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.
Let be a graph with vertices and edges ; we assume that all graphs are simple, undirected, and connected. For notation, we let , , , , and . The open-neighborhood of a vertex , denoted , is the set of all vertices adjacent to : . The closed-neighborhood of a vertex , denoted , is the set of all vertices adjacent to , as well as itself: . A vertex set is a (closed) dominating set if all vertices are within the closed neighborhood of some ; that is, . Similarly, is an open-dominating set if all vertices are within the open neighborhood of some ; that is, .
Assume that 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 can determine only whether or not the intruder is in , then the detection system is called an identifying code (IC) as introduced by Karpovsky, Charkrabarty and Levitin [16]. A dominating set of is an identifying code if for any any two distinct vertices , we have . If a detector at can determine only whether or not the intruder is in but can differentiate from , then the detection system is called a locating-dominating (LD) set, as introduced in Slater [25]. A dominating set of is a locating-dominating set if for any two distinct vertices , we have .
This paper focuses on a detection system where a sensor at vertex can sense intruders at adjacent vertices (i.e., within ), but not at 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 is an open-locating-dominating set if for any two distinct vertices , we have . 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 and , we let denote the (open) dominators of and denote the (open) domination number of . A vertex is -open-dominated by an open-dominating set if . If is an open-dominating set and , and are -distinguished if , where denotes the symmetric difference. If is an open-dominating set and , and are -distinguished if or .
The purpose of -domination in a detection system is to guarantee there are at least 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 is a RED:OLD set if , is an OLD set. For fault-tolerance beyond simple -redundancy, we conceptually establish that all detectors transmit either to represent the absence of an intruder, or to represent the presence of an intruder. A false negative is therefore when a detector incorrectly transmits a instead of a , while a false positive is when a detector incorrectly transmits a instead of a . 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 -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 . Firstly, if there is some 2-dominated vertex , with , then an intruder at coupled with a false-negative at is indistinguishable from the absence of an intruder at coupled with a false-positive at . Thus, 2-domination is insufficient for an ERR:OLD set. And secondly, if there are two vertices with and are -distinguished with , then transmitting could be due to a false-positive at (in which case the intruder is at ) or a false-negative at (in which case the intruder is at ). Thus, we cannot eliminate either of or as the intruder location, meaning -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.
Figure 1 Optimal OLD (a), RED:OLD (b), DET:OLD (c), and ERR:OLD (d) sets. Detectors are darkened.
For finite graphs, the notations , , , and ERR: OLD(G) represent the cardinality of the smallest possible OLD, RED:OLD, DET: OLD, and ERR:OLD sets on graph , respectively [23]. From Figure 1, which shows optimal solutions on the given graph which we will call , we see that , , , and .
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 permits ERR:OLD, then
OLD sets
From Theorem 1.1, by using we can arrive at the following existence criteria for OLD on arbitrary graphs:
Theorem 2.2. [21] A graph has an OLD set if and only if satisfies and, whenever , .
Chellali et al. [2] proved the upper and lower bounds for on arbitrary graphs:
Observation 2.3 [2] If graph has an OLD set, then .
The upper bound of is achievable, and Foucaud et al. [7] later provided a full characterization for this class of extremal graphs:
Definition 2.4 . [6] For any , the half-graph, is given by and .
Seo and Slater established the existence criteria for trees, determined the lower and upper bounds for on trees, and fully characterized the extremal trees which achieve these upper and lower bounds [22].
Theorem 2.6. [22] A tree of order has an OLD set if and only if every vertex is adjacent to at most one leaf.
Henning and Yeo [10] determined the upper bound on OLD(G) when G is a cubic graph.
Theorem 2.8. [10] If is a twin-free cubic graph of order , then .
RED:OLD sets
From Theorem 1.1, by using , we can arrive at the following existence criteria for RED:OLD on arbitrary graphs:
Theorem 2.9. A graph, , has RED:OLD if and only if and for each distinct , .
These existence criteria can be simplified when discussing regular graphs:
Theorem 2.10. [20] Let be an -regular graph with . There exists a RED:OLD set for if and only if there exists an OLD set for .
Further, we have the following results from Seo [20] when discussing cubic graphs specifically:
Theorem 2.11. [20] Let be a cubic graph. If is -free, then is a RED:OLD set.
Theorem 2.12. [20] Let be a cubic graph of order . Then we have
Seo [20] has also constructed an infinite family of finite cubic graphs exhibiting the extremal value of .
DET:OLD sets
Similar to the existence criteria for OLD and RED:OLD sets, one can simply use Theorem 1.1 with 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 permits DET:OLD, then every has at most one degree 2 neighbor.
Further, we have the following results from Seo [20] when discussing cubic graphs specifically:
Theorem 2.15. [20] If is a -free cubic graph, then for any vertex , is a DET:OLD set for .
