Classifying Resolving Lists by Distances between Members

Let G be a connected graph and let w1, · · ·wr be a list of vertices. We refer to the choice of a triple (r;G;w1, · · ·wr), as a metric selection. Let ρ be the shortest path metric ofG. We say that w1, · · ·wr resolves G (metricly) if the function c : V (G) −→ Z given by x 7→ (ρ(w1, x), · · · , ρ(wr, x)) is injective. We refer to c as the code map, and to its image as the codes of the triple (r;G;w1, · · · , wr). This paper proves basic results on the following questions: 1. What sets can be the image of a code map? 2. Given the image of a graph’s code map, what can we determine about the graph?


Introduction and Summary
We consider resolving lists, as used in the definition for metric dimension of finite graphs.
For k ∈ N, let Ind(k) be the set of integers from 1 to k, inclusive.Let G = (V (G), E(G)) be a finite, simple, undirected, and (if necessary) connected graph of order |V (G)| ≥ 2. Let ρ G , or simply ρ if the context is clear, be the function such that ρ G (x, y) is the length of a shortest path in G between vertices x and y.
Define a metric selection to be a triple (r; G; w 1 , • • • , w r ) where r ∈ N, G is a connected graph, and w 1 , • • • , w r is an irredundant list of r vertices of G. Define the code map of this selection to be the function c from V (G) to Z r given by v → (ρ(v, w 1 ), • • • , ρ(v, w r )) .
We refer c(v) as the code of v.We say w 1 , • • • , w r is a resolving list of G, and that the metric selection is resolved, if its code map is injective.
The metric dimension of G, which is denoted here by dim(G), is the smallest r for which a resolving list of length r exists.
Our work arose from discussion of metric dimension, as characterized in Chartrand et al. [1] and Harary et al. [2].The author contributed to Eroh et al. [4] which studies the impact of small changes on metric dimension.A coauthor of the latter paper, Dr. C.X. Kang, posed the question of how much of a graph is classified by its code map.
Hernando et al. [3] established an upper bound on the number of vertices of a graph of metric dimension r and diameter d.The authors demonstrated that the bound is sharp by building a graph based on a list of r vertices such that the distance between any two list members is b = 2d/3 + 1. Essentially, the text looks at all codes that could arise from such a graph, and then builds the graph using codes as vertices.The result is a graph whose order is the desired bound.
This paper does not deal with minimality questions.Instead, we attempt to classify graphs based on the image of the code map.Our classification begins by looking at codes from a list of vertices for which the distances between the members (of the list) are fixed.
Suppose that c is a code map for a list w 1 , • • • , w r that resolves some graph G.For each i, c(w i ) is the only member of the image of c that has 0 for its i-th component.Consequently, if one has just the image of c, one knows which codes represent the resolving list.Furthermore, the components in codes can now be identified as distances from a vertices whose codes are identified.
We start our classification with the basic information from codes of a resolving list.
Definition 1.1.Let r ∈ N. By a baseline matrix of dimension r, we mean a symmetric r × r matrix B such that (1.a) for i, j ∈ Ind(r), B i,j is a non-negative integer, (1.b) for i, j ∈ Ind(r), B i,j = 0 if and only if i = j, and Let B be such a matrix.Define Max(B) = max{B i,j : i, j ∈ Ind(r)} .
Define a metric type to be a triple τ = (r, B, h) where r ∈ N, B is a baseline matrix of dimension r, and h ∈ N such that h ≥ Max(B).
Let r ∈ N and let B be a baseline matrix of dimension r.Define U (B) to be the set of r-tuples (x 1 , • • • , x r ) of non-negative integers such that (2.a) for all i, j ∈ Ind(r), x i + x j ≥ B i,j , and (2.b) for all i, j ∈ Ind(r), x i + B i,j ≥ x j .
The second condition is equivalent to requiring |x i − x j | ≤ B i,j for all i, j ∈ Ind(r).For each i ∈ Ind(r), let e i (B) = (B i,1 , B i,2 , • • • , B i,r ).Trivially, each e i (B) ∈ U (B).
