Error-correcting open-locating-dominating sets

Devin C. Jean1, Suk J. Seo2
1Vanderbilt University
2Middle Tennessee State University

Abstract

An open-locating-dominating set of a graph models a detection system for a facility with a possible “intruder” or a multiprocessor network with a possible malfunctioning processor. A “sensor” or “detector” is assumed to be installed at a subset of vertices where each can detect an intruder or a malfunctioning processor in its neighborhood, but not at its own location. We consider a fault-tolerant variant of an open-locating-dominating set called an error-correcting open-locating-dominating set, which can correct a false-positive or a false-negative signal from a detector. In particular, we prove the problem of finding a minimum error-correcting open-locating-dominating set in an arbitrary graph is NP-complete. Additionally, we characterize the existence criteria for an error-correcting open-locating-dominating set in an arbitrary graph. We also consider extremal graphs that require every vertex to be a detector and minimum error-correcting open-locating-dominating sets in infinite grids.

Keywords: system, open-locating-dominating sets, error-correction, cubic graphs, NP-completeness, extremal graphs, infinite grids

1. Introduction

Let G be a graph with vertices V(G) and edges E(G); we assume that all graphs are simple, undirected, and connected. For notation, we let n=|V(G)|, m=|E(G)|, deg(v)=|{abE(G):v{a,b}}|, δ(G)=minvV(G)deg(v), and Δ(G)=maxvV(G)deg(v). The open-neighborhood of a vertex vV(G), denoted N(v), is the set of all vertices adjacent to v: {wV(G):vwE(G)}. The closed-neighborhood of a vertex vV(G), denoted N[v], is the set of all vertices adjacent to v, as well as v itself: N(v){v}. A vertex set SV(G) is a (closed) dominating set if all vertices are within the closed neighborhood of some vS; that is, vSN[v]=V(G). Similarly, SV(G) is an open-dominating set if all vertices are within the open neighborhood of some vS; that is, vSN(v)=V(G).

Assume that G models a facility with a possible “intruder” (or a multiprocessor network with a possible malfunctioning processor), where we want to be able to precisely determine the location of the intruder by placing (the minimum number of) detectors. Much work has been done on the topic of the graphical parameters involving various types of detectors. If a detector at v can determine only whether or not the intruder is in N[v], then the detection system is called an identifying code (IC) as introduced by Karpovsky, Charkrabarty and Levitin [16]. A dominating set S of V(G) is an identifying code if for any any two distinct vertices u,vV(G), we have N[u]SN[v]S. If a detector at v can determine only whether or not the intruder is in N[v] but can differentiate v from N(v), then the detection system is called a locating-dominating (LD) set, as introduced in Slater [25]. A dominating set S of V(G) is a locating-dominating set if for any two distinct vertices u,vV(G)S, we have N(u)SN(v)S.

This paper focuses on a detection system where a sensor at vertex v can sense intruders at adjacent vertices (i.e., within N(v)), but not at v itself. This type of detection system is called an open-locating-dominating (OLD) set, as introduced by Honkala, Laihonen and Ranto [11] and Seo and Slater [21]. An open-dominating set SV(G) is an open-locating-dominating set if for any two distinct vertices u,vV(G), we have N(u)SN(v)S. Additional recent work on variations of OLD sets with larger detection ranges includes Givens et al. [9]. Jean and Lobstein [12] have maintained a bibliography, currently with more than 500 entries, for work on detection system related topics.

For an OLD set SV(G) and uV(G), we let NS(u)=N(u)S denote the (open) dominators of u and dom(u)=|NS(u)| denote the (open) domination number of u. A vertex vV(G) is k-open-dominated by an open-dominating set S if |NS(v)|k. If S is an open-dominating set and u,vV(G), u and v are k-distinguished if |NS(u)NS(v)|k, where denotes the symmetric difference. If S is an open-dominating set and u,vV(G), u and v are k#-distinguished if |NS(u)NS(v)|k or |NS(v)NS(u)|k.

The purpose of k-domination in a detection system is to guarantee there are at least k detectors monitoring each location in the graph simultaneously, which can allow for a level of basic fault tolerance. A fault-tolerant variant of an OLD set is called a redundant open-locating-dominating (RED:OLD) set which is resilient to a (single) detector being destroyed or going offline [23]. That is, an OLD set SV(G) is a RED:OLD set if vS, S{v} is an OLD set. For fault-tolerance beyond simple k-redundancy, we conceptually establish that all detectors transmit either 0 to represent the absence of an intruder, or 1 to represent the presence of an intruder. A false negative is therefore when a detector incorrectly transmits a 0 instead of a 1, while a false positive is when a detector incorrectly transmits a 1 instead of a 0. An OLD set is called an error-detecting open-locating-dominating (DET:OLD) set if it can tolerant any single false negative [23]. Extending this, an OLD set is called an error-correcting open-locating-dominating (ERR:OLD) set, if it can tolerate any single false positive. The following theorem characterizes OLD, RED:OLD, DET:OLD, and ERR:OLD sets and they are useful in constructing those sets or verifying whether a given set meets their requirements.

Theorem 1.1. An open-dominating set is

(i) an OLD set if and only if every pair of vertices is 1-distinguished [14];

(ii) a RED:OLD set if and only if all vertices are 2-dominated and all pairs are 2-distinguished [23];

(iii) a DET:OLD set if and only if all vertices are 2-dominated and all pairs are 2#-distinguished [23];

(iv) an ERR:OLD set if and only if all vertices are 3-dominated an all pairs are 3-distinguished.

In this paper, we will focus on the ERR:OLD parameter. As can be seen in Theorem 1.1, each new fault-tolerant capability on top of the basic OLD set demands stronger requirements for both the domination and distinguishing properties of the detector set. To see why the requirements for DET:OLD are not sufficient for ERR:OLD, consider the following two simple examples using detector set SV(G). Firstly, if there is some 2-dominated vertex v, with NS(v)={p,q}, then an intruder at v coupled with a false-negative at p is indistinguishable from the absence of an intruder at v coupled with a false-positive at q. Thus, 2-domination is insufficient for an ERR:OLD set. And secondly, if there are two vertices u,v with NS(u)NS(v)={a,b} and (u,v) are 2#-distinguished with NS(v)NS(u)={p,q}, then (p,q,a,b) transmitting (0,1,1,1) could be due to a false-positive at q (in which case the intruder is at u) or a false-negative at p (in which case the intruder is at v). Thus, we cannot eliminate either of u or v as the intruder location, meaning 2#-distinguishing is insufficient for an ERR:OLD. Indeed, it has been shown that an ERR:OLD set, or rather an equivalent error-correcting separating set, requires 3-domination and 3-distinguishing [24]. With both of these stronger properties, we see that two previous problem cases are resolved.

f1-1
Figure 1 Optimal OLD (a), RED:OLD (b), DET:OLD (c), and ERR:OLD (d) sets. Detectors are darkened.

For finite graphs, the notations OLD(G), RED:OLD(G), DET:OLD(G), and ERR: OLD(G) represent the cardinality of the smallest possible OLD, RED:OLD, DET: OLD, and ERR:OLD sets on graph G, respectively [23]. From Figure 1, which shows optimal solutions on the given graph which we will call G, we see that OLD(G)=6, RED:OLD(G)=7, DET:OLD(G)=10, and ERR:OLD(G)=11.

In Section 2, we characterize the existence criteria for error-correcting open-locating-dominating sets in an arbitrary graph. We also consider extremal graphs that require every vertex to be a detector. In Section 3, we present a proof that the problem of finding a minimum error-correcting open-locating-dominating set in an arbitrary graph is NP-complete. We conclude with Section 4 by establishing bounds on the minimum densities of error-correcting open-locating-dominating sets on the infinite square grid, infinite triangular grid, and infinite king grid.

2. Existence of ERR:OLD sets and extremal graphs

In this section, we discuss the existence criteria of ERR:OLD sets and some extremal graphs. But before this, we will first present previous results on the existence criteria and some extremal graphs for the OLD, RED:OLD, and DET:OLD parameters. From Theorem 1.1, it is not hard to see the following relationships between the minimum cardinality of the four fault-tolerant OLD sets:

Observation 2.1. If G permits ERR:OLD, then OLD(G)RED:OLD(G)DET:OLD(G)ERR:OLD(G).

OLD sets

From Theorem 1.1, by using S=V(G) we can arrive at the following existence criteria for OLD on arbitrary graphs:

Theorem 2.2. [21] A graph G has an OLD set if and only if G satisfies δ(G)1 and, whenever uv, N(u)N(v).

Chellali et al. [2] proved the upper and lower bounds for OLD(G) on arbitrary graphs:

Observation 2.3 [2] If graph G has an OLD set, then 2OLD(G)n.

The upper bound of OLD(G)=n is achievable, and Foucaud et al. [7] later provided a full characterization for this class of extremal graphs:

Definition 2.4 . [6] For any kN, the half-graph, Hk is given by V={v1,,vk}{w1,,wk} and E={viwj:ij}.
Theorem 2.5. [7] OLD(G)=n if and only if G is a half-graph.

Seo and Slater established the existence criteria for trees, determined the lower and upper bounds for OLD(G) on trees, and fully characterized the extremal trees which achieve these upper and lower bounds [22].

Theorem 2.6. [22] A tree T of order n2 has an OLD set if and only if every vertex is adjacent to at most one leaf.
Theorem 2.7. [22] If tree T has an OLD set, then n/2+1OLD(G)n1.