Theorem 2.16. [20] A cubic graph permits DET:OLD if and only if it is -free.
Theorem 2.17. [13,20] If is a -free cubic graph, then .
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 permits an ERR:OLD set if and only if and for every subgraph, , .
Proof. First, assume to the contrary that there exists a vertex with ; then, cannot be 3-dominated, a contradiction. Similarly, if we assume there exists with , then and could not be -distinguished, a contradiction.
For the converse, suppose satisfies both properties of the theorem statement. To demonstrate the existence of ERR:OLD, we will conservatively let be the detector set. Clearly, guarantees all vertices will be 3-dominated, so we need only show that all vertex pairs are -distinguished. Let with . If then and will have non-intersecting neighborhoods and thus be -distinguished; otherwise, we can assume . Suppose , and let be a path from to . If , then and must be 4-distinguished, and we would be done, so assume with . Then we have a subgraph, , so by hypothesis and are -distinguished. Finally, suppose . If there is some , then and will be 3-distinguished by , and we would be done; otherwise, we assume that . Because , it must be that with . Then we have a subgraph , so by hypothesis and must be -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 -free graph, , permits an ERR:OLD set if and only if .
Two distinct vertices are said to be twins if (closed twins) or (open twins). From Theorem 2.18, we see that if has twins and , then it cannot have an ERR:OLD set since or and form the corners of a subgraph with .
Corollary 2.20. If permits an ERR:OLD set, then is twin-free.
Next, we will show that there are no graphs with ERR:OLD sets on vertices.
Theorem 2.21. If has an ERR:OLD set, then .
Proof. Firstly, we know that cannot be acyclic because an ERR:OLD set requires . Thus, 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 , as otherwise we would be done. If the smallest cycle is for , then every vertex in the cycle must have a third dominator, and these additional vertices must all be distinct to avoid creating smaller cycles; thus , a contradiction. Suppose the smallest cycle is a subgraph , then each vertex in the cycle must have a third dominator, , , , and adjacent to , , , and , respectively. We see that ; however, because it must be that and . We see that and cannot have any additional edges without creating a smaller cycle, but they are open twins and so cannot be -distinguished, a contradiction. Thus, we can assume that has a triangle.
Next, suppose there exists a 4-Cycle with , which we call a “diamond” subgraph. If then distinguishing and will result in , a contradiction; thus, we can assume that , meaning we have a subgraph. By symmetry, we can assume that , , and have third dominators , , and . If , , and are all distinct, then and we would be done; otherwise without loss of generality assume . Suppose ; then we require to be able to distinguish and , and by symmetry we have . We now have a subgraph plus an extra vertex . Distinguishing and requires without loss of generality , but then and cannot be -distinguished, a contradiction. Otherwise, we can assume ; we see that distinguishing and requires, without loss of generality, . If then to distinguish and ; however, and can no longer be -distinguished, a contradiction, so we assume and . If then and cannot be -distinguished, a contradiction, so by symmetry we assume and . Thus, 3-dominating and requires , but then and cannot be -distinguished, a contradiction. Thus, the existence of a diamond subgraph implies that .
At this point, we know that has a triangle, say . To 3-dominate the vertices in , assume , , and . Additionally, we can assume , , and are distinct because otherwise we produce a diamond subgraph. To 3-dominate , , and without producing a diamond subgraph, it must be that . At this point we can no longer add any edges without producing a diamond subgraph, but and are not -distinguished, a contradiction, completing the proof.
Figure 2 shows the first graphs with an ERR:OLD set in the lexicographic ordering of tuples; i.e., the graphs with the smallest number of edges given the smallest number of vertices. For ERR:OLD, this is and .
Figure 2 The two smallest graphs supporting ERR:OLD, which have and
Next, we will show the relation between the existence of ERR:OLD sets and DET:OLD sets for regular graphs.
Observation 2.22. If and are sets with , then implies .
Proof. Suppose . Then . Therefore , completing the proof.
Suppose is a regular graph with DET:OLD. We can conservatively let be the set of detectors. Then every vertex pair in must be -distinguished; thus, Theorem 2.22 yields that every vertex pair is -distinguished, which is sufficient for the distinguishing requirement of ERR:OLD.
Proposition 2.23. A regular graph with 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 -free; thus, we have the following existence criteria.
Corollary 2.24. A cubic graph has an ERR:OLD set if and only if is -free.
Now we consider extremal graphs with with least . If a graph has an ERR:OLD set, then Theorem 2.18 yields that , which serves as a lower bound for the least , namely . Figures 2 and 3 show extremal graphs with with the fewest number of edges when . For even , we will show that cubic graphs exist which meet the theoretical lower bound of . For odd , this bound of is trivially unobtainable, but we introduce a new class of “almost-cubic” graphs meeting the lower bound of . In either case, we provide an extremal graph for each .
