Minimal resolving sets for the hypercube

📝 Abstract

For a given undirected graph $G $, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in $S $. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let $Q^n$ denote the $n $-dimensional hypercube with vertex set ${0,1}^n $. In Erd"os and Renyi (Erdos & Renyi, 1963) it was shown that a particular set of $n$ vertices forms a resolving set for the hypercube. The main purpose of this note is to prove that a proper subset of that set of size $n-1$ is also a resolving set for the hypercube for all $n \ge 5$ and that this proper subset is a minimal resolving set.

💡 Analysis

For a given undirected graph $G $, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in $S $. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let $Q^n$ denote the $n $-dimensional hypercube with vertex set ${0,1}^n $. In Erd"os and Renyi (Erdos & Renyi, 1963) it was shown that a particular set of $n$ vertices forms a resolving set for the hypercube. The main purpose of this note is to prove that a proper subset of that set of size $n-1$ is also a resolving set for the hypercube for all $n \ge 5$ and that this proper subset is a minimal resolving set.

📄 Content

arXiv:1106.3632v3 [cs.DM] 16 Jan 2012 Minimal resolving sets for the hypercube Ashwin Ganesan∗ Abstract For a given undirected graph G, an ordered subset S = {s1, s2, . . . , sk} ⊆V of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in S. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let Qn denote the n-dimensional hypercube with vertex set {0, 1}n. In Erd¨os and Renyi [5] it was shown that a particular set of n vertices forms a resolving set for the hypercube. The main purpose of this note is to prove that a proper subset of that set of size n −1 is also a resolving set for the hypercube for all n ≥5 and that this proper subset is a minimal resolving set.

1. Introduction Let G = (V, E) be a simple, undirected graph. An ordered subset S = {s1, s2, . . . , sk} ⊆ V of vertices is a resolving set for the graph if the vertices of the graph are distin- guishable by their vector of distances to the vertices in S. In other words, if d(v, S) denotes the vector (d(v, s1), d(v, s2), . . . , d(v, sk)) of distances in the graph from v to the elements of S, then S is a resolving set for G if and only if d(u, S) ̸= d(v, S) whenever u ̸= v. In this manner, the vertices in S ‘resolve’ or are able to distinguish between the vertices of the graph using information on the distances to these vertices. Note that this definition immediately implies that a superset of any resolving set is again a resolving set, but a proper subset of a resolving set is not necessarily a resolv- ing set. A well-studied problem has been to obtain resolving sets which are minimum (i.e. of minimal size) or which are minimal (i.e. which have no proper subsets that are resolving sets). The concept of resolving sets arises both in puzzles such as the coin weighing problem and strategies for the Mastermind game, as well as practical applications such as chemical compounds and drug discovery [4], network discovery and verification [1], and robot navigation. Thus, it is of interest to have descriptions of resolving sets which are minimal; see [3] for some recent results. ∗Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Ettimadai, Coimbatore

641 105, India. Email: ashwin.ganesan@gmail.com, g ashwin@cb.amrita.edu. 1 The n-dimensional hypercube Qn is the graph whose vertices are the 2n 0-1 se- quences of length n, with two vertices being adjacent whenever the corresponding two sequences differ in exactly one coordinate. Resolving sets for hypercubes were studied in Erd¨os and Renyi [5], Lindestr¨om, and many other papers (see [3] for some recent results), and this problem still remains open, though some asymptotic bounds are known. A lower bound due to Erd¨os and Renyi and an upper bound due to Linde- str¨om imply that the minimum size of a resolving set of Qn asymptotically approaches 2n/ log n. The problem of determining minimum resolving sets for the n-dimensional hypercube for finite values of n is still open. Resolving sets for more general graphs were first introduced in Harary and Melter [6]; the minimum size of a resolving set of a graph is also called the metric dimension of the graph. Notation: We use the following notation for hypercubes and level sets of the poset of subsets of an n-element set [2]. Let X denote the n-element set {1, 2, . . . , n}. With a slight abuse of notation, we use the same symbol v ∈V (Qn) for a vertex of the hypercube to represent both a binary string of length n as well as a subset of {1, 2, . . . , n}. The operation + denotes binary addition. Thus, for n = 5, we can write x = 11001 = {1, 2, 5}, y = 10100 = {1, 3} ∈V , and |x+y| = |01101| = |{2, 3, 5}| = 3, where the operation of binary addition of two strings is equivalent to that of taking the symmetric difference of the two corresponding sets. Also, ei ∈V is the binary string that has a 1 in the i-th coordinate and a 0 in the remaining coordinates; thus, e2 = 010 . . . 0 = {2}. The level set X(k) denotes the set of all subsets of X of size k. For v ∈V and an ordered subset S = {s1, s2, . . . , sk} ⊆V , we define the distance vector d(v, S) := (d(v, s1), d(v, s2), . . . , d(v, sk)), where the distance d(x, y) between two vertices x and y in Qn is just the number of coordinates where the two sequences differ, which equals the size of their symmetric difference. Thus, S is a resolving set for Qn if u ̸= v implies d(u, S) ̸= d(v, S). For x ∈V and S ⊆V , S + x denotes the (ordered) set obtained by adding x to each element of S. 1.1. Prior work and our results In Erd¨os and Renyi [5], the set of 4 vertices {{1, 2, 3, 4, 5}, {1, 2, 3}, {2, 4}, {2, 3, 5}} was shown to be a resolving set for Q5; they provided the results of an exhaustive computation of the di

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