The general position problem and strong resolving graphs

Sandi Klavžar; Ismael G. Yero

doi:10.1515/math-2019-0088

Article Open Access

The general position problem and strong resolving graphs

Sandi Klavžar and Ismael G. Yero

Published/Copyright: October 13, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(G_SR), where G_SR is the strong resolving graph of G, and ω(G_SR) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp(G ⊠ H) ≥ gp(G)gp(H), and asked whether the equality holds for arbitrary connected graphs G and H. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

Keywords: general position problem; strong resolving graph; graph products

MSC 2010: 05C12; 05C76

1 Introduction

The general position problem was recently and independently introduced in [1, 2]. If G = (V(G), E(G)) is a graph, then S ⊆ V(G) is a general position set if no triple of vertices from S lie on a common geodesic in G. The general position problem is to find a largest general position set of G, the order of such a set is the general position number gp(G) of G. A general position set of G of order gp(G) is shortly called gp-set. The general position problem has been further studied in a sequence of very recent papers [3, 4, 5, 6, 7].

A vertex u of a connected graph G is maximally distant from a vertex v if every w ∈ N(u) satisfies d_G(v, w) ≤ d_G(u, v), where N(u) is the open neighborhood of u. If u is maximally distant from v, and v is maximally distant from u, then u and v are mutually maximally distant (MMD for short). The strong resolving graph G_SR of G has V(G) as the vertex set, two vertices being adjacent in G_SR if they are MMD in G. The notion of the strong resolving graph was introduced in [8] as a tool to study the strong metric dimension. There it was proved that the problem of finding the strong metric dimension of a graph G can be transformed to the problem of finding the vertex cover number of G_SR. Further on, the strong resolving graph itself was remarked as a kind of graph transformation in [9], and several characterizations and realizations of it were described.

Now, one of the open problems presented in [9] concerns finding applications for the strong resolving graph construction, other than that of computing the strong metric dimension of graphs. In this paper we give a partial answer to this problem by establishing a connection between the general position number of a graph G and the clique number of the graph G_SR. More precisely, in Theorem 3.1 we prove that gp(G) ≥ ω(G_SR) holds for any connected graph G. Then we demonstrate with different infinite families of graphs, including direct products of complete graphs, that the bound is sharp. We also show that for any integers r ≥ t ≥ 2, there exists a graph G such that gp(G) = r and ω(G_SR) = t. In Section 4 we focus on strong products of graphs. We prove that gp(G ⊠ H) ≥ gp(G)gp(H) holds for connected graphs G and H and that the bound is again sharp. In particular, if gp(G) = ω(G_SR), then gp(G ⊠ K_n) = n ⋅ gp(G) = ω((G ⊠ K_n)_SR). We close the section with a question on whether actually the equality gp(G ⊠ H) = gp(G)gp(H) holds for arbitrary connected graphs G and H. In Section 5 we give additional large families of graphs, based on the generalized lexicographic product, for which the equality in Theorem 3.1 holds. In the final section we determine the general position number for different rooted product graphs and relate the values with the corresponding clique numbers of strong resolving graphs.

Before giving our results, we list in the next section definitions and concepts not yet given, as well as some results needed later.

2 Preliminaries

For a positive integer k we will use the notation [k] = {1, …, k}. If G = (V(G), E(G)) is a graph, then n(G) = |V(G)| and m(G) = |E(G)|. If X ⊆ V(G), then the subgraph of G induced by X is denoted 〈X〉.

The distance d_G(u, v) between vertices u and v of a graph G is the number of edges on a shortest u, v-path. A subgraph H of a graph G is isometric if d_H(u, v) = d_G(u, v) holds for all u, v ∈ V(H). A set of subgraphs {H₁, …, H_k} of a graph G is an isometric cover of G if each H_i, i ∈ [k], is isometric in G and ⋃i=1k V(H_i) = V(G). With this concept in hand we can recall the following.

Theorem 2.1

[1, Theorem 3.1] If {H₁, …, H_k} is an isometric cover of G, then

gp(G)≤∑i=1kgp(Hi).

If G is a connected graph, S ⊆ V(G), and 𝓟 = {S₁, …, S_p} a partition of S, then 𝓟 is distance-constant (alias “distance-regular” [10, p. 331]) if for any i, j ∈ [p], i ≠ j, the distance d_G(u, v), where u ∈ S_i and v ∈ S_j, is independent of the selection of u and v. This distance is then the distance d_G(S_i, S_j) between the parts S_i and S_j. A distance-constant partition 𝓟 is in-transitive if d_G(S_i, S_k) ≠ d_G(S_i, S_j) + d_G(S_j, S_k) holds for pairwise different indices i, j, k ∈ [p]. With these concepts, general position sets can be characterized as follows.

Theorem 2.2

[3, Theorem 3.1] Let G be a connected graph. Then S ⊆ V(G) is a general position set if and only if the components of 〈S〉 are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S.

Let η(G) denote the maximum order of an induced complete multipartite subgraph of the complement G of a graph G. Then we have:

Theorem 2.3

[3, Theorem 4.1] If diam(G) = 2, then gp(G) = max{ω(G), η(G)}.

Vertices u and v of a graph G are true twins if N[u] = N[v], where N[u] is the closed neighborhood of u. Note that only adjacent vertices can be true twins.

Proposition 2.4

If G has no true twins and diam(G) = 2, then gp(G) = ω(G_SR) if and only if gp(G) = α(G).

Proof

Since G is true-twin free, G_SR is isomorphic to the complement G of G. Hence ω(G_SR) = α(G) and thus the conclusion.□

The Petersen graph P is a sporadic example of a graph without true twins and of diameter 2 for which gp(P) ≠ ω(P_SR). Indeed, gp(P) = 6 and ω(P_SR) = α(P) = 4.

Let G and H be graphs. Among the four standard graph products, we will consider the direct product G × H, the strong product G ⊠ H, and the lexicographic product G[H]. The vertex set of all these products is V(G) × V(H). Let (g, h), (g′, h′) ∈ V(G) × V(H). In G × H, the vertices (g, h) and (g′, h′) are adjacent if gg′ ∈ E(G) and hh′ ∈ E(H). In G ⊠ H, the vertices (g, h) and (g′, h′) are adjacent if one of the following three conditions hold: (i) gg′ ∈ E(G) and h = h′, (ii) g = g′ and hh′ ∈ E(H), (iii) gg′ ∈ E(G) and hh′ ∈ E(H). Finally, in G[H] the vertices (g, h) and (g′, h′) are adjacent if either gg′ ∈ E(G), or g = g′ and hh′ ∈ E(H). We note that the lexicographic product is also denoted with G ∘ H to emphasize the associativity of the operation, but here we use G[H] to be consistent with the generalized lexicographic product (to be defined later). If G ∗ H is one of the above products and h ∈ V(H), then the subgraph of G ∗ H induced by {(g, h) : g ∈ V(G)} is called a G-layer. Analogously H-layers are defined. In G × H, each G-layer is an edgeless graph of order n(G). In all other above products, each G-layer is isomorphic to G. If X is a set of vertices of G ∗ H, then the projection of X to G is the set {g ∈ V(G) : (g, h) ∈ X for some h ∈ V(H)}. Analogously the projection of X to H is defined. For more information on the standard graph products see the book [11], here we just recall the following well-known result (cf. [11, Proposition 5.4]) needed later.

Proposition 2.5

If (g, h) and (g′, h′) are vertices of a strong product G ⊠ H, then

dG⊠H(g,h),(g′,h′)=max{dG(g,g′),dH(h,h′)}.

3 The lower bound and equality cases

In this section we first prove the key result that connects the general position problem with the strong resolving graphs.

Theorem 3.1

If G is a connected graph, then gp(G) ≥ ω(G_SR). Moreover, equality holds if and only if G contains a gp-set that induces a complete subgraph of G_SR.

Proof

Let S ⊆ V(G_SR) induce a complete subgraph of G_SR. This means that any two vertices x, y ∈ S are MMD in G. We now consider the vertices of S in the graph G. If there are three distinct vertices x, y, z ∈ S lying on a common geodesic, say y lies in an x, z-geodesic, then neither x, y nor y, z are MMD in G, which is a contradiction. Thus, any three vertices of S do not lie in a common geodesic of G, and therefore, S is a general position set in G. Selecting S to be a complete subgraph G_SR of order ω(G_SR) leads to the desired bound.

Suppose now that gp(G) = ω(G_SR). By the above, any complete subgraph of G_SR of order ω(G_SR) yields a gp-set. Conversely, let S be a gp-set of G that forms a complete subgraph of G_SR. Then, using the already proven inequality gp(G) ≥ ω(G_SR), we have

|S|≤ω(GSR)≤gp(G)=|S|,

from which we conclude that ω(G_SR) = gp(G).□

One would immediately think of characterizing the class of graphs achieving the equality in Theorem 3.1. However, such a characterization seems to be elusive because of the great variety of different structures that can appear. In the following we justify this variety and begin a couple of simple examples that were implicitly known previously.

Block graphs, in particular complete graphs and trees.

Indeed, in [1] it was observed that in block graphs the set of simplicial vertices forms a gp-set. Since simplicial vertices of a graph G also form a set of MMD vertices of a graph (equivalently, they form a complete subgraph of G_SR), Theorem 3.1 implies that gp(G) = ω(G_SR) if G is a block graph.
Complete multipartite graphs.

Let G = K_{n₁,…,n_k}, where k ≥ 2, n₁ ≥ n₂ ≥ ⋯ ≥ n_k ≥ 2, and n₁ ≥ k. Then it is easy to see that the vertices of the n₁-partite set form a maximum general position set. Moreover, the vertices of this set also form a set of mutually maximally distant vertices of G. Hence, gp(G) = ω(G_SR) by Theorem 3.1.

Let G and H be graphs where V(G) = {v₁, … , v_n}. The corona G ⊙ H of graphs G and H is obtained from the disjoint union of G and n disjoint copies of H, say H₁, …, H_n, where for all i ∈ [n], the vertex v_i ∈ V(G) is adjacent to each vertex of H_i. Then we have another equality case:

Proposition 3.2

If H = ⋃_i K_{n_i}, n_i ≥ 1, then for every graph G, gp(G ⊙ H) = ω((G ⊙ H)_SR).

Proof

From [4, Theorem 4.3], it can be noticed that gp(G ⊙ H) = n(G)∑_i n_i, and also that the union of the sets of vertices of all copies of H in G ⊙ H form a gp-set S of G ⊙ H. Every two vertices belonging to one copy of H are MMD, as well as are MMD every two vertices belonging to two different copies of H. Hence S forms a complete subgraph of (G ⊙ H)_SR. Thus we deduce the equality by Theorem 3.1.□

We note in passing that Proposition 3.2 remains valid in a more general setting when different disjoint unions of complete graphs are attached to the vertices of G.

In the next result we provide a family of direct product graphs for which the equality holds in Theorem 3.1. To this end, we locally use the Cartesian product of graphs. As the other standard graph products, the Cartesian product G ◻ H of graphs G and H has vertex set V(G) × V(H), and two vertices (g, h) and (g′, h′) are adjacent in G ◻ H if one of the following two conditions hold: (i) gg′ ∈ E(G) and h = h′, (ii) g = g′ and hh′ ∈ E(H).

Proposition 3.3

If n₁ ≥ n₂ ≥ 3, then

gp(Kn1×Kn2)=ω((Kn1×Kn2)SR)=n1=α((Kn1×Kn2)SR).

Proof

We first note that ω(K_n₁ × K_n₂) = min{n₁, n₂} = n₂. On the other hand, since every two vertices of K_n₁ × K_n₂ belonging to two different copies of K_n₁ and of K_n₂ are adjacent, every maximal induced complete multipartite subgraph of K_n₁ × K_n₂ is formed by the set of vertices of one copy of K_n₁ or of K_n₂. Thus, η(K_n₁ × K_n₂) = max{n₁, n₂} = n₁. Now, since n₁ ≥ n₂ ≥ 3, it follows that diam(K_n₁ × K_n₂) = 2 and hence Theorem 2.3 yields gp(K_n₁ × K_n₂) = max{η(K_n₁ × K_n₂), ω(K_n₁ × K_n₂)} = n₁. From [9, 12] it is known that (K_n₁ × K_n₂)_SR ≅ K_n₁ ◻ K_n₂ and since ω(K_n₁ ◻ K_n₂) = max{n₁, n₂} = n₁, the first two equalities follows. The last equality then follows by Proposition 2.4.□

Note that if we consider n₁ > n₂ = 2 in the result above, then K_n₁ × K₂ is of diameter 3, and its strong resolving graph is K_n₁ ◻ K₂. Thus, ω((K_n₁ × K₂)_SR) = 2. Since gp(K_n₁ × K₂) = n₁ > 2, there is no equality as in the proposition.

Another example of direct products for which the equality in Theorem 3.1 does not hold is K_r,t × K_n, where r ≥ t ≥ 2 and n ≥ 3. Since diam(K_r,t × K_n) = 3, [3, Theorem 5.1] implies that gp(K_r,t × K_n) = α(K_r,t × K_n). Since it is not difficult to verify that α(K_r,t × K_n) = rn, we get gp(K_r,t × K_n) = rn. On the other hand, from [9, Theorem 35] we know that (K_r,t × K_n)_SR ≅ ⋃i=1n K_r+t, and so ω((K_r,t × K_n)_SR) = r + t. As r ≥ t ≥ 2 and n ≥ 3 we have rn > r + t.

Based on the above special cases we pose the following question about a possible dichotomy in direct product.

Problem 3.4

Is it true that gp(G × H) = ω((G × H)_SR) can only hold in the case when diam(G × H) = 2?

To conclude the section we give the following realization result which intuitively indicates that one cannot expect some natural upper bound on gp(G) in terms of ω(G_SR).

Proposition 3.5

For any integers r ≥ t ≥ 2, there exists a graph G such that gp(G) = r and ω(G_SR) = t.

Proof

Since r ≥ t, there exists a non-negative integer q such that r = t + q. We now consider a graph G_q defined as follows. We begin with q copies of the cycle graph C₄ and t–q copies of the graph P₂. Then we add an extra vertex z and one edge between z and exactly one vertex of each copy of C₄ and of P₂. We observe that the components of the strong resolving graph (G_q)_SR are: one complete graph of order t, q complete graphs K₂, and t + 1 isolated vertices. Thus ω((G_q)_SR) = t. On the other hand, a set formed by two non-adjacent vertices of each copy of the cycle C₄ (those ones not adjacent to z), and one vertex of each copy of the path P₂, used to construct G_q, is a general position set of G_q, and so, gp(G_q) ≥ 2q + t – q = t + q = r. We can readily observe that such set is indeed a gp-set of G_q, and therefore gp(G) = r, which completes the proof.□

4 Strong products

If G and H are connected graphs, then each G-layer and each H-layer of G ⊠ H is an isometric subgraph of G ⊠ H. Hence Theorem 2.1 gives the following upper bound.

Corollary 4.1

If G and H are connected graphs, then

gp(G⊠H)≤min{n(G)gp(H),n(H)gp(G)}.

We will later see that the bound of Corollary 4.1 is tight. On the other hand we have the following lower bound.

Theorem 4.2

If G and H are connected graphs, then gp(G ⊠ H) ≥ gp(G)gp(H).

Proof

Let S_G and S_H be gp-sets of G and H, respectively. We claim that S_G × S_H is a general position set of G ⊠ H. To prove it, consider arbitrary pairwise different vertices of S_G × S_H, say (g, h), (g′, h′), (g″, h″), and assume on the contrary that in G ⊠ H there exists a shortest (g, h), (g″, h″)-path P that passes through (g′, h′). We now distinguish several cases.

Suppose first that g = g′ = g″. Since (g, h), (g′, h′), (g″, h″) are pairwise different vertices of G ⊠ H, the vertices h, h′, h″ are then pairwise different. But then the projection of P to H is a shortest h, h″-path that contains h′, a contradiction. Similarly, if g, g′, g″ are pairwise different, then the projection of P to G is a shortest g, g″-path that contains g′.

Suppose next that g = g′ and g″ ≠ g. Then clearly h ≠ h′. If h″ is different from both h and h′, then, as above, consider the projection of P to H to get a contradiction. The other subcase is that h″ = h′ (the subcase h″ = h is treated analogously). Let d_G(g, g″) = k and d_H(h, h″) = ℓ. By Proposition 2.5, d_G⊠H((g, h), (g″, h″)) = max{k, ℓ}. Denoting by P′ the (g, h), (g, h′)-subpath of P and by P″ the (g, h′), (g″, h″)-subpath of P, we get that max{k, ℓ} = |P| = |P′| + |P″| ≥ ℓ + k, a contradiction since k ≥ 1 and ℓ ≥ 1. The case g = g″, g′ ≠ g, and the case g′ = g″, g ≠ g′, are treated analogously.□

In [6, Theorem 3.3] it was proved that gp(P_∞ ⊠ P_∞) = 4. Since the strong grid P_n ⊠ P_m is an isometric subgraph of P_∞ ⊠ P_∞ for each n, m ≥ 2, it follows that gp(P_n ⊠ P_m) ≤ 4. On the other hand, as P_n ⊠ P_m contains K₄ we also have gp(P_n ⊠ P_m) ≥ 4. We conclude that

gp(Pn⊠Pm)=4,n,m≥2. (1)

This result shows that the bound in Theorem 3.1 is sharp. More sharpness examples are provided with the next result which also shows the tightness of Corollary 4.1.

Proposition 4.3

If G is a connected graph and n ≥ 1, then gp(G ⊠ K_n) = n ⋅ gp(G). Moreover, if gp(G) = ω(G_SR), then gp(G ⊠ K_n) = ω((G ⊠ K_n)_SR).

Proof

The first assertion follows by combining Theorem 4.2 with Corollary 4.1.

Suppose now that in addition gp(G) = ω(G_SR) holds. In view of Theorem 3.1, there exists a gp-set S_G of G that induces a complete subgraph of G_SR. By the proof of Theorem 4.2, S_G × V(K_n) is a gp-set of G ⊠ K_n. The components of the subgraph of G ⊠ K_n induced by S_G × V(K_n) are of the form Q ⊠ K_n, where Q is a complete component induced by S_G.

If (g, x) and (g′, x′) belong to different components Q ⊠ K_n and Q′ ⊠ K_n induced by S_G × V(K_n), then with Proposition 2.5 in mind we have d_{G⊠K_n}((g, x), (g′, x′)) = d_G(g, g). Since g and g′ are MMD, this implies that also (g, x) and (g′, x′) are MMD.

Suppose next that (g, x) and (g′, x′) belong to the same component Q ⊠ K_n. If g = g′, then (g, x) and (g′, x′) are clearly MMD. Suppose now that g ≠ g′. Then, since g and g′ are adjacent and MMD in G, the vertices g and g′ are true twins. But then it follows that (g, x) and (g′, x′) are MMD in G ⊠ K_n. The second assertion now follows from Theorem 2.2.□

Since gp(T) = t for every tree T with t leaves, we have gp(T ⊠ P_n) ≥ 2t by Theorem 4.2. We next show that this becomes an equality for an infinite number of trees. To this end, we say that a tree T belongs to a family 𝓣 if there exits a finite sequence T₁, …, T_r, r ≥ 1, of trees such that,

T₁ is a path on at least three vertices;
T₂ is obtained from T₁ by adding a path P of order at least 3 and joining by an edge one not leaf vertex of P with one not leaf vertex of T₁;
for every i ∈ {3, …, r}, T_i is obtained from T_i–1 by adding a path P of order at least 3 and joining by an edge one not leaf vertex of P with one vertex of degree larger than two of T_i–1; and
T = T_r.

Note that if T ∈ 𝓣 is obtained by the above construction in r steps, then T has exactly 2r leaves.

Proposition 4.4

If T ∈ 𝓣 and has 2r leaves, then gp(T ⊠ P_n) = 4r = ω((T ⊠ P_n)_SR).

Proof

From Theorem 4.2 we get gp(T ⊠ P_n) ≥ 4r. We next show that this is also the exact value.

Let P_{n_i}, i ∈ [r], be the path used to generate T in the i^th step of the construction of T. Let S_i = V(P_{n_i}) × V(P_n), i ∈ [r]. Then note that S₁, …, S_r form a partition of V(T ⊠ P_n), where each S_i induces a graph isomorphic to the strong grid graph P_{n_i} ⊠ P_n.

Since {S₁, …, S_r} form an isometric cover of T ⊠ P_n, Theorem 2.1 and (1) imply that

gp(T⊠Pn)≤∑i=1rgp(〈Si〉)=4r,

hence the first equality follows.

From [9, Theorem 40] we know that T_SR ⊠ (P_n)_SR is a subgraph of (T ⊠ P_n)_SR. Since T_SR ⊠ (P_n)_SR contains a clique of size 4r, we then also have such a clique in (T ⊠ P_n)_SR and so ω((T ⊠ P_n)_SR) ≥ 4r. Theorem 3.1 completes the argument.□

Proposition 4.5

If r₁ ≥ t₁ ≥ 1 and r₂ ≥ t₂ ≥ 1, then

gp(Kr1,t1⊠Kr2,t2)=r1r2=ω((Kr1,t1⊠Kr2,t2)SR)=α(Kr1,t1⊠Kr2,t2).

Proof

We first observe that diam(K_r₁,t₁ ⊠ K_r₂,t₂) = 2 and thus Theorem 2.3 applies.

The set obtained from the Cartesian product of the partite sets of cardinality r₁ and r₂ of K_r₁,t₁ and K_r₂,t₂, respectively, forms a maximal induced complete multipartite subgraph of the complement of K_r₁,t₁ ⊠ K_r₂,t₂ of cardinality r₁r₂. Since ω(K_r₁,t₁ ⊠ K_r₂,t₂) = 4, we deduce that gp(K_r₁,t₁ ⊠ K_r₂,t₂) = r₁r₂.

On the other hand, since K_r₁,t₁ ⊠ K_r₂,t₂ has diameter two and has not true twin vertices, the strong resolving graph (K_r₁,t₁ ⊠ K_r₂,t₂)_SR is just the complement of K_r₁,t₁ ⊠ K_r₂,t₂. Thus, we obtain that ω((K_r₁,t₁ ⊠ K_r₂,t₂)_SR) = α(K_r₁,t₁ ⊠ K_r₂,t₂) = r₁r₂, hence the last two equalities.□

Theorem 4.6

If r ≥ 2 and t ≥ 1, then 6 ≤ gp(P_r ⊠ C_2t+1) ≤ 7. Moreover, if t ∈ [2] or r = 2, then gp(P_r ⊠ C_2t+1) = 6.

Proof

If t = 1, then by Proposition 4.3, gp(P_r ⊠ C₃) = gp(P_r ⊠ K₃) = 3gp(P_r) = 6. Hence, from now on we may assume t ≥ 2.

Let U = {u₁, …, u_r} and V = {v₁, …, v_2t+1} be the vertex sets of P_r and C_2t+1, respectively, with natural adjacencies. From Theorem 4.2, we know that gp(P_r ⊠ C_2t+1) ≥ 6. A subpath P of C_2t+1 which is of length at most t is an isometric subgraph of C_2t+1, hence U × P induces an isometric subgraph of P_r ⊠ C_2t+1. In particular this implies that the set {U × {v₁, …, v_t+1}, U × {v_t+2, …, v_2t+1}} forms an isometric cover of P_r ⊠ C_2t+1 consisting of two strong grids. Hence, again using Theorem 2.1 together with (1) we infer that gp(P_r ⊠ C_2t+1) ≤ 8.

We now suppose that gp(P_r ⊠ C_2t+1) = 8 and let S be a gp-set of P_r ⊠ C_2t+1. Let S′ be the projection of S onto C_2t+1 and consider the following situations.

|S′| = 8.

This means that for each v_i ∈ S′ we have |(U × {v_i}) ∩ S| = 1. Without loss of generality we can assume that v₁ ∈ S′. Consider now a partition of V given by the sets V₁ = {v₁, …, v_t+1} and V₂ = {v_t+2, …, v_2t+1}. As noted above, U × V₁ and U × V₂ induce strong grids that are isometric subgraphs of P_r ⊠ C_2t+1. Thus, by (1) and since we have assumed gp(P_r ⊠ C_2t+1) = 8, we deduce |S ∩ (U × V₁)| = 4 and |S ∩ (U × V₂)| = 4. Analogously, if V1′ = {v₂, …, v_t+1} and V2′ = {v_t+2, …, v_2t+1, v₁}, then also U × V1′ and U × V2′ induce two strong grids that are isometric subgraphs of P_r ⊠ C_2t+1. Since |S ∩ (U × V2′ )| = 5, we get a contradiction.
4 ≤ |S′| ≤ 7.

This means that there exists at least one vertex v_i ∈ S′ such that |(U × {v_i}) ∩ S| = 2. Note that for every v_i ∈ S′, it must happen |(U × {v_i}) ∩ S| ≤ 2, otherwise we find a geodesic containing three vertices of S. Without loss of generality we can assume that v₁ ∈ S′ satisfies that |(U × {v₁}) ∩ S| = 2. A similar argument as in Case 1 leads to the partition of V given by V1′ and V2′ (as in Case 1), and such that U × V1′ and U × V2′ induce strong grids that are isometric subgraphs of P_r ⊠ C_2t+1 for which |S ∩ (U × V2′ )| = 6, which is again not possible.
|S′| ≤ 3.

Since for every v_i ∈ S′, it must happen |(U × {v_i}) ∩ S| ≤ 2, we deduce that |S| = ∑_{v_i∈S′}|(U × {v_i}) ∩ S| ≤ 2|S′| ≤ 6. This is a final contradiction proving that |S| = 8 is not possible.

We have thus proved that gp(P_r ⊠ C_2t+1) ≤ 7. Let next t = 2. Then we consider again the projection S′ as defined above, but in this case we clearly have |S′| ≤ 5. Now, if |S| = 7, then we get a contradiction along the same lines as above. Hence gp(P_r ⊠ C₅) = 6. Finally, if r = 2, then the situation in which |S′| = 7 leads to the existence of seven vertices lying in different layers of the factor graph P₂. But then there are three of such vertices lying on the same geodesic, which is not possible and so gp(P₂ ⊠ C_2t+1) = 6.□

Upper bounds on the general position number of the cylinder P_r ⊠ C_2t and of the torus C_r ⊠ C_t, can be deduced by using similar techniques as in the proof above, except that in the last two cases we split the torus into two cylinders. On the other hand, lower bounds can be obtained from Theorem 4.2. That is next stated.

Remark 4.7

Let r, t be two integers.

If r ≥ 2 and t ≥ 3, then 6 ≤ gp(P_r ⊠ C_2t) ≤ 8.
If r ≥ 5 and t ≥ 3, then 9 ≤ gp(C_r ⊠ C_2t) ≤ 16.
If r ≥ 4 and t ≥ 2, then 9 ≤ gp(C_r ⊠ C_2t+1) ≤ 14.

Using similar approach as in the proof of Theorem 4.6, the last upper bound 14 from Remark 4.7 can be lowered to 13.

Since C₄ is a complete bipartite graph and satisfies gp(C₄) = 2, from Proposition 4.5, we obtain that gp(C₄ ⊠ C₄) = 4. Note that it also occurs the equality gp(C₄ ⊠ C₄) = 4 = ω((C₄ ⊠ C₄)_SR) (for information on the structure of (C₄ ⊠ C₄)_SR see [9]).

Based on the results of this section we pose the following:

Problem 4.8

Is it true that if G and H are arbitrary connected graphs, then

gp(G⊠H)=gp(G)gp(H)?

Assuming that the answer to the problem is positive, if gp(G) = ω(G_SR) and gp(H) = ω(H_SR), then gp(G ⊠ H) = ω((G ⊠ H)_SR).

5 Generalized lexicographic products

Let G be a graph with V(G) = {g₁, …, g_n} and let H_i, i ∈ [n], be pairwise disjoint graphs. Then the generalized lexicographic product G[H₁, …, H_n] has the vertex set

⋃i∈[n]{(gi,h):h∈V(Hi)},

and the edge set

{(gi,h)(gj,h′):gigj∈E(G),h∈E(Hi),h′∈E(Hj)}∪⋃i∈[n]{(gi,h)(gi,h′):hh′∈E(Hi)}.

In words, G[H₁, …, H_n] is obtained from G by replacing each vertex v_i ∈ V(G) with the graph H_i, and each edge g_ig_j ∈ E(G) with all possible edges between H_i and H_j. From this reason we will say that v_i ∈ V(G) expands to H_i in G[H₁, …, H_n].

The generalized lexicographic product was introduced by Sabidussi back in [13]. If all the graphs H_i, i ∈ [n], are isomorphic to a graph H, then the generalized lexicographic product G[H₁, …, H_n] = G[H, …, H] becomes the standard lexicographic product G[H].

Theorem 5.1

Let G be a graph with V(G) = {v₁, …, v_n} and let k_i, i ∈ [n], be positive integers. If S is a gp-set of G that induces a complete subgraph of G_SR, and min{k_i : v_i ∈ S} ≥ max{k_i : v_i ∉ S}, then

gp(G[Kk1,…,Kkn])=∑i:vi∈Ski=ω((G[Kk1,…,Kkn])SR).

Proof

Let G and its gp-set S be as stated in the theorem. Then gp(G) = ω(G_SR) by Theorem 3.1. To simplify the notation, let Ĝ = G[K_k₁, …, K_{k_n}] in the rest of the proof. Moreover, if a vertex v_i ∈ V(G) expands to R = K_{k_i} in Ĝ, and v̂ ∈ V(R), then we will write v_i = g(v̂). That is, if v̂ ∈ Ĝ, then g(v̂) is the vertex of G that expands to the complete subgraph of Ĝ to which v̂ belongs.

If x̂, ŷ ∈ V(Ĝ), x̂ ≠ ŷ, then by the construction of Ĝ we infer that

dG^(x^,y^)=1;g(x^)=g(y^),dG(g(x^),g(y^));g(x^)≠g(y^). (2)

By Theorem 2.2, the components of 〈S〉 are complete subgraphs of G, denote them with Q₁, …, Q_r. Then gp(G) = ∑i=1r |V(Q_i)|. Since each vertex of Q_i expands to a complete subgraph of Ĝ, the complete subgraph Q_i expands to a complete subgraph of Ĝ, we will denote it with Q̂_i.

We first claim that Ŝ = ⋃i=1r V(Q̂_i) is a general position set of Ĝ. If x̂ ∈ V(Q̂_i) and x̂′ ∈ V(Q̂_i′), where i, i′ ∈ [r], i ≠ i′, then d_Ĝ(x̂, x̂′) = d_G(g(x̂), g(x̂′)) holds by (2). Therefore, since {Q₁, …, Q_r} form an in-transitive, distance-constant partition of S, the complete subgraphs {Q̂₁, …, Q̂_r} form an in-transitive, distance-constant partition of Ŝ. Hence, in view of Theorem 2.2, Ŝ is a general position set of Ĝ.

We next claim that Ŝ is a gp-set of Ĝ. Assume on the contrary that there exists a general position set T̂ of Ĝ such that |T̂| > |Ŝ|. Applying Theorem 2.2 again we know that the components of 〈T̂〉 are complete graphs. Let T = {g(x̂) : x̂ ∈ T̂}. Since T̂ is a general position set and because of (2) we infer that T is a general position set of G. But since min{k_i : v_i ∈ S} ≥ max{k_i : v_i ∉ S} and |T̂| > |Ŝ| it follows that |T| > |S| = gp(G), a contradiction.

We have thus proved that gp(Ĝ) = ∑_{i:v_i∈S} k_i. To complete the proof we need to show that also ω(Ĝ_SR) = ∑_{i:v_i∈S} k_i. Since S is a complete subgraph of G_SR and because of (2) we get that Ŝ is a set of MMD vertices of Ĝ. By the equality part of Theorem 3.1 we thus have ω(Ĝ_SR) = gp(Ĝ) = ∑_{i:v_i∈S} k_i.□

6 Rooted product graphs

By a rooted graph we mean a connected graph having one fixed vertex called the root of the graph. Consider now a connected graph G of order n, and let H be a rooted graph with root v. The rooted product graph G ∘_v H is the graph obtained from G and n copies of H, say H₁, …, H_n, by identifying the root of H_i with the i^th vertex of G, see [14, 15]. To formulate the following result, the notion of an interval between vertices u and v of a graph G, defined as I_G(u, v) = {w : d_G(u, v) = d_G(u, w) + d_G(w, v)}, will be useful.