Henning and Yeo [10] determined the upper bound on OLD(G) when G is a cubic graph.

Theorem 2.8. [10] If G is a twin-free cubic graph of order n, then OLD(G)34n.

RED:OLD sets

From Theorem 1.1, by using S=V(G), we can arrive at the following existence criteria for RED:OLD on arbitrary graphs:

Theorem 2.9. A graph, G, has RED:OLD if and only if δ(G)2 and for each distinct u,vV(G), |N(u)N(v)|2.

These existence criteria can be simplified when discussing regular graphs:

Theorem 2.10. [20] Let G be an r-regular graph with r2. There exists a RED:OLD set for G if and only if there exists an OLD set for G.

Further, we have the following results from Seo [20] when discussing cubic graphs specifically:

Theorem 2.11. [20] Let G be a cubic graph. If G is C4-free, then V(G) is a RED:OLD set.
Theorem 2.12. [20] Let G be a cubic graph of order n. Then we have 23nRED:OLD(G)n.

Seo [20] has also constructed an infinite family of finite cubic graphs exhibiting the extremal value of RED:OLD(G)=23n.

DET:OLD sets

Similar to the existence criteria for OLD and RED:OLD sets, one can simply use Theorem 1.1 with S=V(G) to arrive at the existence criteria for DET:OLD sets in arbitrary graphs. While there are no simpler existence criteria on arbitrary graphs, the following necessary conditions have been found:

Observation 2.13. [13] If G permits DET:OLD, then every vV(G) has at most one degree 2 neighbor.
Theorem 2.14. [13] If Gn,m has DET:OLD, then m3nn22.

Further, we have the following results from Seo [20] when discussing cubic graphs specifically:

Theorem 2.15. [20] If G is a C4-free cubic graph, then for any vertex xV(G), V(G){x} is a DET:OLD set for G.
Theorem 2.16. [20] A cubic graph permits DET:OLD if and only if it is C4-free.
Theorem 2.17. [13,20] If G is a C4-free cubic graph, then 67DET:OLD\%(G)3031.

The lower bound from Theorem 2.17 is achieved by the infinite hexagonal grid [23] and an infinite family of finite cubic graphs [20].

ERR:OLD Sets

Because an ERR:OLD set requires 3-domination, many graphs do not allow an ERR:OLD set to exist. In light of this, our next theorem characterizes all graphs that permit an ERR:OLD set.

Theorem 2.18. A graph G permits an ERR:OLD set if and only if δ(G)3 and for every C4 subgraph, abcd, |N(a)N(c)|3.

Proof. First, assume to the contrary that there exists a vertex v with deg(v)2; then, v cannot be 3-dominated, a contradiction. Similarly, if we assume there exists a,cV(G) with |N(a)N(c)|2, then a and c could not be 3-distinguished, a contradiction.

For the converse, suppose G satisfies both properties of the theorem statement. To demonstrate the existence of ERR:OLD, we will conservatively let S=V(G) be the detector set. Clearly, δ(G)3 guarantees all vertices will be 3-dominated, so we need only show that all vertex pairs are 3-distinguished. Let u,vV(G) with uv. If d(u,v)3 then u and v will have non-intersecting neighborhoods and thus be 3-distinguished; otherwise, we can assume d(u,v)2. Suppose d(u,v)=2, and let uxv be a path from u to v. If |N(u)N(v)|=1, then u and v must be 4-distinguished, and we would be done, so assume {x,y}N(u)N(v) with xy. Then we have a C4 subgraph, uxvy, so by hypothesis |N(u)N(v)|3 and (u,v) are 3-distinguished. Finally, suppose d(u,v)=1. If there is some p(N(u)N(v)){u,v}, then u and v will be 3-distinguished by {u,v,p}, and we would be done; otherwise, we assume that N(u)N(v)={u,v}. Because δ(G)3, it must be that p,qN(u)N(v) with pq. Then we have a C4 subgraph upvq, so by hypothesis u and v must be 3-distinguished, completing the proof. ◻

From Theorem 2.18, it is clear that checking whether or not a graph permits an ERR:OLD set can be done in polynomial time.

Corollory 2.19. A C4-free graph, G, permits an ERR:OLD set if and only if δ(G)3.

Two distinct vertices u,vV(G) are said to be twins if N[u]=N[v] (closed twins) or N(u)=N(v) (open twins). From Theorem 2.18, we see that if G has twins u and v, then it cannot have an ERR:OLD set since deg(u)=deg(v)2 or u and v form the corners of a C4 subgraph with |N(u)N(v)|2.

Corollary 2.20. If G permits an ERR:OLD set, then G is twin-free.

Next, we will show that there are no graphs with ERR:OLD sets on n6 vertices.

