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Some Results on Strong Edge Geodetic Problem in Graphs Communications in Mathematics and Applications Vol. 11, No. 3, pp. 403–413, 2020 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v11i3.1385 Research Article Some Results on Strong Edge Geodetic Problem in Graphs D. Antony Xavier , Deepa Mathew* , Santiagu Theresal and Eddith Sarah Varghese Department of Mathematics, Loyola College (University of Madras), Chennai, India *Corresponding author: [email protected] Abstract. For a graph G(V(G),E(G)), the problem to find a S V(G) where every edge of the graph ⊆ G is covered by a unique fixed geodesic between the pair of vertices in S is called the strong edge geodetic problem and the cardinality of the smallest such S is the strong edge geodetic number of G. In this paper the strong edge geodetic problem for product graphs are studied and also some results for general graphs are derived. Keywords. Strong edge geodetic number; Strong geodetic number; Edge geodetic number; Geodetic set MSC. 05C12; 68Q17; 94C15 Received: April 4, 2020 Accepted: June 2, 2020 Copyright © 2020 D. Antony Xavier, Deepa Mathew, Santiagu Theresal and Eddith Sarah Varghese. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Consider a graph G(V(G),E(G)), with order V(G) and size E(G) . An (x y) geodesic is the j j j j ¡ 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 [1]. In [2] the geodetic number of a graph is defined as follows, 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] S I[u,v]. The set S of vertices of G is called a geodetic set of G, if I[S] V . A geodetic Æ u,v S Æ set of minimum2 cardinality is called minimum geodetic set of G. The cardinality of the minimum 404 Some Results on Strong Edge Geodetic Problem in Graphs: D. A. Xavier et al. 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 g1(G) [13]. Strong geodetic problem is a variation of geodetic problem and is defined in [11] as follows. For a graph G(V (G),E(G)), given a set S V(G), for each pair of vertices (x, y) S, x y, let ge(x, y) be a selected fixed shortest ⊆ ⊆ 6Æ S path between x and y. Let Ie(S) {ge(x, y): x, y S} and V(Ie(S)) V(Pe). If V(Ie(S)) V Æ 2 Æ Pe Ie(S) Æ 2 for some Ie(S), then S is called a strong geodetic set. The cardinality of the minimum strong geodetic set is the strong geodetic number of G and is denoted by sg(G). The edge version of the strong geodetic problem is defined in [10] i.e. for a graph G(V (G),E(G)), a set S V (G) is called ⊆ a strong edge geodetic set if for any pair x, y S a shortest path Pxy can be assigned such that 2 S E(P ) E(G). The cardinality of the smallest strong edge geodetic set of G is called {x,y} ( ) xy [ 2 2 Æ the strong edge geodetic number and is denoted as sge(G). 2. Strong Edge Geodetic Number of Graphs Theorem 1. For a graph G(V(G),E(G)) with diameter d 2, the strong edge geodetic number, ¸ l d pd2 8d E m sge(G) Å Å j j . ¸ 2d Proof. Consider a graph G(V(G),E(G)) with diameter d 2 and assume that the set S V(G) ¸ ⊆ ¡ S ¢ forms a minimum strong edge geodetic set for G. Then the edges of G are covered with j2j ¡ S ¢ geodesics and each geodesic covers at most d edges. Thus E j j d. Considering only the j j · 2 l d pd2 8d E m positive and the integer roots, the inequality reduces to S Å Å j j . j j ¸ 2d The above bound attains equality when G Pn (n 2). Æ» ¸ Theorem 2. For a graph G with order n 2, sge(G) 2 if and only if G Pn. ¸ Æ Æ» Proof. For Pn v0v1 ...vn, the geodesic between the vertices vo and vn covers all the edges of Æ Pn and thus sge(Pn) 2. Conversely let G be a graph with sge(G) 2 and let S {u,v} be the Æ Æ Æ minimum strong edge geodetic set of G. Then all the edges of G lies on the u v geodesic. Thus ¡ G Pn. Æ» A graph is geodetic if for every pair for vertices the shortest path between them is unique. An undirected graph G (V,E) is said to be geodetic, if between any pair of vertices x, y V (G) Æ 2 there is a unique shortest path [5]. Theorem 3. For a connected non-complete geodetic graph G on n vertices with minimum cutset of independent vertices C V(G), sge(G) n c where C c. ½ · ¡ j j Æ Proof. Consider a connected non-complete geodetic graph G with minimum cutset of independent vertices C {c1, c2,... cc}, such that the V(G)\C splits G into at least two Æ components. It is clear that every vertex in C is adjacent to at least one vertex in each component of G\C. Let S V(G)\C. Consider an edge xy E(G). This edge xy can be of two forms. Æ 2 Communications in Mathematics and Applications, Vol. 11, No. 3, pp. 403–413, 2020 Some Results on Strong Edge Geodetic Problem in Graphs: D. A. Xavier et al. 405 Case i: Suppose xy is an edge in any one component of G\C. Then xy itself forms a unique fixed geodesic joining x and y, where {x, y} S. 2 Case ii: Suppose xy is an edge such that y C and x S. Since G is a geodetic graph, there 2 2 exists a vertex z G\C such that the edge xy lies on a unique fixed geodesic xyz. It is 2 clear that the vertices x and z lies on two different components of G\C. Thus S forms a strong edge geodetic set of G and therefore sge(G) n c. · ¡ Theorem 4 ([13]). For a graph G of order n and exactly one vertex of degree (n 1), g1(G) n 1. ¡ Æ ¡ Theorem 5. For a graph G of order n and exactly one vertex of degree (n 1), sge(G) n 1. ¡ Æ ¡ Proof. For any connected graph G, g1(G) sge(G) and from Theorem 4, sge(G) n 1. Let · ¸ ¡ v V(G) be the unique vertex of degree n 1. Let vx E(G). This implies that there exists at 2 ¡ 2 least at least one vertex x0 where x and x0 are not adjacent. Let Q V (G)\{v}. Clearly, the edge Æ vx lies on the unique fixed geodesic xvx0. Also, any edge of G which are not incident with v lies on a fixed geodesic between the vertices of Q. Thus sge(G) n 1 Hence for a graph G of order · ¡ n and exactly one vertex of degree (n 1), sge(G) n 1. ¡ Æ ¡ The converse of the above theorem need not be true. For G C4, sgeG 3, but G does not have Æ Æ any vertex of degree 3. Corollary 1. For a graph G of order n 3 with a cut vertex of degree n 1, sge(G) n 1. ¸ ¡ Æ ¡ Proof. Consider a connected graph G of order n 3 with a cut vertex x of degree n 1. ¸ ¡ This implies that x is the only vertex of degree n 1. Then it follows from Theorem 5 that ¡ sge(G) n 1. Æ ¡ Theorem 6. For a split graph G(V,E) with complete set K, maximum stable set T, sge(G) 2 2 ¸ 2 E s1 2s2 3s1 2s1s2 j jÅ Å Å Å , where s1 T and s2 denotes number of simplicial vertices in complete 2(2s1 s2) Æ j j set K. Å Proof. Let S be a minimum strong edge geodetic set of G and let A be set of simplicial vertices in K with A s2. Also, let B be set of non simplicial vertices in S, B s3. Clearly, T S, j j Æ j j Æ ⊆ A S and S s1 s2 s3. ⊆ j j Æ Å Å The strong edge geodetic set contains three components including the stable set, the set of simplicial vertices in K and the set of non-simplicial vertices. It is clear that s1 and s2 are fixed, hence to get the minimum strong edge geodetic set, s3 should be minimized.
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