Minimal doubly resolving sets and the strong metric dimension of some convex polytopes q Jozef Kratica a,⇑ , Vera Kovac ˇevic ´ -Vujc ˇic ´ b , Mirjana C ˇ angalovic ´ b , Milica Stojanovic ´ b a Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36/III, 11000 Belgrade, Serbia b Faculty of Organizational Sciences, University of Belgrade, Jove Ilic ´a 154, 11000 Belgrade, Serbia article info Keywords: Minimal doubly resolving set Strong metric dimension Convex polytopes abstract In this paper we consider two similar optimization problems on graphs: the strong metric dimension problem and the problem of determining minimal doubly resolving sets. We prove some properties of strong resolving sets and give an integer linear programming for- mulation of the strong metric dimension problem. These results are used to derive explicit expressions in terms of the dimension n, for the strong metric dimension of two classes of convex polytopes D n and T n . On the other hand, we prove that minimal doubly resolving sets of D n and T n have constant cardinality for n > 7. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The metric dimension problem (MDP), introduced independently by Slater [1] and Harary and Melter [2], has been widely investigated [3–8]. It arises in many diverse areas including network discovery and verification [3], geographical routing pro- tocols [9], the robot navigation, connected joints in graphs, chemistry, etc. Given a simple connected undirected graph G ¼ðV ; EÞ, where V ¼f1; 2; ... ; ng; jEj¼ m; dðu; v Þ denotes the distance between vertices u and v, i.e. the length of a shortest u  v path. A vertex x of the graph G is said to resolve two vertices u and v of G if dðu; xÞ – dðv ; xÞ. A vertex set B ¼fx 1 ; x 2 ; ... ; x k g of G is a resolving set of G if every two distinct vertices of G are resolved by some vertex of B. Given a vertex t, the k-tuple rðt; BÞ¼ðdðt; x 1 Þ; dðt; x 2 Þ; ... ; dðt; x k ÞÞ is called the vector of metric coordinates of t with respect to B.A metric basis of G is a resolving set of the minimum cardinality. The metric dimension of G, denoted by bðGÞ, is the cardinality of its metric basis. Caceres et al. [4] define the notion of a doubly resolving set as follows. Vertices x, y of the graph G (n P 2) are said to dou- bly resolve vertices u, v of G if dðu; xÞ dðu; yÞ – dðv ; xÞ dðv ; yÞ. A vertex set D ¼fx 1 ; x 2 ; ... ; x l g of G is a doubly resolving set of G if every two distinct vertices of G are doubly resolved by some two vertices of D. The minimal doubly resolving set problem consists of finding a doubly resolving set of G with the minimum cardinality, denoted by wðGÞ. Note that if x, y doubly resolve u, v then dðu; xÞ dðv ; xÞ – 0 or dðu; yÞ dðv ; yÞ – 0, and hence x or y resolves u, v. Therefore, a doubly resolving set is also a resolving set and bðGÞ 6 wðGÞ. The strong metric dimension problem (SMDP) was introduced by Sebö and Tannier [10] and further investigated by Oel- lermann and Peters-Fransen [11]. Recently, the strong metric dimension of distance hereditary graphs has been studied by May and Oellermann [12]. A vertex w strongly resolves two vertices u and v if u belongs to a shortest v  w path or v belongs to a shortest u  w path. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S.A strong metric basis of G is a strong resolving set of the minimum cardinality. Now, the strong metric 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.047 q This research was partially supported by Serbian Ministry of Education and Science under the grant no. 174033. ⇑ Corresponding author. E-mail addresses: jkratica@mi.sanu.ac.rs (J. Kratica), verakov@fon.rs (V. Kovac ˇevic ´ -Vujc ˇic ´), canga@fon.rs (M. C ˇ angalovic ´), milicas@fon.rs (M. Stojanovic ´). Applied Mathematics and Computation 218 (2012) 9790–9801 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc