On the metric dimension of Cartesian powers of a graph
A set of vertices resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in . The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph on vertices, and let be the distance matrix of . We prove that if there exists such that and the vector , after sorting its coordinates, is an arithmetic progression with nonzero common difference, then the metric dimension of the Cartesian product of copies of is . In the special case that is a complete graph, our results close the gap between the lower bound attributed to Erdős and Rényi and the upper bound by Chvátal. The main tool is the Möbius function of a certain partially ordered set on .
A set of vertices resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in . The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. The Cartesian product of graphs is the graph with vertex set such that and are adjacent whenever there exists such that for all and is adjacent to in .
For a graph and , denote by the Cartesian product of copies of , and by the metric dimension of . This paper undertakes the study of the asymptotic behavior of when the connected graph is fixed and tends to infinity, especially when is a complete graph on vertices, which we denote by . In this context, the definition of a resolving set can be rephrased in the following way. Denote the distance between vertices in by . Given a subset of , define by for every and . The set of vertices is a resolving set of if and only if is an injection.
The concept of resolving set and that of metric dimension date back to the 1950s — they were defined by Bluementhal [Blu53] in the context of metric space. These notions were introduced to graph theory by Harany and Melter [HM76] and Slater [Sla75] in the 1970s.
Under the guise of a coin weighing problem, the metric dimension of a hypercube was first studied by Erdős and Rényi. The coin weighing problem, posed by Söderberg and Shapiro [SS63], assumes coins of weight or , where and are known, and an accurate scale. Söderberg and Shapiro asked the question of how many weighings are needed to determine which of coins are of weight and which of weight if the numbers of each are not known. The variant of the problem, where the family of weighings has to be given in advance, is connected to the metric dimension of the hypercube . It was observed that the minimum number of weighings differs from by at most (see [ST04, Section 1]). A lower bound on the number of weighings by Erdős and Rényi [ER63] and an upper bound by Lindström [Lin64] and independently by Cantor and Mills [CM66] imply that .
The metric dimension of is also connected to the Mastermind game. Mastermind is a deductive game for two players, the codemaker and the codebreaker
Motivated by the above applications, we establish a lower bound and an upper bound on for every connected graph in Section 2 and Section 3 respectively. For certain families of graphs, the lower bound and the upper bound are asymptotically equivalent. In particular, we show that for all in Section 4. We emphasize that the situation is totally different when varies and is fixed. For example, Cáceres et al. [CHM07, Theorem 6.1] showed that . We conclude with a generalization to integer matrices and some open problems in Section 5.
2 An upper bound on
We establish the following upper bound on .
Given a connected graph on vertices, let be the distance matrix of . For every , the metric dimension of is at most
where is defined by
Clearly, . It is less clear that . We claim that there exists such that and for all . Denote the th row of by . For , because , the equation , or , defines a subspace of different from . In other words, defines a -codimensional subspace of , and so is nonempty. Finally, we scale properly so that it becomes a vector in .
Our construction of a resolving set of is inspired by the upper bound for the coin weighing problem by Lindström [Lin65]. Among various constructions such as the recursive construction by Cantor and Mills [CM66] and the construction by Bshouty [Bsh09] based on Fourier transform, we find the one using the theory of Möbius functions by Lindström [Lin71] best suits our needs.
We recall the basics of Möbius functions. Let be a locally finite partially ordered set. The Möbius function can be defined inductively by the following relation:
The classical Möbius function in number theory is essentially the Möbius function of the set of natural numbers partially ordered by divisibility. For our purpose, we instead partially order in the following way: if and only if , where is the bitwise AND operation. The Möbius function is thus
where is the number of ones in the binary expansion of . With the binary operator , the partially ordered set is indeed a meet-semilattice — a partially ordered set in which any pair of elements has the greatest lower bound. We need the following identity for our meet-semilattice.
Lemma 2 (Lemma of Lindström [Lin69]).
Let be a locally finite meet-semilattice with Möbius function . Let and . Let be defined for all with values in a commutative ring with unity. Then we have
The last ingredient is the following estimation on the partial sum of .
Theorem 3 (Theorem 1 of Bellman and Shapiro [Bs48]).
We now construct a resolving set for Theorem 1 using the Möbius function of .
Proof of Theorem 1.
Let be such that , the coordinates of are distinct integers
and set . For each , let be the largest integer such that
that is, .
Let be the set of the first elements of under the lexicographical order. We label the copies of in by , namely each vertex of is an element of , where is the vertex set of . Set .
Our resolving set will be described by a matrix whose rows and columns are indexed by and respectively with entries from . Note that each row of is an element of , thus can be seen as a vertex of . For and , we denote the entry of on row and column by .
We claim that a matrix can be chosen to satisfy the following properties.
For example, when , , we take and . In the table below, we supply the values of for . The readers can verify (4a) in this case.
In general, pick arbitrary . On the left hand side of (4a), the summation consists of terms, moreover of them has (respectively ). Since and by (3), it is easy to assign one of to for all , possibly in many ways, to satisfy (4a). For , we take . For every and , as , the left hand side of (4b) equals by Lemma 2.