Theorem 2.21. If G has an ERR:OLD set, then n7.

Proof. Firstly, we know that G cannot be acyclic because an ERR:OLD set requires δ(G)3. Thus, G has a cycle, which we will exploit in casing. We will proceed by casing on the size of the smallest cycle, which implies that edges cannot be added between the vertices of said smallest cycle. We assume n6, as otherwise we would be done. If the smallest cycle is Ck for 5k6, then every vertex in the cycle must have a third dominator, and these additional k vertices must all be distinct to avoid creating smaller cycles; thus n2k10>6, a contradiction. Suppose the smallest cycle is a C4 subgraph abcd, then each vertex in the cycle must have a third dominator, a, b, c, and d adjacent to a, b, c, and d, respectively. We see that abcda; however, because n6 it must be that a=c and b=d. We see that a and c cannot have any additional edges without creating a smaller cycle, but they are open twins and so cannot be 3-distinguished, a contradiction. Thus, we can assume that G has a triangle.

Next, suppose there exists a 4-Cycle abcd with bdE(G), which we call a “diamond” subgraph. If acE(G) then distinguishing a and c will result in n7, a contradiction; thus, we can assume that acE(G), meaning we have a K4 subgraph. By symmetry, we can assume that a, b, and c have third dominators a, b, and c. If a, b, and c are all distinct, then n7 and we would be done; otherwise without loss of generality assume a=b. Suppose a=b=c; then we require daE(G) to be able to distinguish d and a, and by symmetry we have xN(d)N(a){a,b,c,d,a}. We now have a K5 subgraph plus an extra vertex xN(d). Distinguishing a and b requires without loss of generality xN(a), but then a and d cannot be 3-distinguished, a contradiction. Otherwise, we can assume a=bc; we see that distinguishing a and b requires, without loss of generality, bcE(G). If adE(G) then dcE(G) to distinguish a and d; however, b and d can no longer be 3-distinguished, a contradiction, so we assume adE(G) and dcE(G). If acE(G) then b and c cannot be 3-distinguished, a contradiction, so by symmetry we assume acE(G) and caE(G). Thus, 3-dominating a and c requires acE(G), but then c and a cannot be 3-distinguished, a contradiction. Thus, the existence of a diamond subgraph implies that n7.

At this point, we know that G has a triangle, say abc. To 3-dominate the vertices in abc, assume aN(a){b,c}, bN(b)={a,c}, and cN(c){a,b}. Additionally, we can assume a, b, and c are distinct because otherwise we produce a diamond subgraph. To 3-dominate a, b, and c without producing a diamond subgraph, it must be that {ab,bc,ca}E(G). At this point we can no longer add any edges without producing a diamond subgraph, but a and b are not 3-distinguished, a contradiction, completing the proof. ◻

Figure 2 shows the first graphs with an ERR:OLD set in the lexicographic ordering of (n,m) tuples; i.e., the graphs with the smallest number of edges given the smallest number of vertices. For ERR:OLD, this is n=7 and m=12.

f2-1
Figure 2 The two smallest graphs supporting ERR:OLD, which have n=7 and m=12

Next, we will show the relation between the existence of ERR:OLD sets and DET:OLD sets for regular graphs.

Observation 2.22. If A and B are sets with |A|=|B|, then |AB|=k implies |AB|=2k.

Proof. Suppose |AB|=k. Then |AB|=|A|k=|B|k. Therefore |AB|=|AB|+|BA|=k+(|B||BA|)=k+(|B|(|B|k))=k+k, completing the proof. ◻

Suppose G is a regular graph with DET:OLD. We can conservatively let S=V(G) be the set of detectors. Then every vertex pair in G must be 2#-distinguished; thus, Theorem 2.22 yields that every vertex pair is 4-distinguished, which is sufficient for the distinguishing requirement of ERR:OLD.

Proposition 2.23. A regular graph G with δ(G)=Δ(G)3 has a DET:OLD set if and only if it has an ERR:OLD set.

A graph is cubic or 3-regular if every vertex is of degree 3. Jean and Seo [13] have previously proven that a cubic graph has a DET:OLD set if and only if the graph is C4-free; thus, we have the following existence criteria.

Corollary 2.24. A cubic graph G has an ERR:OLD set if and only if G is C4-free.

Now we consider extremal graphs with ERR:OLD(G)=n with least m. If a graph G has an ERR:OLD set, then Theorem 2.18 yields that δ(G)3, which serves as a lower bound for the least m, namely m3n2. Figures 2 and 3 show extremal graphs with ERR:OLD(G)=n with the fewest number of edges when 7n9. For even n10, we will show that cubic graphs exist which meet the theoretical lower bound of m=3n2. For odd n11, this bound of m=3n2 is trivially unobtainable, but we introduce a new class of “almost-cubic” graphs meeting the lower bound of m=3n+12. In either case, we provide an extremal graph for each n10.

f3-1
Figure 3 Extremal graphs with ERR:OLD(G)=n with the fewest number of edges when n=8 and n=9.
Definition 2.25. G is quasi-cubic if every vertex is degree 3 aside from one degree-4 vertex.