Figure 3 Extremal graphs with with the fewest number of edges when and .
Definition 2.25. is quasi-cubic if every vertex is degree 3 aside from one degree-4 vertex.
When , cubic and quasi-cubic graphs together form the entire family of extremal graphs with which have the minimum number of edges for a given . If is an ERR:OLD set, we know that if with . Because every vertex in a cubic or quasi-cubic graph is adjacent to a degree-3 vertex, these classes or graphs require if they permit ERR:OLD.
Theorem 2.26. If is cubic or quasi-cubic, then has an ERR:OLD set if and only if is -free.
Proof. Suppose has a subgraph, . Because is cubic or quasi-cubic, at most one vertex in can be degree 4, and all others will be degree 3. By symmetry, assume . Then and cannot be 3-distinguished, implying does not have ERR:OLD.
For the converse, suppose does not have ERR:OLD. Because , we know that an ERR:OLD set not existing implies where and are not 3-distinguished. Regardless of whether or not , we see the only way to have and not be 3-distinguished is by having , which creates a subgraph, completing the proof.
Suppose we have a cubic graph that permits an ERR:OLD set. Then, we can construct a quasi-cubic graph from as shown in the next theorem.
Theorem 2.27. Let be a cubic graph with ERR:OLD where and such that and are not part of a triangle or the terminal edges of a subgraph. Construct quasi-cubic graph from by deleting edges and and adding a new vertex with . Then has an ERR:OLD set.
Proof. By Theorem 2.26, we know that is -free and we need only show that is likewise -free. Suppose for a contradiction that has a subgraph; then due to being -free, the subgraph must involve vertex , which is an endpoint of every new edge in . By symmetry, we can reduce the problem to two possible cases: the includes edges or edges , and we will consider to be the opposite vertex of in the subgraph. If we are in the case, we see that any choice of results in forming a triangle in , a contradiction. Otherwise, in the case we see that and because , so causes and to be the ends of a subgraph in , 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 be a set of variables. Let be a conjunction of clauses, where each clause is a disjunction of three literals from distinct variables of . QUESTION: Is there is an assignment of values to such that is true?
ERR:OLD Minimization
INSTANCE: A graph and integer with . QUESTION: Is there an ERR:OLD set with ? Or equivalently, is ERR:OLD() ?
Figure 4 Subgraph
Lemma 3.1. Given a graph , let be the subgraph shown in Figure 4, where vertices have no additional edges in the larger graph but are permitted to have additional edges to vertices outside of this subgraph. If is an ERR:OLD set for , then .
Proof. To 3-distinguish and , we require . To 3-dominate , we additionally require . By symmetry, we have that , completing the proof.
Figure 5 Variable () and clause () graphs
Figure 6 Construction of from with , ,
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, 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 be an instance of the 3-SAT problem with clauses on variables. We will construct a graph, , as follows. For each variable , create an instance of the graph (Figure 5); this includes a vertex for and its negation . For each clause of , create a new instance of the graph (Figure 5). For each clause , create an edge from the vertex to , , and from the variable graphs, each of which is either some or ; for an example, see Figure 6. The resulting graph has precisely vertices and edges, and can be constructed in polynomial time. To complete the problem instance, we define .
Suppose is an ERR:OLD set on with . By Lemma 3.1, we require at least all of the detectors shown by the shaded vertices in Figure 5. Additionally, in each the vertices and require or in order to be 3-dominated. Thus, we find that , implying that , so for each . For each , we see that is not 3-dominated unless it is adjacent to at least one additional detector from its three neighbors in the graphs; therefore, is satisfiable.
For the converse, suppose we have a solution to the 3-SAT problem ; we will show that there is a ERR:OLD, , on with . We construct by first including all of the vertices as shown in Figure 5. Next, for each variable, , if is true then we let the vertex ; otherwise, we let . Thus, the fully-constructed has . It can be shown that every vertex pair is now 3-distinguished, and that every vertex which is not a clause vertex is 3-dominated. Because we selected each or based on a satisfying truth assignment for , each must be adjacent to at least one additional detector vertex from the graphs. Thus, all vertices are 3-dominated, implying is an ERR:OLD set for with , 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 to denote the minimum density of an ERR:OLD set on , rather than the cardinality [23]. Formally, where , a subset on a locally-finite (i.e. finite for any finite ) (connected) graph has density for any . Notably, the choice of center vertex 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, for some , 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 . Similarly, we obtain and from Figure 7 (b) and (c), respectively.
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 , denoted , representing its total contribution to the domination of its neighbors. Because is an open-dominating set, every vertex is 1-open-dominated, and each vertex with contributes precisely to the share of each of its dominators, for a total of per vertex in . Therefore, an upper bound for the inverse average share of all detectors forms a lower bound for the density of in .