Let r and B be as before.Define µ B on U (B) 2 by Then µ B is a well-known metric.Define Let τ = (r, B, h) be a metric type.We define U (B, h), or U (τ ), to be the subset of all tuples (x 1 , • • • , x r ) ∈ U (B) such that x i ≤ h for all i ∈ Ind(r).For each i ∈ Ind(r), e i (B) ∈ U (τ ); in this context, we refer to e i (B) as e i (τ ) or, if the meaning is clear from context, e i .Likewise, we refer to the restriction of µ B to U (τ ) as µ τ , or simply µ.Finally, define C(B, h) to be the graph whose vertex set is U (B, h) and whose set of edges consists of all pairs of U (B, h) members that belong to E(B).
Note: Each member of U (B, h) serves two roles here.Literally, each is a tuple (x 1 , • • • , x r ).However, we want to build graphs in which each tuple is treated as vertex.We shall write "x = (x 1 , • • • , x r )" as a sign that we treat the whole as an individual (usually a vertex) x, but need to cite the components x i in our discussion.
(3.a) ∀i, j ∈ Ind(r), B i,j = ρ G (w i , w j ), and We leave it to the reader to verify that (4.a) if (r; G; w 1 , • • • , w r ) has type τ , then the image of its code map is contained in U (τ ), and All of the inequalities required between members of B and coordinates of a tuple in U (B) follow from the Triangle Inequality for ρ.
Our main result is Proposition 2.1, which effectively characterizes which sets of tuples can appear as codes for a graph with a given baseline matrix.We postpone that statement until we have more language in place.A consequence is Corollary 1.1.Let τ be a metric type.
Then there is a metric selection (r; H; v 1 , • • • , v r ) of type τ whose the code map p is injective and has the same image set as f .
Let f and f be their respective code maps.Then there exists (r; of type τ whose code map p is injective and whose image is the union of the images of f and f . We conclude this introduction with facts about the C(B, h) graphs.
As noted earlier, the proof of [3][Lemma 3.3] can be roughly paraphrased as: for r, d ∈ N, the graph of metric dimension r and diameter d that has the greatest number of vertices is based on a resolving list in which the distance between any two different members is roughly 2d/3.This spurred the current author to ask: for a fixed baseline matrix B, is there a tight bound on the growth of |U (B, h)| as h → ∞?In fact, the growth is linear.
Theorem 1.4.Let r ∈ N and let B be a baseline matrix of dimension r.Put h 0 = Max(B) and let h 1 = (h 0 + 1)/2 .Put h 2 = h 0 + h 1 , and let We conclude our discussion of C(B, h) by showing that, under minimal assumptions, (6.a) C(B, h) has metric dimension r and (6.b) every automorphism of this graph preserves the set {e 1 , • • • , e r }.
However, these results are limited in the following sense.Suppose that (r; G; w 1 , • • • , w r ) and (r, H; z 1 , • • • , z r ) are two resolved metric selections, both of which have type τ = (r, B, h).Furthermore, suppose that the images of their respective code maps happen to agree.It is possible that the diameters, the metric dimensions, and the automorphism groups of G and H, respectively, disagree.
Let W ⊆ V (C(B, h)), and let E ⊆ E(C(B, h)) such that the endpoints of each e ∈ E belong to W . Definition 2.1 formulates the subgraph coordinate condition on the pair (W, E).Proposition 2.1 shows that ( 1) each e i ∈ W and (2) the code map of (r, (W, E); e 1 , • • • , e r ) is the identity function.
It is easy to see that if W is fixed, there remains a huge variety of choices of E which satisfy this criterion.A few examples show that different choices produce wildly different graphs.

Proof of Theorem 1.2
Suppose τ = (r, B, h) is a metric type.Let (r; G; w 1 , • • • , w k ) be a metric selection of type τ , and let f : G −→ U (B, h) be its code map.If v, w are adjacent vertices of G, then for each i ∈ Ind(r), |ρ(w

This observation is the basis of our definition of E(B).
Next, consider a subset of U (B, h) which is assigned a set of edges to make it a graph.The next theorem characterizes when such a graph creates a metric selection whose code map is the identity.Definition 2.1.Let τ = (r, B, h) be a metric type.Let H be a graph whose vertex set is a subset of U (B, h) and whose edge set is a subset of E(B).We say that H satisfies the subgraph coordinate condition for τ if the following is true: Suppose W is a subset of U (τ ).Let W * be the graph on W induced from C(B, h).If W * satisfies (7), we say that W satisfies the subset coordinate condition for τ .
We show that the subgraph coordinate condition on a graph H implies that H is connected.That is, connectedness is not assumed.Lemma 2.1.Let τ be a metric type.If x = (x 1 , • • • , x r ) ∈ U (τ ) and i ∈ Ind(r) such that x i = 0, then x = e i .
We can now prove that the subgraph coordinate condition implies that each tuple, as a vertex, is also its image under the code map.The full statement is Proposition 2.1.Let τ be a metric type, and suppose that H satisfies the subgraph coordinate condition for τ .Then Proof.Put W = V (H) and ρ = ρ H .For each i ∈ Ind(r), there is x ∈ V (H) whose i-th coordinate is least amongst all members of H.The subgraph coordinate condition, plus Lemma 2.1, implies x = e i .
For the next assertion, fix i ∈ Ind(r) and let D be the connected component of H that contains e i .We induct on a double-implication: For each k ∈ N ∪ {0}, our proposition is that for We test the implications one at a time for k + 1.First, suppose y i = k + 1.By the subgraph coordinate condition, there is a neighbor If ρ(e i , y) ≤ k, the second part of the inductive hypothesis would imply y i ≤ k, which is false.Therefore, ρ(e i , y) = k + 1.
As before, assume y = (y 1 , • • • , y r ) ∈ W .This time, we also assume that y ∈ D and ρ(e i , y) ≤ k + 1.Our goal is to show that ρ(e i , y) = y i .If ρ(e i , y) ≤ k, the inductive hypothesis applies to y, and we are done.Suppose ρ(e i , y) = k +1.Then y must have a neighbor x ∈ D such that ρ(e i , x) = k.Expand x = (x 1 , • • • , x r ).Apply the inductive hypothesis to x to deduce that x i = k.Now xy ∈ E(H), which means µ(x, y) ≤ 1. Therefore y i ∈ {k − 1, k, k + 1}.If y i ≤ k, the other half of the inductive hypothesis implies that ρ(e i , y) = y i = k + 1.Therefore, Consequently, ρ(x, y) ≥ µ(x, y).
We add an informal corollary.Suppose G is a graph and w 1 , • • • , w r resolves it.Let f be the corresponding code map.Suppose we augment G by adding edges.That is, let E 1 be a set of ordered pairs of vertices of G, and create a new graph G 1 by adding E 1 to the existing edges of G. Consider the question: when is f also the code map of G 1 ?From Proposition 2.1, f remains the code map for G 1 if and only if for every new edge ab ∈ E 1 , µ(f (a), f (b)) ≤ 1.
Examples.The above proposition characterizes which graphs can produce a specified set of codes.It also reveals that many aspects of the graph are not determined by the codes.We consider two simple examples.Let n be an integer ≥ 3.
• Model the path graph Z 2n as consisting of integers (1 − n) ≤ i ≤ n, where successive integers are connected by an edge.Choose e 1 = 0 and e 2 = 1.For 1 ≤ i ≤ n, the code for i is (i, i − 1) and the code for 1 − i is (i − 1, i).By 2.1, the code map will be unchanged if we add one more edges between i and 1 − i.Now Z 2n has a unique non-trivial automorphism σ (in this case, i → 1 − i) which preserves {e 1 , e 2 }.Adding one or more of the allowed edges produces a graph with the same automorphism; however, any one extra edge reduces the diameter from n − 1 to a small value.If one adds just the edge from 1 − n to n, graph mutates into C 2n .This new graph now has a transitive automorphism group.
• This time, consider the cycle C 2n+1 , and choices e 1 and e 2 which are distance n apart.The codes from this choice consist of all pairs (i, j) in which i + j is either n or n + 1. Points with codes (i, n − i) form one arc from e 1 to e 2 , and the remaining codes form the other.Without altering the image of the code map, edges can be attached between (i, n − i) and (i + 1, n − i) and/or (i, n + 1 − i).