Theorem 6.1

Let G be any connected graph of order n ≥ 2, and let H be a rooted graph with root v.

gp(G ∘_v H) = n = ω((G ∘_v H)_SR) if and only if H is a path and v is a leaf of H.
If H contains a gp-set S not containing v and such that for each pair of vertices u, w ∈ S neither u ∈ I_H(v, w) nor w ∈ I_H(v, u), then gp(G ∘_v H) = n ⋅ gp(H). Moreover, if in addition S is a maximum clique in H_SR, then gp(G ∘_v H) = ω((G ∘_v H)_SR).
Suppose H is not a path rooted in one of its leaves. If every gp-set S of H either contains the root v, or contains two vertices x, y such that (x ∈ I_H(v, y) or y ∈ I_H(v, x)), then 2n ≤ gp(G ∘_v H) ≤ n(gp(H) – 1). Particularly, if every gp-set of H contains the root v, then gp(G ∘_v H) = n(gp(H) – 1).

Proof

If G is P₂ and H is a path rooted in a leaf v, then G ∘_v H is also a path, and so gp(G ∘_v H) = 2 = ω((G ∘_v H)_SR). In this sense, from now we may assume G is different from P₂.

If H is a path and v is a leaf, then clearly the set formed by the remaining leaves of all copies of H forms a general position set of G ∘_v H, and so gp(G ∘_v H) ≥ n. Now, suppose gp(G ∘_v H) > n and let S be gp-set of G ∘_v H. In consequence, by the pigeon hole principle there exists a copy, say H_i, of H such that |S ∩ V(H_i)| ≥ 2, and indeed, it must happen |S ∩ V(H_i)| = 2. But then, the two vertices of S ∩ V(H_i) and any other distinct vertex of S lie on a common geodesic, which is not possible. Therefore, gp(G ∘_v H) ≤ n and the first equality follows. On the other hand, it can be easily observed that the strong resolving graph of G ∘_v H is formed by a component isomorphic to a complete graph K_n, and the remaining vertices of it are isolated ones. Thus, ω((G ∘_v H)_SR) = n, which gives the second equality.

On the other hand, assume gp(G ∘_v H) = n = ω((G ∘_v H)_SR). If H is not a path rooted in one of its leaves, then there are at least two vertices of H, say a, b, such that d_H(a, v) = d_H(b, v). In consequence, the set formed by the union of the copies of a and b in each copy of H is a general position set of G ∘_v H of cardinality 2n, which is not possible. Thus, H must be a path rooted in one of its leaves.
Let A_i, i ∈ [n], be a gp-set of H_i satisfying the statement of the item, and let A = ⋃i=1n A_i. Then A is a general position set of G ∘_v H, and so gp(G ∘_v H) ≥ n ⋅ gp(H). Hence, suppose gp(G ∘_v H) > n ⋅ gp(H) and let B be a gp-set of G ∘_v H. Thus, again by the pigeon hole principle, there must be a copy H_j of H such that |B ∩ V(H_j)| > gp(H), but this is impossible since each copy of H is an isometric subgraph of G ∘_v H and B ∩ V(H_j) is a general position set of the graph induced by H_j. Consequently, gp(G ∘_v H) ≤ n ⋅ gp(H) and the equality follows.

On the other hand, assume that A_i is a maximum clique in H_SR. Hence gp(H) = ω(H_SR). Thus, from the above we get that gp(G ∘_v H) = n ⋅ ω(H_SR). It remains only to prove that ω((G ∘_v H)_SR) = n ⋅ω(H_SR). Since any two vertices u, w ∈ A_i satisfy that neither u ∈ I_H(w, v) nor w ∈ I_H(u, v), we see that A = ⋃i=1n A_i (defined as above) is also a clique in (G ∘_v H)_SR, and so ω((G ∘_v H)_SR) ≥ n ⋅ω(H_SR). Clearly, if we suppose that ω((G ∘_v H)_SR) > n ⋅ω(H_SR), then we obtain that some (H_j)_SR contains a clique of cardinality larger than ω(H_SR), which is not possible. Therefore, the required equality follows.
If every gp-set S of H either contains v or contains two vertices x, y such that without loss of generality v belongs to an x, y-geodesic, then in order to construct a general position set of G ∘_v H from the union of the gp-sets S in each copy of H, we need to remove some vertices from each copy of S including v if it is the case. Clearly, the maximum number of vertices we may remove from S is |S| – 1, since a set formed by one vertex from each copy of H is a general position set of G ∘_v H. However, as we next show by removing from S at most |S| – 2 vertices or removing |S| – 1 and adding one other vertex not from S, we also obtain a general position set. Since H is not a path rooted in one of it leaves, there are two vertices x_i, y_i ∈ V(H_i) such that d_{H_i}(x_i, v) = d_{H_i}(y_i, v), i ∈ [n]. Thus, the set Q = ⋃i=1n {x_i, y_i} is a general position set of cardinality 2n in G ∘_v H, and the lower bound follows. On the other hand, let D be a gp-set of G ∘_v H and for every i ∈ [n], let D_i = D ∩ V(H_i). If there is a set D_j such that |D_j| = gp(H) (note that |D_j| ≤ gp(H) since V(H_j) induces an isometric subgraph of G ∘_v H), then either v ∉ D_j or for any two vertices x, y of D_j it must happen that v does not belong to an x, y-geodesic nor to a y, x-geodesic, but this is a contradiction with our assumption. Therefore, for every i ∈ [n], |D_i| ≤ gp(H) – 1, which implies the upper bound.

We now consider the particular case in which every gp-set of H contains the root v. Let S_i be a gp-set of the copy H_i of H and let S = ⋃i=1n (S_i ∖ {v}). Since v belongs to S_i, it happens that v does not belong to any x, y-geodesic for every x, y ∈ S ∖ {v}. Thus, no three vertices of S lie on the same geodesic of G ∘_v H, and so, S is a general position set of G ∘_v H. Therefore, gp(G ∘_v H) = n(gp(H) – 1).□

Equality in the lower bound given in item (iii) of Theorem 6.1 above can be noticed if H is a path rooted in a vertex of degree two, where gp(G ∘_v H) = 2n. On the other hand, as we next observe the value of gp(G ∘_v H) can be very far from both bounds given above.

Proposition 6.2

There is a graph G of order n and a graph H rooted in a vertex v such that n ≪ gp(G ∘_v H) ≪ n(gp(H) – 1).

Proof

We consider a graph H obtained as follows. We begin with a complete graph K_r. Next we add a vertex v and join it by an edge to exactly t vertices of K_r, where 2 ≤ t ≤ r – 2, and choose v as the root of this graph. Note that H has only one gp-set S formed by the set of vertices of the complete graph K_r. Also, note that if x is adjacent to v, then x ∈ I_H(y, v) for any y not adjacent to v.

Now, let G be a connected graph and let A be a gp-set of G ∘_v H. Thus, if A_i = A ∩ V(H_i), then either every vertex of A_i is adjacent to v or no vertex of A_i is adjacent to v, and so |A_i| ≤ max{t, r – t}. As a consequence, gp(G ∘_v H) = |A| = ∑i=1n |A_i| ≤ n ⋅ max{t, r – t}. On the other hand, the union of all neighbors of v in each copy of H in G ∘_v H, or the union of all not neighbors of v in each copy of H in G ∘_v H is clearly a general position set of G ∘_v H, and so gp(G ∘_v H) ≥ n ⋅ max{t, r – t}, which implies the equality gp(G ∘_v H) = n ⋅ max{t, r – t}. Since gp(H) = r, the difference n(gp(H) – 1) – gp(G ∘_v H) = n(r – 1 – max{t, r – t}) can be arbitrarily large, as well as the difference gp(G ∘_v H) – n = n(max{t, r – t} – 1).□

Acknowledgements

We acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-1693, J1-9109, N1-0095, N1-0108). This research was done while the second author was visiting the University of Ljubljana, Slovenia, supported by "Ministerio de Educación, Cultura y Deporte", Spain, under the "José Castillejo" program for young researchers (reference number: CAS18/00030).

References

[1] Manuel P., Klavžar S., A general position problem in graph theory, Bull. Aust. Math. Soc., 2018, 98, 177–187.10.1017/S0004972718000473Search in Google Scholar

[2] Ullas Chandran S.V., Jaya Parthasarathy G., The geodesic irredundant sets in graphs, Int. J. Math. Combin., 2016, 4, 135–143.Search in Google Scholar

[3] Anand B.S., Ullas Chandran S.V., Changat M., Klavžar S., Thomas E.J., Characterization of general position sets and its applications to cographs and bipartite graphs, Appl. Math. Comput., 2019, 359, 84–89.10.1016/j.amc.2019.04.064Search in Google Scholar

[4] Ghorbani M., Klavžar S., Maimani H.R., Momeni M., Rahimi-Mahid F., Rus G., The general position problem on Kneser graphs and on some graph operations, arXiv:1903.04286 [math.CO] (17 Mar 2019).10.7151/dmgt.2269Search in Google Scholar

[5] Klavžar S., Patkós B., Rus G., Yero I.G., On general position sets in Cartesian products, arXiv:1907.04535v3 [math.CO] (25 Jul 2019).10.1007/s00025-021-01438-xSearch in Google Scholar

[6] Manuel P., Klavžar S., The graph theory general position problem on some interconnection networks, Fund. Inform., 2018, 163, 339–350.10.3233/FI-2018-1748Search in Google Scholar

[7] Patkós B., On the general position problem on Kneser graphs, arXiv:1903.08056 [math.CO] (19 Mar 2019).10.26493/1855-3974.1957.a0fSearch in Google Scholar

[8] Oellermann O.R., Peters-Fransen J., The strong metric dimension of graphs and digraphs, Discrete Appl. Math., 2007, 155, 356–364.10.1016/j.dam.2006.06.009Search in Google Scholar

[9] Kuziak D., Puertas M.L., Rodríguez-Velázquez J.A., Yero I.G., Strong resolving graphs: the realization and the characterization problems, Discrete Appl. Math., 2018, 236, 270–287.10.1016/j.dam.2017.11.013Search in Google Scholar

[10] Kanté M.M., Sampaio R.M., dos Santos V.F., Szwarcfiter J.L., On the geodetic rank of a graph, J. Comb., 2017, 8, 323–340.10.4310/JOC.2017.v8.n2.a5Search in Google Scholar

[11] Hammack R., Imrich W., Klavžar S., Handbook of Product Graphs, Second Edition, 2011, CRC Press, Boca Raton, FL.10.1201/b10959Search in Google Scholar

[12] Rodríguez-Velázquez J.A., Yero I.G., Kuziak D., Oellermann O.R., On the strong metric dimension of Cartesian and direct products of graphs, Discrete Math., 2014, 335, 8–19.10.1016/j.disc.2014.06.023Search in Google Scholar

[13] Sabidussi G., Graph derivatives, Math. Z., 1961, 76, 385–401.10.1007/BF01210984Search in Google Scholar

[14] Godsil C.D., McKay B.D., A new graph product and its spectrum, Bull. Aust. Math. Soc., 1978, 18, 21–28.10.1017/S0004972700007760Search in Google Scholar

[15] Schwenk A.J., Computing the characteristic polynomial of a graph, In: Bari R.A., Harary F. (Eds.), Graphs and Combinatorics, Lecture Notes Math., 1974, 406, 153–172.10.1007/BFb0066438Search in Google Scholar

Received: 2019-04-24

Accepted: 2019-08-23

Published Online: 2019-10-13

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0088

Keywords for this article

general position problem; strong resolving graph; graph products

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