Locating-dominating Set Research Articles

Extremal digraphs for open neighbourhood location-domination and identifying codes

A set S of vertices of a digraph D is called an open neighbourhood locating-dominating set if every vertex in D has an in-neighbour in S, and for every pair u,v of vertices of D, there is a vertex in S that is an in-neighbour of exactly one of u and v. The smallest size of an open neighbourhood locating-dominating set of a digraph D is denoted by γOL(D). We study the class of digraphs D whose only open neighbourhood locating-dominating set consists of the whole set of vertices, in other words, γOL(D) is equal to the order of D. We call those digraphs extremal. By considering digraphs with loops allowed, our definition also applies to the related (and more widely studied) concept of identifying codes. We extend previous studies from the literature for both open neighbourhood locating-dominating sets and identifying codes of both undirected and directed graphs. These results all correspond to studying open neighbourhood locating-dominating sets on special classes of digraphs. To do so, we prove general structural properties of extremal digraphs, and we describe how they can all be constructed. We then use these properties to give new proofs of several known results from the literature. We also give a recursive and constructive characterization of the extremal di-trees (digraphs whose underlying undirected graph is a tree).

Stable Locating-Dominating Sets in the Edge Corona and Lexicographic Product of Graphs

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs.

Bilangan Dominasi-Lokasi pada Graf Hasil Kali Operasi Comb Graf Lintasan dan Graf Siklus

Dominating set of graph G is subset D⊆V(G) which for every vertex v∈V(G)\D those vertices has neighbour in D. If for every pairs of vertice x and y their neighbour set different than we called D locating-dominating set. As for the minimum cardinality of possible dominating set of G is called the locating-dominating number of G. We determine the value of the locating-dominating number for comb product path (Pn) and cycle (Cn) graph.

Locating-dominating sets: From graphs to oriented graphs

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u,v∉S, the neighbourhood of u and v on S are distinct (i.e. N(u)∩S≠N(v)∩S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set S such that for each w∈V∖S, N−(w)∩S≠∅ and for each pair of distinct vertices u,v∈V∖S, N−(u)∩S≠N−(v)∩S. We consider the following two parameters. Given an undirected graph G, we look for γ→LD(G) (Γ→LD(G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation of G, then γ→LD(G)≤γLD(D)≤Γ→LD(G) where γLD(D) is the minimum size of a locating-dominating set of D.For the best orientation, we prove that, for every twin-free graph G on n vertices, γ→LD(G)≤n/2 which proves a “directed version” of a widely studied conjecture on the location-domination number. As a side result we obtain a new improved upper bound for the location-domination number in undirected trees. Moreover, we give some bounds for γ→LD(G) on many graph classes and drastically improve the value n/2 for (almost) d-regular graphs by showing that γ→LD(G)∈O(log⁡d/d⋅n) using a probabilistic argument.While γ→LD(G)≤γLD(G) holds for every graph G, we give some graph classes such as outerplanar graphs for which Γ→LD(G)≥γLD(G) and some for which Γ→LD(G)≤γLD(G) such as complete graphs. We also give general bounds for Γ→LD(G) such as Γ→LD(G)≥α(G). Finally, we show that for many graph classes Γ→LD(G) is polynomial on n but we leave open the question whether there exist graphs with Γ→LD(G)∈O(log⁡n).

Progress on fault-tolerant locating-dominating sets

A locating-dominating set is a subset of vertices representing “detectors” in a graph G which can locate an “intruder” given that each detector covers its closed neighborhood and can distinguish its own location from its neighbors. We explore a fault-tolerant variant of locating-dominating sets, called error-detecting locating-dominating (DET:LD) sets, which can tolerate one false negative. The concept of DET:LD sets was originally introduced in 2002, along with several properties in various classes of graphs; we extend this work by fully characterizing DET:LD sets for general graphs and establishing existence criteria. We prove that the problem of determining the minimum density of a DET:LD set in arbitrary graphs is NP-complete. Additionally, we determine that the minimum density in cubic graphs is at least [Formula: see text], and we show an infinite family of cubic graphs achieving this density.

