Open-independent, Open-locating-dominating Sets

A distinguishing set for a graph G = (V, E) is a dominating set D, each vertex $v \in D$ being the location of some form of a locating device, from which one can detect and precisely identify any given "intruder" vertex in V(G). As with many applications of dominating sets, the set $D$ might be required to have a certain property for <D>, the subgraph induced by D (such as independence, paired, or connected). Recently the study of independent locating-dominating sets and independent identifying codes was initiated. Here we introduce the property of open-independence for open-locating-dominating sets.


Introduction
For a graph G = (V, E) that represents a facility, an "intruder" in the system might be a thief, saboteur or fire. If G represents a multiprocessor network with each vertex representing one processor, an "intruder" might be a malfunctioning processor. We assume that certain vertices will be the locations of detectors, each detector having some capability to identify the location of an intruder vertex.
For u, v ∈ D, let d(u, v) denote the distance in G between u and v. Some detectors, like sonar devices, can be assumed to determine the distance to the intruder vertex x anywhere in the system. Much work has been done on locating sets as introduced in Slater [36] (and also called metric bases as independently introduced in Harary and Melter [11]). An (ordered) set X = {x 1 , x 2 , ..., x k } ⊆ V (G) is a locating set if for every w ∈ V (G) the ordered k-tuple (d(x 1 , w), d(x 2 , w), ..., d(x k , w)) uniquely determines w. We say that a vertex x resolves vertices u and v if d(x, u) = d(x, v). Then X is locating if for every two vertices u and v at least one x i ∈ X resolves u and v. For the recently introduced centroidal bases described in Foucaud, Klasing and Slater [9] the set of detectors in X provide just an ordering of the relative distances to an intruder vertex, not the exact distances.
For the case in which a detector at v can determine if the intruder is at v or if the intruder is in N (v) (but which element in N (v) can not be determined), as introduced in Slater [37,38,39], a locating-dominating set L ⊆ V (G) is a dominating set for which, given any two vertices u and v in V (G) − L, one has N (u) ∩ L = N (v) ∩ L, that is, for any two distinct vertices u and v (including ones in L) there is a vertex x ∈ L with d(x, u) ∈ {0, 1} and d(x, u) = d(x, v) or d(x, v) ∈ {0, 1} and d(x, u) = d(x, v). Every graph G has a locating-dominating set, namely V (G), and the locating-dominating number LD(G) is the minimum cardinality of such a set. See, for example, [3,8,17].
As introduced by Karpovsky, Charkrabarty and Levitin [22], an identifying code C ⊆ V (G) is a dominating set for which given any two vertices u and v in V (G) one has See, for example, [2,4,25]. Graph G has an identifying code when for every pair of vertices u and v we have N [u] = N [v], and the identifying code number IC(G) is the minimum cardinality of such a set.
When a detection device at vertex v can determine if an intruder is in N (v) but will not/can not report if the intruder is at v itself, then we are interested in open-locating-dominating sets as introduced for the k-cubes Q k by Honkala, Laihonen and Ranto [21] and for all graphs by Seo and Slater [26,27]. An open dominating set u). A graph G has an open-locating-dominating set when no two vertices have the same open neighborhood, and OLD(G) is the minimum cardinality of such a set. See, for example, [5,16,21,28,29,30,31,32,33]. Lobstein [24] maintains a bibliography, currently with more than 300 entries, for work on these topics.
Dominating sets D have many applications (see Haynes, Hedetniemi and Slater [12,13]), and in many cases the subgraph generated by D, denoted D , is required to have an additional property such as independence, paired, or connected. Recently, independent locating-dominating sets and independent identifying codes have been introduced in Slater [42]. Not all graphs have independent locating-dominating sets (respectively, independent identifying codes), and there is no forbidden subgraph characterization of such graphs. In fact, we have the following.
Theorem A (Slater [42]) Simply deciding, for a given input graph G, if G has an independent locating-dominating set is NP-complete.
Theorem B (Slater [42]) Simply deciding, for a given input graph G, if G has an independent identifying code is NP-complete.
Note that, by definition, an open dominating set S can not be independent, each v ∈ S must be open dominated by some x ∈ N (v). In this paper we consider "open-independence" and introduce open-independent, open-locating-dominating sets.

Open-independent sets; open-independent-dominating sets; open-independent, open dominating sets
Assuming every vertex is the possible location of an intruder and that a detector at vertex v can not detect an intruder at w ∈ V (G) if d(v, w) ≥ 2, in order for every intruder to be detectable we require a dominating set for the detectors.
is defined to be enclaveless if every vertex in E is adjacent to at least one vertex V (G) − E.). Also, S ⊆ V (G) is independent if no two vertices in S are adjacent. Now, R ⊆ V (G) is dominating when condition (1) below holds, and R is independent when (2) below holds. ( For open domination, one assumes that a vertex v does not dominate itself. An intruder (thief, saboteur, fire) at v might prevent its own detection; a malfunctioning processor might not detect its own miscalculations.
The open-independence number for a graph G denoted by OIN D(G) is the maximum cardinality of an open-independent set for G. Note that OIN D(G) ≥ β(G), where β(G) denotes the maximum cardinality of an independent set for G.

www.ejgta.org
Open-independent open-locating-dominating sets | Suk J. Seo and Peter J. Slater The domination number γ(G) is the minimum cardinality of a dominating set, a dominating set of cardinality γ(G) being called a γ(G)-set whereas any dominating set is called a γ-set. Similar terminology is used for other parameters. The independent domination number (which could be denoted γ IN D (G)) is traditionally denoted by i(G) and is the minimum cardinality of a dominating set D for which every component of D is a singleton. We let γ OIN D (G) denote the minimum cardinality of an open-independent dominating set D, a dominating set D for which each component of D has cardinality at most two, The . Open-independent, open dominating sets have been considered in another context by Studer, Haynes, and Lawson [43]. As introduced in Haynes and Slater [14,15], a paired dominating set D is a dominating set for which D has a perfect matching. Studer, et al. [43] define an openindependent, open dominating set as an induced-paired dominating set.
As noted, in this paper we are interested in distinguishing sets and will consider open-independent, open-locating-dominating sets.