From the perspective of the other arc, (j, n + 1 − j) can be linked to (j, n − j) and/or (j − 1, n + 1 − j).Therefore, the same code map applies to a variety of graphs based on C 2n+1 , with a potentially large number of links between the two arcs.None of the new edges change the diameter, which remains n; however, most additions create a graph whose automorphism group is trivial.
We can now prove assertions made in the introduction.
Proof of Corollary 1.1 Fix a type τ = (r, B, h).For Part (A), let (r; G, w 1 , • • • , w r ) be a metric selection of type τ , and let f be its code map.Let W be the image of f , and let E be the set of edges of the form f (x)f (y) where xy ∈ E(G) (and f (x) = f (y)).
Clearly H = (W, E) satisfies the subgraph coordinate condition By Proposition 2.1, the code map for (r; H; e 1 , • • • , e r ) is the identity.In other words, the latter is resolved and has the same image as the code map for G. Now suppose that (W 1 , E 1 ), (W 2 , E 2 ) both satisfy the subgraph coordinate condition for τ .Obviously, (W 1 ∪ W 2 , E 1 ∪ E 2 ) satisfies the same condition.An obvious variation on the previous paragraph proves Part (B).
The following lemma implies most of the remaining verifications.It extends [3] Then z ∈ U (τ ).
Proof.We run through the inequalities required of the coordinates of z.Obviously, z i ≤ h for each index i.Now suppose i, j ∈ Ind(r) so i = j.We claim that z i +z j ≥ B i,j .First, if z i = x i +1 or z j = x j + 1, then z i + z j ≥ x i + x j ≥ B i,j .
Now suppose z i ≤ x i and z j ≤ x i .Then z i ≥ y i , z j ≥ y j , and This completes one set of conditions.Again, suppose i, j ∈ Ind(r) so i = j.This time, we argue that z i + B i,j ≥ z j .When z i ≥ z j , this assertion is vacuous.Now assume z j > z i .First, if z j = x j − 1, From now on, assume z j ≥ x j .
If z i = x i + 1, z i + B i,j = x i + 1 + B i,j ≥ x j + 1 ≥ z j .
Now assume z i ≤ x i .
In the remaining case, y j ≥ z j and y i ≤ z i .Then z i + B i,j ≥ y i + B i,j ≥ y j ≥ z j .
Proof of Theorem 1.2 Let x ∈ U (B, h) and i ∈ Ind(r) such that x i = 0. Let z be as in Lemma 2.2 for the ordered pair x, e i .Then z i = x i − 1.Hence, the set U (B, h) satisfies the subgraph coordinate condition.Let x, y be two different members of U (B, h), and let z be the construction of Lemma 2.2.Then Based on this step, it is easy to set up an inductive proof that µ(x, y) = ρ(x, y).We leave the details to the reader.

Theorem 1 . 2 .
Let r, h ∈ N, and let B be a baseline matrix of dimension r such that h ≥ Max(B).Then (5.a) C(B, h) has diameter ≤ h, (5.b) the shortest distance metric of C(B, h) equals the restriction of µ, (5.c) the tuple (r; C(B, h); e 1 , • • • , e r ) is resolved, and (5.d) the code map of (r; C(B, h); e 1 , • • • , e r ) is the identity function.Consequently, the triple (r; C(B, h); e 1 , • • • , e r ) has a universal property among all metric selections of type (r, B, h).Corollary 1.3.Let τ = (r, B, h) be a metric type, and let (r; G; w 1 , • • • , w r ) be a resolved metric selection of type τ .Let f be the code map of (r; G; w 1 , • • • , w r ).Then the image of f lies in the set of vertices of C(B, h).Furthermore, if xy (8.a) H is a connected graph, (8.b) e 1 , • • • , e r ∈ V (H), (8.c) the code map of (r, H; e 1 , • • • , e r ) is the identity function, and By the inductive hypothesis, x ∈ D and ρ(e i , x) = k.Consequently, y ∈ D and ρ(e i , y) ∈ {k − 1, k, k + 1} .