To show that resolves , it suffices to demonstrate that every is uniquely determined by the vector
Suppose this vector is provided. We shall gradually uncover for . Assume that is known for every . We extend the distance function of to the bilinear form
where and are in . Observe that
Since both and are known, we are able to determine
Let . We can thus deduce from (5) the value of
3 A lower bound on
A straightforward generalization of the lower bound on the coin weighing problem by Erdős and Rényi gives a lower bound on the metric dimension of (see Moser [Mos70] and Pippenger [Pip77] for different proofs using the second moment method and the information-theoretic method).
Given a connected graph on vertices, for every , the metric dimension of is at least
Let be a resolving set of of size , where is the vertex set of . For every , let be independent random variables defined by , where the independent random variables are chosen uniformly at random from , and define
where is the diameter of the graph. Since each is bounded by , Hoeffding’s inequality provides an upper bound on the cardinality of the complement of :
From the equivalent definition of a resolving set mentioned in Section 1, the function , defined by for every and , is injective. Since the image of under is contained in a cube of side length in and by Theorem 1, we obtain
Taking logarithm gives
4 Asymptotically tight cases
Given a connected graph on vertices, let be the distance matrix of . The following statements are equivalent.
The technical parameter defined by (1) equals .
There exists such that and the vector , after sorting its coordinates, is an arithmetic progression with nonzero common difference.
There exists a permutation on such that
where the column space is understood as a subspace of , and is the -dimensional all-ones column vector.
Let be a vector such that and the coordinates of are distinct integers, and let . Clearly, , and equality holds if and only if the vector , after sorting its coordinates, is an arithmetic progression with common difference . This shows the implication from Statement 1 to Statement 2. The converse is evident.
Lastly, we demonstrate the equivalence between Statement 2 and Statement 3. Suppose that there exists such that and the vector , after sorting its coordinates, is an arithmetic progression with nonzero common difference. Thus there exists with and a permutation such that
We obtain that
which implies Statement 3. Reversing the argument, one can show that Statement 3 indicates the existence of satisfying the conditions in Statement 2. However, one can always scale properly so that it becomes a vector in . ∎
Given a connected graph on vertices, let be the distance matrix of . If is a complete graph, a path, a cycle or a complete bipartite graph, or the matrix
is invertible, then the metric dimension of is
When is invertible, Statement 3 in Lemma 5 applies here. When is a complete graph, a path or a cycle, by Statement 2 in Lemma 5, it suffices to construct a vector such that and the vector , after sorting its coordinates, is an arithmetic progression with nonzero common difference. We list the construction of below and leave the verification to the readers.
|complete graph||path||even cycle||odd cycle|
Lastly, because is a cycle of length , for a complete bipartite graph , it suffices to check that is invertible for . Denote by the -dimensional all-ones matrix, and by the -dimensional identity matrix. Recall that has eigenvalues and . As , is invertible and , hence . Using row operations and Schur complements, we have the following matrix equivalence:
Notice that has eigenvalues and , which are nonzero. Therefore is invertible for a complete bipartite graph. ∎
Sebő and Tannier [ST04, Section 1] claimed that , where is the path on vertices, and they thought “this upper bound is probably the asymptotically correct value”. Our result confirms their conjecture.
5 Open problems
has a solution for some permutation on . Using McKay’s dataset [McK17] of connected graphs up to 10 vertices, we find 1 graph on 6 vertices, 4 graphs on 9 vertices and 1709 graphs on 10 vertices for which .
The graph on vertices is . We give a simple argument for in Appendix A. We believe that our construction of a resolving set can be significantly improved for such graphs.
Given a connected graph on vertices, the metric dimension of is . In particular, .
In the proofs of Theorem 1 and Theorem 4, we have made use of one property of graph distance, that is, it is integer valued. In addition, we used some other properties of graph distance in Remark 1 just to show that Theorem 1 is not vacuously true for any connected graph. In this sense, our results are more related to integer matrices than graphs.
Given a integer matrix and , is the minimum cardinality of a subset of such that , defined by for every and , is an injection.
Given a integer matrix with , if none of the diferences between two rows of is parallel to , then for every ,
where is defined by
It is conceivable that the generalization of Conjecture A to integer matrices holds.
Given a integer matrix with , if none of the diferences between two rows of is parallel to , then .
Appendix A Proof of
Let be the distance matrix of . For every such that , the vector , after sorting its coordinates, is never an arithmetic progression with nonzero common difference. Moreover, there exists such that .
Label the vertices of of degree by and those of degree by , and let be the distance matrix of :
Assume for the sake of contradiction that there exists such that
and the vector , after sorting its coordinates, is an arithmetic progression with nonzero common difference . This means , or equivalently,
Notice that shifting by the same amount will affect neither (8) nor (9). Without out loss of generality, we may assume that . Under the assumption , (9c) becomes for all . Set for . Together with (9a) and (9b), we know that the pairwise distances among are in , and so there exists and a permutation on such that . Now implies that