Extremal cubic graphs for fault-tolerant locating domination

A locating-dominating set is a subset of vertices in a graph, which is used to model a detection system for a facility or network. A detector placed at each vertex v in the subset is assumed to detect an “intruder” at v as well as in its neighborhood. Any system that is modeled by a locating-dominating set can precisely determine the exact location of the intruder from the information provided by these detectors. We explore two fault-tolerant variants of locating-dominating sets, namely redundant locating-dominating (RED:LD) and error-correcting locating-dominating (ERR:LD) sets. The problems of determining the minimum cardinality of RED:LD and ERR:LD sets in arbitrary graphs are known to be NP-complete. In this paper we determine the lower and upper bounds on values of the minimum densities of RED:LD and ERR:LD sets in cubic graphs, and present cubic graphs that achieve the extremal values on these parameters.

On some variations of dominating identification in graphs

In this study, we introduce the locating-dominating value and the location-domination polynomial of graphs and location-domination polynomials of some families of graphs were identified. Locatingdominating set of graph G is defined as the dominating set which locates all the vertices of G. And, location-domination number G is the minimum cardinality of a locating-dominating set in G.

On Locating-Dominating Set of Regular Graphs

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

Stable Locating-Dominating Sets in Graphs

A set S ⊆ V(G) of a (simple) undirected graph G is a locating-dominating set of G if for each v ∈ V(G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S= NG(y)∩S for any distinct vertices x and y in V(G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γsl(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs.

On trees with unique locating kernels

A locating-dominating set of a graph G is a set D of vertices such that for every two vertices x,yin V(G)setminus D the sets N(x)cap D and N(y)cap D are non-empty and different. In this paper, we define the locating kernel of a graph G, i.e., a subset of its vertex set which is independent and a locating-dominating set. We provide a constructive characterization of trees with a unique locating kernel.

Characterizing extremal graphs for open neighbourhood location-domination

An open neighbourhood locating-dominating set is a set S of vertices of a graph G such that each vertex of G has a neighbour in S, and for any two vertices u,v of G, there is at least one vertex in S that is a neighbour of exactly one of u and v. We characterize those graphs whose only open neighbourhood locating-dominating set is the whole set of vertices. More precisely, we prove that these graphs are exactly the graphs for which all connected components are half-graphs (a half-graph is a special bipartite graph with both parts of the same size, where each part can be ordered so that the open neighbourhoods of consecutive vertices differ by exactly one vertex). This corrects a wrong characterization from the literature.

Computing Open Locating-Dominating Number of Some Rotationally-Symmetric Graphs

Location detection is studied for many scenarios, such as pointing out the flaws in multiprocessors, invaders in buildings and facilities, and utilizing wireless sensor networks for monitoring environmental processes. The system or structure can be illustrated as a graph in each of these applications. Sensors strategically placed at a subset of vertices can determine and identify irregularities within the network. The open locating-dominating set S of a graph G=(V,E) is the set of vertices that dominates G, and for any i,j∈ V(G) N(i)∩S≠N(j)∩S is satisfied. The set S is called the OLD-set of G. The cardinality of the set S is called open locating-dominating number and denoted by γold(G). In this paper, we computed exact values of the prism and prism-related graphs, and also the exact values of convex polytopes of Rn and Hn. The upper bound is determined for other classes of convex polytopes. The graphs considered here are well-known from the literature.

Parameterized algorithms for locating-dominating sets

A locating-dominating set D of a graph G is a dominating set of G where each vertex not in D has a unique neighborhood in D, and the Locating-Dominating Set problem asks if G contains such a dominating set of bounded size. This problem is known to be NP-hard even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set is W[1]-hard when parameterized by the size of a minimum clique cover. We present an exponential kernel for the distance to cluster parameterization and show that, unless NP ⊆ coNP/poly, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by either of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

Domination and location in twin-free digraphs

A dominating set D in a digraph is a set of vertices such that every vertex is either in D or has an in-neighbour in D. A dominating set D of a digraph is locating-dominating if every vertex not in D has a unique set of in-neighbours within D. The location-domination number γL(G) of a digraph G is the smallest size of a locating-dominating set of G. We investigate upper bounds on γL(G) in terms of the order of G. We characterize those digraphs with location-domination number equal to the order or the order minus one. Such digraphs always have many twins: vertices with the same (open or closed) in-neighbourhoods. Thus, we investigate the value of γL(G) in the absence of twins and give a general method for constructing small locating-dominating sets by the means of special dominating sets. In this way, we show that for every twin-free digraph G of order n, γL(G)≤4n5+1 holds, and there exist twin-free digraphs G with γL(G)=2(n−2)3. Improved bounds are proved for certain special cases. In particular, if G is twin-free and a tournament, or twin-free and acyclic, we prove γL(G)≤⌈n2⌉, which is tight in both cases.