When n10, cubic and quasi-cubic graphs together form the entire family of extremal graphs with ERR:OLD(G)=n which have the minimum number of edges for a given n. If S is an ERR:OLD set, we know that vS if wN(v) with deg(w)=3. Because every vertex in a cubic or quasi-cubic graph is adjacent to a degree-3 vertex, these classes or graphs require RED:OLD(G)=n if they permit ERR:OLD.

Theorem 2.26. If G is cubic or quasi-cubic, then G has an ERR:OLD set if and only if G is C4-free.

Proof. Suppose G has a C4 subgraph, abcd. Because G is cubic or quasi-cubic, at most one vertex in abcd can be degree 4, and all others will be degree 3. By symmetry, assume deg(a)=deg(c)=3. Then a and c cannot be 3-distinguished, implying G does not have ERR:OLD.

For the converse, suppose G does not have ERR:OLD. Because δ(G)3, we know that an ERR:OLD set not existing implies u,vV(G) where u and v are not 3-distinguished. Regardless of whether or not uvE(G), we see the only way to have u and v not be 3-distinguished is by having |N(u)N(v)|2, which creates a C4 subgraph, completing the proof. ◻

Suppose we have a cubic graph G that permits an ERR:OLD set. Then, we can construct a quasi-cubic graph from G as shown in the next theorem.

Theorem 2.27. Let G be a cubic graph with ERR:OLD where a,b,c,dV(G) and ab,cdE(G) such that ab and cd are not part of a triangle or the terminal edges of a P5 subgraph. Construct quasi-cubic graph G from G by deleting edges ab and cd and adding a new vertex x with N(x)={a,b,c,d}. Then G has an ERR:OLD set.

Proof. By Theorem 2.26, we know that G is C4-free and we need only show that G is likewise C4-free. Suppose for a contradiction that G has a C4 subgraph; then due to G being C4-free, the C4 subgraph must involve vertex x, which is an endpoint of every new edge in G. By symmetry, we can reduce the problem to two possible cases: the C4 includes edges {ax,xb} or edges {ax,xc}, and we will consider yV(G) to be the opposite vertex of x in the C4 subgraph. If we are in the {ax,xb} case, we see that any choice of y results in aby forming a triangle in G, a contradiction. Otherwise, in the {ax,xc} case we see that yb and yd because ab,cdE(G), so y causes ab and cd to be the ends of a P5 subgraph in G, a contradiction, completing the proof. ◻

3. NP-Completeness of ERR:OLD minimization

It has been shown that determining many graphical parameters related to detection systems, such as the minimum cardinality of an IC, LD, or OLD set, are NP-complete problems on arbitrary graphs [1,3,4,21]. Similarly, the problems involving RED:OLD and DET:OLD parameters have been known to be NP-complete [5,13]. We will now prove that the problem of determining the smallest ERR:OLD set is also NP-complete. For additional information about NP-completeness, see Garey and Johnson [8].

3-SAT

INSTANCE: Let X be a set of N variables. Let ψ be a conjunction of M clauses, where each clause is a disjunction of three literals from distinct variables of X.
QUESTION: Is there is an assignment of values to X such that ψ is true?

ERR:OLD Minimization

INSTANCE: A graph G and integer K with 7K|V(G)|.
QUESTION: Is there an ERR:OLD set S with |S|K? Or equivalently, is ERR:OLD(G) K?

f4-1
Lemma 3.1. Given a graph G, let G7 be the subgraph shown in Figure 4, where vertices {b,c,d,e} have no additional edges in the larger graph but {a,f,g} are permitted to have additional edges to vertices outside of this subgraph. If S is an ERR:OLD set for G, then V(G7)S.

Proof. To 3-distinguish b and c, we require {b,c,d}S. To 3-dominate b, we additionally require {a,f}S. By symmetry, we have that V(G7)S, completing the proof. ◻

f5-1
f6-1
Figure 6 Construction of G from (x1¯x2¯x3)(x1x2x4¯)(x1x3¯x4) with N=4, M=3, K=109
Theorem 3.2. The ERR:OLD Minimization problem is NP-complete.

Proof. Clearly, the ERR:OLD minimization is NP, as every possible candidate solution can be generated nondeterministically in polynomial time (specifically, O(n) time), and each candidate can be verified in polynomial time using Theorem 1.1 (iv). To complete the proof, we will now show a reduction from 3-SAT to ERR:OLD minimization.