Open-independent, open-locating-dominating sets
For an open-locating-dominating set S each v ∈ V (G) has a distinct set of detectors, N (v) ∩ S.        There is no forbidden subgraph characterization of the set of graphs which have OLD OIN Dsets, nor of the set of graphs which do not have OLD OIN D -sets. In fact, simply deciding for a given graph G if G has an OLD OIN D -set is an NP-complete problem. As noted in Garey and Johnson [10], Problem 3-SAT is NP-complete.

Theorem 3.2. Simply deciding, for a given graph G, if G has an open-independent, OLD-set is an NP-complete decision problem. That is, XOIOLD is NP-complete.
Proof. One can easily verify in polynomial time if a given set S ⊆ V (G) is an OLD OIN D -set, so XOIOLD ∈ N P . We can reduce the known NP-complete 3-SAT problem to XOIOLD in polynomial time as follows. For each u i ∈ U let G i be the 6-vertex graph illustrated in Figure 5 Interconnect the clause components and literal components by adding edges d j c j,1 , d j c j,2 and d j c j,3 for 1 ≤ j ≤ m where c j = {c j,1 , c j,2 , c j,3 } ∈ C, as illustrated in Figure 5. Let G be the resulting graph of order 6n + 3m.
Assume there is a satisfying truth assignment S ⊆ U ∪ U . Form W ⊆ V (G) by letting otherwise the literal u i ∈ S and one adds u i to S. Then <W > consists of 4n + 2m vertices inducing 2n + m independent edges. Note that N ( It is easily seen that G has W as an open-independent, OLD-set.
Assume G has an OLD OIN D -set W . By Proposition 3.1 we have That is, S must be a satisfying truth assignment.

Theorem 3.3. If the girth of G satisfies g(G) ≥ 5 and W ⊆ V (G), then W is an OLD OIN D -set if and only if (1) each v ∈ W is open-dominated exactly once, and (2) each v /
∈ W is open-dominated at least twice.
Assume conditions (1) and (2) Assume W is an OLD OIN D -set for path P n : v 1 , v 2 , ..., v n . By Proposition 3.1 we have A t Figure 6. Some trees with OLD OIN D -sets.

Theorem 3.4. (Seo and Slater [26]) A tree T has an OLD-set if and only if no two endpoints of T have the same neighbor.
Similar to the characterization given in Studer, et al. [43] for open-independent dominating sets in trees, we can recursively define the collection of pairs (T, W ) where T is a tree and W is the unique OLD OIN D (T )-set. First note that the tree A t of order 2t + 1 in Figure 6 has Theorem 3.5. If T n is a tree of order n with an OLD OIN D -set, then the OLD OIN D (T )-set W is unique and T n can be obtained recursively from P 2 by a sequence of operations OP1 and OP2 defined as follows.
(OP1) Let T * be a tree with OLD OIN D (T )-set W and let z ∈ W . The tree T is obtained from T * by adding a P 3 : x, w, v and adding the edge zx.
(OP2) Let T * be a tree with OLD OIN D (T )-set W and let z be any vertex in T * . The tree T is obtained from T * by adding an A t with t ≥ 2 and adding the edge zx.
Proof. We first observe that if T is obtained from T * by (OP1), then W ∪ {w, v} is an OLD OIN Dset for T , and if T is obtained from T * by (OP2), then W ∪ {w 1 , v 1 , ..., w t , v t } is an OLD OIN D -set for T .
Assume tree T has OLD OIN D -set W . If T is a path, then Proposition 3.2 shows that T can be obtained from P 2 by a sequence of (OP1)-operations and there is a unique OLD OIN D -set for T . If T is not a path, select a vertex y with deg y ≥ 3 where all or all but one of the branches at y are paths. Suppose y, u 1 , u 2 , ..., u j is a branch path with j ≥ 3. By Proposition 1 we must Because W is an OLD-set, y can not be the support vertex of two or more endpoints. If y is adjacent to an endpoint x and y, u 1 , u 2 is a branch path, Proposition 1 would imply that {u 1 , u 2 } ⊆ W and {x, y} ⊆ W , so W would not be open-independent. Now y can be assumed to have deg y − 1 = b branch paths of length two. We have a subgraph A b with vertices {y, w 1 , v 1 , ..., w b , v b } with b ≥ 2. Let N (y) = {w 1 , w 2 , ..., w b , z}, and T can be obtained from T * = T − {y, w 1 , v 1 , ..., w b , v b } by (OP2).

OLD OIN D % for infinite grids
Much work has been done on distinguishing sets (LD-sets, IC-sets and OLD-sets) in infinite grids (hexagonal, square, triangular, tumbling block, etc). See, for example, [1,6,7,18,19,20,22,23,26,28,29,30,40,41]. Similarly, the open share sh op (x; D) is defined in Seo and Slater [26] for open dominating set D. Specifically, if D is open dominating and x ∈ D then, for each y ∈ N (x), let  In this paper, we will focus on open-locating-dominating sets along with open-shares of vertices.

Open independent sets
In this paper we focused on open-independence for OLD-sets. Of interest is the parameter OIN D itself, as well as the lower open independence parameter oind where oind(G) is the minimum cardinality of a maximally open-independent set.