The locating dominating set (LDS) of generalized of corona product of path graph and any graphs

Graph G is usually written by G = (V, E) is graph G where V(G) is vertex set on graph G and E(G) is edge set on graph G. Graph G used in this study is only on simple and undirected graphs. Dominating set (DS) is graph G which have a vertex set D, where each vertex in D can dominate the neighboring vertices, in other words every vertex from u ∈ V(G) − D is adjacent to verted v ∈ D. The minimum cardinality of dominating set is called by domination number, symbolized by γ(G). Locating dominating set (LDS) is dominating set with additional condition. A graph G = (V, E) is said to be locating dominating set if the set of vertex dominator D satisfies every vertex that is not D, that is V − D has a different intersection set with D. The minimum cardinality of locating dominating set is called by locating domination number, symbolized by γL (G). In this paper we will determine the LDS on edge corona product. The edge corona product of graph is development of corona product graph. The edge corona of two graphs G and H is obtained by taking one copy of G and |E(G)| copies of H and joining each end vertices of i-th edge of G to every vertex in the i-th copy of H, symbolized by G ⋄ H. The results in this study are shown that there is a relation between the locating dominating set on the basic graph and its operation.

Domination parameters in hypertrees and sibling trees

A locating-dominating set (LDS) of a graph G is a dominating set S of G such that for every two vertices u and v in V(G)∖S, N(u)∩S≠N(v)∩S. The locating-domination number γL(G) is the minimum cardinality of a LDS of G. Further if S is a total dominating set then S is called a locating-total dominating set. In this paper we determine the domination, total domination, locating-domination and locating-total domination numbers for hypertrees and sibling trees.

Locating-dominating sets of functigraphs

A locating-dominating set of a graph G is a dominating set of G such that every vertex of G outside the dominating set is uniquely identified by its neighborhood within the dominating set. The location-domination number of G is the minimum cardinality of a locating-dominating set in G. Let G1 and G2 be two disjoint copies of a graph G and f:V(G1)→V(G2) be a function. A functigraph FGf consists of vertex set V(G1)∪V(G2) and edge set E(G1)∪E(G2)∪{uv:v=f(u)}. In this paper, we study the variation of location-domination number in passing from G to FGf and find its sharp lower and upper bounds. We also study location-domination number of functigraphs of complete graphs for all possible definitions of function f. We also obtain location-domination number of functigraphs of a family of spanning subgraph of the complete graphs.

Locating domination in bipartite graphs and their complements

A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G, λ(G), is the minimum cardinality of a locating-dominating set. In this work we study relationships between λ(G) and λ(G¯) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying λ(G¯)=λ(G)+1. To this aim, we define an edge-labeled graph GS associated with a distinguishing set S that turns out to be very helpful.

Location-domination in line graphs

A set D of vertices of a graph G is locating if every two distinct vertices outside D have distinct neighbors in D; that is, for distinct vertices u and v outside D, N(u)∩D≠N(v)∩D, where N(u) denotes the open neighborhood of u. If D is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of G. A graph G is twin-free if every two distinct vertices of G have distinct open and closed neighborhoods. It is conjectured (Garijo et al., 2014 [15]) and (Foucaud and Henning, 2016 [12]) respectively, that any twin-free graph G without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.

Identification, Location-Domination and Metric Dimension on Interval and Permutation Graphs. II. Algorithms and Complexity

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (OPEN) LOCATING-DOMINATING SET and METRIC DIMENSION) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter $2$ and permutation graphs of diameter $2$. While IDENTIFYING CODE and (OPEN) LOCATING-DOMINATING SET are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting METRIC DIMENSION is $W[2]$-hard. We show that for interval graphs, this parameterization of METRIC DIMENSION is fixed-parameter-tractable.