Let ψ be an instance of the 3-SAT problem with M clauses on N variables. We will construct a graph, G, as follows. For each variable xi, create an instance of the Fi graph (Figure 5); this includes a vertex for xi and its negation xi¯. For each clause yj of ψ, create a new instance of the Hj graph (Figure 5). For each clause yj=αβγ, create an edge from the yj vertex to α, β, and γ from the variable graphs, each of which is either some xi or xi¯; for an example, see Figure 6. The resulting graph has precisely 25N+8M vertices and 51N+17M edges, and can be constructed in polynomial time. To complete the problem instance, we define K=22N+7M.

Suppose SV(G) is an ERR:OLD set on G with |S|K. By Lemma 3.1, we require at least all of the 21N+7M detectors shown by the shaded vertices in Figure 5. Additionally, in each Fi the vertices pi and qi require xiS or xi¯S in order to be 3-dominated. Thus, we find that |S|22N+7M=K, implying that |S|=K, so |{xi,xi¯}S|=1 for each i{1,,N}. For each Hj, we see that yj is not 3-dominated unless it is adjacent to at least one additional detector from its three neighbors in the Fi graphs; therefore, ψ is satisfiable.

For the converse, suppose we have a solution to the 3-SAT problem ψ; we will show that there is a ERR:OLD, S, on G with |S|K. We construct S by first including all of the 21N+7M vertices as shown in Figure 5. Next, for each variable, xi, if xi is true then we let the vertex xiS; otherwise, we let xi¯S. Thus, the fully-constructed S has |S|=22N+7M=K. It can be shown that every vertex pair is now 3-distinguished, and that every vertex which is not a yj clause vertex is 3-dominated. Because we selected each xiS or xi¯S based on a satisfying truth assignment for ψ, each yj must be adjacent to at least one additional detector vertex from the Fi graphs. Thus, all vertices are 3-dominated, implying S is an ERR:OLD set for G with |S|K, completing the proof. ◻

4. ERR:OLD on infinite grids

In this section we will explore constructions of minimum-sized ERR:OLD sets on three popular infinite grids explored in literature: the infinite square grid (SQR), the infinite triangular grid (TRI), and the infinite king grid (KNG). Given this goal, a natural question is how we are to define such a “minimum-sized” infinite subset of an infinite vertex set. For this, we employ the concept of the density of a set of vertices and employ notations such as ERR:OLD\%(G) to denote the minimum density of an ERR:OLD set on G, rather than the cardinality [23]. Formally, where Br(v)={uV(G):d(u,v)r}, a subset SV(G) on a locally-finite (i.e. Br(v) finite for any finite r) (connected) graph G has density lim supr|Br(v)S||Br(v)| for any vV(G). Notably, the choice of center vertex v can affect the result of this calculation, in which case the density of the graph as a whole is not well-defined in practice. However, it has been proven that any graph satisfying the “slow-growth” property, limr|Br+1(v)||Br(v)|=1 for some vV(G), have the additional property that the density is invariant of the choice of center point [18].

Figure 7 shows an ERR:OLD set with the minimum density we have found so far for the three infinite grids. One can verify that each of these ERR:OLD sets satisfies the domination and distinguishing requirements in Theorem 1.1. Figure 7 (a) shows an ERR:OLD set with a tiling of SQR where seven out of eight vertices are detectors, so ERR:OLD\%(SQR)78. Similarly, we obtain ERR:OLD\%(TRI)47 and ERR:OLD\%(KNG)49 from Figure 7 (b) and (c), respectively.

f7-1
Figure 7 Our best constructions of ERR:OLD sets on SQR (a), TRI (b), and KNG (c).

To establish lower bound densities for domination-based parameters in infinite graphs, we employ a concept termed a share argument, originally introduced by Slater [26] and extensively utilized by other literature [14,19]. Essentially, this technique inverts the problem: instead of directly seeking a lower bound for the density of an ERR:OLD set, we determine an upper bound on the average share of a detector vertex vS, denoted sh(v)=uN(v)1dom(u), representing its total contribution to the domination of its neighbors. Because S is an open-dominating set, every vertex is 1-open-dominated, and each vertex with dom(u)=k contributes precisely 1k to the share of each of its k dominators, for a total of 1 per vertex in V(G). Therefore, an upper bound for the inverse average share of all detectors forms a lower bound for the density of S in V(G).

As an example of how share could be used to produce a lower bound for ERR:OLD, suppose G is k-regular and let vS be an arbitrary detector vertex in an ERR:OLD set SV(G). We know that v has k neighbors and each neighbor must be 3-dominated due to S being an ERR:OLD set. Therefore, sh(v)13+13++13+13=k3. Because v was selected arbitrarily, this value of k3 serves as an upper bound for the average share of all detectors, and so its inverse, 3k, represents a lower bound for the minimum density of an ERR:OLD in G. Applying this finding to the infinite grids we will cover, we have the following naive lower bounds: ERR:OLD\%(SQR)34, ERR:OLD\%(TRI)12, and ERR:OLD\%(KNG)38.

