove that T ∪ X forms a strong doubly edge geodetic set set forḠ. For v ∈ T and u ∈ N(v), the geodesics are the fixed geodesics between the vertices of T ∪ X and they will cover all the edges inĒ(Ḡ) at least twice. Thus T ∪ X forms a strong doubly edge geodetic set forḠ.
Conversely, assume that A is a strong doubly edge geodetic set ofḠ. This bound is sharp for the graphs in Figure 3.  Two edges e, f ∈ E(G) are geodesic if they belong to some shortest path of G and are otherwise called non-geodesic edges [11]. Proof. Let the components in G − A be G 1 , G 2 , . . . , G a and S be the strong doubly edge geodetic set of G. Also let S i be the strong doubly edge geodetic set of G i and S i ∩ S j = φ where i, j ∈ . From this we get,|S| ≥ a a 2|A| ( a 2 ) . On solving this inequality and considering only the integral part, |S| ≥ a a 4|A| a(a−1) .
If a = 2 in the above theorem then the set A is a convex edge cut and hence the following theorem.
Corollary 1. For a graph G with convex edge cut A, sdg e (G) ≥ 2 2|A| .
This bound is sharp for Glued binary trees without randomization GT 1 (r) and GT 2 (r). The glued binary tree GT 2 (r) is obtained by adding cross edges to each leaves of GT 1 (r) (Refer Figure 4 and Figure 5). For GT 1 (r), A = 2 r and it can be easily verified that sdg e (GT 1 (r)) = 2 √ 2 r+1 . Similarly for GT 2 (r), A = 2 r+1 and it can be easily verified that sdg e (GT 2 (r)) = 4 √ 2 r .   Clearly, the set {w 1 , w 2 , . . . , w a−2 , u r+1 , v d−r } forms a strong doubly edge geodetic set for G.
Thus the graph G is obtained where sdg e (G) = a, radius(G) = G and diameter(G) = d.  Proof. Let G be a graph obtained from the path P 3 : p 1 , u 1 , p 3 and adding new vertices u 2 , u 3 , . . . , u a−1

2
, v 1 , v 2 , . . . , v b−a in such a way that each u i , 2 ≤ i ≤ a−1 2 and v j , 1 ≤ j ≤ b − a are joined to the vertices p 1 and p 3 respectively. Also, join the vertices w 1 , w 2 , . . . w a−1 to the vertex p 1 (Refer Figure: 7). It is straightforward to see that S = {w 1 , w 2 , . . . , w a−1 , p 3 } forms a doubly edge geodetic set for G. But S does-not form a strong doubly edge geodetic set for G.
It can be easily seen that T = S ∪ {v 1 , v 2 , . . . , v b−a } forms a strong doubly edge geodetic set for G. Thusdg e (G) = a and sdg e (G) = b − a + a − 1 + 1 = b.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.