As an example of how share could be used to produce a lower bound for ERR:OLD, suppose is -regular and let be an arbitrary detector vertex in an ERR:OLD set . We know that has neighbors and each neighbor must be 3-dominated due to being an ERR:OLD set. Therefore, . Because was selected arbitrarily, this value of serves as an upper bound for the average share of all detectors, and so its inverse, , represents a lower bound for the minimum density of an ERR:OLD in . Applying this finding to the infinite grids we will cover, we have the following naive lower bounds: , , and .
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 denote for some sequence of single-character symbols, . Thus, , , and so on.
Figure 8 SQR labeling scheme. Vertices are labeled as
Theorem 4.1. The infinite square grid, SQR, has .
Proof. Let be an ERR:OLD set on SQR and let be an arbitrary detector; we will use the labeling scheme shown in Figure 8. We see that in order to distinguish vertices and we require . By symmetry, we also find that for any , . Suppose ; then we require due to overlapping -sets, as well as to 3-dominate and . Then and we would be done; otherwise, we can assume that and by symmetry as well. Suppose ; then we also require due to overlapping -sets. Then and we would be done; otherwise, we can assume that and by symmetry as well. Then , completing the proof.
Figure 9 TRI labeling scheme. Vertices are labeled as
Theorem 4.2. The infinite triangular grid, TRI, has .
Proof. Let be an ERR:OLD set on TRI and let be an arbitrary detector; we will use the labeling scheme shown in Figure 9. We know that every vertex must be 3-dominated; suppose are all exactly 3-dominated. If , then we see that and (which are exactly 3-dominated) cannot be -distinguished, a contradiction; otherwise we can assume that and by symmetry as well. Then to 3-dominated , we require . If at least two vertices in are 4-dominated, then and we would be done; otherwise, without loss of generality assume that . We see that is already adjacent to three detectors, implying , and by symmetry as well. Then cannot be 3-dominated, a contradiction. Otherwise, we know that there must be at least one 4-dominated vertex in , and symmetry yields a second 4-dominated vertex in , giving and completing the proof.
Figure 10 KNG labeling scheme. Vertices are labeled as
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 .
Proof. Let be an ERR:OLD set on KNG and let be an arbitrary detector; we will use the labeling scheme shown in Figure 10. To begin, suppose that each vertex has . We see that due to ; if this intersect reaches size 2, then and will not be able to be -distinguished. Thus, we can assume that ; that is, , and by symmetry as well. We see that cannot be 3-dominated, a contradiction; therefore, we can assume that at least one vertex in is 4-dominated, and by symmetry we also have that for any , there is at least one vertex in which is 4-dominated.
In order to simultaneously satisfy all four -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 and with all other vertices in being (exactly) 3-dominated. Similar to before, we see that , implying that and by symmetry as well. Applying the same reasoning to and gives us , and by symmetry as well. Applying this again to and gives , and by symmetry . To 3-dominate each vertex in , we require . We see that , so and we would be done. Otherwise, we know that there must be at least three vertices in which are 4-dominated. If has at least four vertices which are 4-dominated, then 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 and we would be done; otherwise, we may assume contains exactly three vertices which are exactly 4-dominated, and all other vertices in 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 .
Case 1: Suppose and all other vertices in are exactly 3-dominated. We first see that , implying that . Applying this same logic to and gives us , and by symmetry as well. Applying this to and gives us , and by symmetry. We see that cannot be 3-dominated, a contradiction, completing this case.
Case 2: Suppose and all other vertices in are exactly 3-dominated. Applying similar logic as before, from and , we see that , and by symmetry as well. Applying this to and gives us , and by symmetry. Next, applying this to and gives us . Finally, applying this to and gives us . To 3-dominated each vertex in , we require . To distinguish and , we require . Because is now 4-dominated, it must be that , and by symmetry. Then to 3-dominate and we require . However, this causes to be 5-dominated, a contradiction, completing this case.
Case 3: Suppose and all other vertices in are exactly 3-dominated. Applying the usual reasoning to and yields , and by symmetry as well. Applying this again to and gives , and by symmetry as well. With and , we have , and by symmetry as well. With and , we have . To dominate each vertex in , we require . To 3-dominate , we require . Then and become 4-dominated, implying , contradicting and completing this case.
Case 4: Suppose and all other vertices in are exactly 3-dominated. Applying the usual logic to and yields . Applying this to and yields . Vertices and yield . Vertices and yield , and vertices and yield . To 3-dominate and , we require . If , then would become 4-dominated, implying , contradicting that is 3-dominated; otherwise, we may assume that . To distinguish and , we require . We see that is 4-dominated, implying that . Finally, we find that 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 on several infinite grids:
(i) For SQR, .
(ii) For TRI, .
(iii) For KNG, .
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 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
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