In what follows, we will augment these same naive share arguments with a few additional pieces of information in order to arrive at much higher lower bounds. As a shorthand, we will let σA denote kA1k for some sequence of single-character symbols, A. Thus, σa=1a, σab=1a+1b, and so on.

f8-1
Theorem 4.1. The infinite square grid, SQR, has ERR:OLD%(SQR)67.

Proof. Let S be an ERR:OLD set on SQR and let xS be an arbitrary detector; we will use the labeling scheme shown in Figure 8. We see that in order to distinguish vertices v1 and v2 we require |{v5,v7,v8,v12}|3. By symmetry, we also find that for any A{{v8,v9,v11,v12},{v5,v6,v10,v11},{v6,v7,v9,v10}}, |AS|3. Suppose v6S; then we require {v5,v7,v9,v10,v11}S due to overlapping A-sets, as well as {v8,v12}S to 3-dominate v1 and v2. Then sh(x)σ3344=76 and we would be done; otherwise, we can assume that v6S and by symmetry {v8,v10,v12}S as well. Suppose v5S; then we also require {v7,v11}S due to overlapping A-sets. Then sh(x)σ3344=76 and we would be done; otherwise, we can assume that v5S and by symmetry {v7,v9,v11}S as well. Then sh(x)σ4444<76, completing the proof. ◻

f9-1
Theorem 4.2. The infinite triangular grid, TRI, has ERR:OLD%(TRI)611.

Proof. Let S be an ERR:OLD set on TRI and let xS be an arbitrary detector; we will use the labeling scheme shown in Figure 9. We know that every vertex must be 3-dominated; suppose {v1,v3,v5} are all exactly 3-dominated. If v2S, then we see that v1 and v3 (which are exactly 3-dominated) cannot be 3-distinguished, a contradiction; otherwise we can assume that v2S and by symmetry v4,v6S as well. Then to 3-dominated x, we require {v1,v3,v5}S. If at least two vertices in {v2,v4,v6} are 4-dominated, then sh(x)σ344333=116 and we would be done; otherwise, without loss of generality assume that dom(v2)=dom(v4)=3. We see that v6 is already adjacent to three detectors, implying v7S, and by symmetry v9S as well. Then v1 cannot be 3-dominated, a contradiction. Otherwise, we know that there must be at least one 4-dominated vertex in {v1,v3,v5}, and symmetry yields a second 4-dominated vertex in {v2,v4,v6}, giving sh(x)σ443333=116 and completing the proof. ◻

f10-2
f11-1
Figure 11 The four non-isomorphic valid configurations to distribute three 4-dominated vertices in $N(x)$. Red and blue vertices represent 3- and 4-dominated vertices, respectively
Theorem 4.3. The infinite king grid, KNG, has ERR:OLD%(KNG)512.
Proof. Let S be an ERR:OLD set on KNG and let xS be an arbitrary detector; we will use the labeling scheme shown in Figure 10. To begin, suppose that each vertex v{v1,v2,v3} has dom(v)=3. We see that |NS(v1)NS(v2)|1 due to x; if this intersect reaches size 2, then v1 and v2 will not be able to be 3-distinguished. Thus, we can assume that ((N(v1)N(v2)){x})S=; that is, {v3,v9,v10}S=, and by symmetry {v1,v12,v13}S= as well. We see that v2 cannot be 3-dominated, a contradiction; therefore, we can assume that at least one vertex in {v1,v2,v3} is 4-dominated, and by symmetry we also have that for any A{{v3,v4,v5},{v5,v6,v7},{v7,v8,v1}}, there is at least one vertex in A which is 4-dominated.

In order to simultaneously satisfy all four A-sets (“corners”), we require at least two vertices which are 4-dominated. There exists only one non-isomorphic configuration with the minimum of two such vertices, so without loss of generality let dom(v1)4 and dom(v5)4 with all other vertices in N(x) being (exactly) 3-dominated. Similar to before, we see that xNS(v2)NS(v3), implying that {v1,v12,v13}S= and by symmetry {v5,v14,v20,v21,v22}S= as well. Applying the same reasoning to v2 and v4 gives us v3S, and by symmetry v7S as well. Applying this again to v2 and v8 gives v9S, and by symmetry v17S. To 3-dominate each vertex in {v2,v3,v4,v6,v7,v8}, we require {v2,v4,v6,v8,v10,v11,v15,v16,v18,v19,v23,v24}S. We see that dom(v1)=dom(v5)=5, so sh(x)=σ55333333=125 and we would be done. Otherwise, we know that there must be at least three vertices in N(x) which are 4-dominated. If N(x) has at least four vertices which are 4-dominated, then sh(x)σ44443333<125 and we would be done; otherwise, we may assume there are precisely three such vertices. Additionally, if any of these three vertices is 5-dominated, then sh(x)σ54433333<125 and we would be done; otherwise, we may assume N(x) contains exactly three vertices which are exactly 4-dominated, and all other vertices in N(x) are exactly 3-dominated. There are four non-isomorphic configurations which satisfy these constraints, as shown in Figure 11. We will now show that each of these cases has sh(x)125.

Case 1: Suppose dom(v1)=dom(v4)=dom(v6)=4 and all other vertices in N(x) are exactly 3-dominated. We first see that xN(v3)N(v7), implying that {v1,v5}S=. Applying this same logic to v2 and v5 gives us v3S, and by symmetry v7S as well. Applying this to v3 and v5 gives us v4S, and v6S by symmetry. We see that x cannot be 3-dominated, a contradiction, completing this case.

Case 2: Suppose dom(v3)=dom(v5)=dom(v7)=4 and all other vertices in N(x) are exactly 3-dominated. Applying similar logic as before, from v1 and v2, we see that {v3,v9,v10}S=, and by symmetry {v7,v24}S= as well. Applying this to v2 and v4 gives us v13S, and v21S by symmetry. Next, applying this to v4 and v6 gives us {v5,v17}S=. Finally, applying this to v2 and v8 gives us v1S. To 3-dominated each vertex in {v1,v2,v8}, we require {v2,v8,v11,v12,v22,v23}S. To distinguish v1 and x, we require {v4,v6}S. Because v3 is now 4-dominated, it must be that v14S, and v20S by symmetry. Then to 3-dominate v4 and v6 we require {v15,v16,v18,v19}S. However, this causes v5 to be 5-dominated, a contradiction, completing this case.

Case 3: Suppose dom(v2)=dom(v5)=dom(v7)=4 and all other vertices in N(x) are exactly 3-dominated. Applying the usual reasoning to v3 and v4 yields {v5,v13,v14}S=, and by symmetry {v7,v9,v24}S= as well. Applying this again to v4 and v6 gives v17S, and by symmetry v21S as well. With v1 and v4, we have v3S, and by symmetry v1S as well. With v1 and v3, we have v2S. To dominate each vertex in {v1,v2,v3,v4,v8}, we require {v4,v8,v10,v11,v12,v15,v16,v22,v23}S. To 3-dominate x, we require v6S. Then v5 and v7 become 4-dominated, implying v18,v20S, contradicting dom(v6)=3 and completing this case.

Case 4: Suppose dom(v1)=dom(v2)=dom(v5)=4 and all other vertices in N(x) are exactly 3-dominated. Applying the usual logic to v3 and v4 yields {v5,v13,v14}S=. Applying this to v6 and v7 yields {v20,v21}S=. Vertices v7 and v8 yield {v1,v22}S=. Vertices v6 and v8 yield v7S, and vertices v4 and v6 yield v17S. To 3-dominate v6 and v7, we require {v6,v8,v18,v19}S. If v4S, then v5 would become 4-dominated, implying v3,v16S, contradicting that v4 is 3-dominated; otherwise, we may assume that v4S. To distinguish v7 and x, we require v2,v3S. We see that v1 is 4-dominated, implying that {v9,v10,v24}S=. Finally, we find that v8 cannot be 3-dominated, a contradiction, completing the proof. ◻

The following theorem summarizes bounds for minimum ERR:OLD set densities in three infinite grids: the infinite square grid (SQR), the infinite triangular grid (TRI), and the infinite king grid (KNG).

Theorem 4.4. The following are upper and lower bounds for ERR:OLD\%(G) on several infinite grids:

(i) For SQR, 67ERR:OLD\%(SQR)78.

(ii) For TRI, 611ERR:OLD\%(TRI)47.

(iii) For KNG, 512ERR:OLD\%(KNG)49.

Table 1 concludes this section with the upper and lower bounds for other fault-tolerant OLD sets on four of the most studied infinite grids including the infinite hexagonal grid (HEX). Note that ERR:OLD\%(HEX)=1 because HEX is cubic, which requires every vertex to be a detector.

Table 1 Minimum densities for Fault-tolerant OLD sets in the four infinite grids
parameter hexagonal square triangular king
OLD [17,19,21] 12 25 413 [625,14]
RED:OLD [15,23] 23 12 38 [310,13]
DET:OLD [14,23] 67 34 12 [38,1330]
ERR:OLD 1 [67,78] [611,47] [512,49]

Declarations

The authors declare no conflict of interest.

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