Location-domination Number Research Articles

On Resolvability of Graphs Associated with Vector Spaces

Let be a n-dimensional vector space over the field F with a basis . In this paper, we obtain the resolving parameters like metric dimension and partition dimension of graphs associated with vector space. Also, found the values of metric-locating-domination number, locating-domination number, and bipartite decomposition of kind of graph associated with vector space.

Global Stable Location-Domination in Graphs

In this paper, we introduce and investigate the concept of global stable location-domination in graphs. We also characterize the global stable locating-dominating sets in the join, edge corona, corona, and lexicographic product of graphs and determine the exact value or sharp bound of the corresponding global stable location-domination number.

Bounds and Extremal Graphs for Total Dominating Identifying Codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. The smallest size of an identifying code of $G$ is denoted $\gamma^{\text{ID}}(G)$. When every vertex of $G$ also has a neighbour in $C$, it is said to be a total dominating identifying code of $G$, and the smallest size of a total dominating identifying code of $G$ is denoted by $\gamma_t^{\text{ID}}(G)$.
 Extending similar characterizations for identifying codes from the literature, we characterize those graphs $G$ of order $n$ with $\gamma_t^{\text{ID}}(G)=n$ (the only such connected graph is $P_3$) and $\gamma_t^{\text{ID}}(G)=n-1$ (such graphs either satisfy $\gamma^{\text{ID}}(G)=n-1$ or are built from certain such graphs by adding a set of universal vertices, to each of which a private leaf is attached).
 Then, using bounds from the literature, we remark that any (open and closed) twin-free tree of order $n$ has a total dominating identifying code of size at most $\frac{3n}{4}$. This bound is tight, and we characterize the trees reaching it. Moreover, by a new proof, we show that this upper bound actually holds for the larger class of all twin-free graphs of girth at least 5. The cycle $C_8$ also attains the upper bound. We also provide a generalized bound for all graphs of girth at least 5 (possibly with twins).
 Finally, we relate $\gamma_t^{\text{ID}}(G)$ to the similar parameter $\gamma^{\text{ID}}(G)$ as well as to the location-domination number of $G$ and its variants, providing bounds that are either tight or almost tight.

On distance-s locating and distance-t dominating sets in graphs

Let [Formula: see text] and [Formula: see text] be positive integers, and let [Formula: see text] be a simple and connected graph with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text] and let [Formula: see text]. Let [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] locating set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] location number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] locating sets of [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] dominating set of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] dominating sets of [Formula: see text]. The [Formula: see text]-location-domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all sets [Formula: see text] such that [Formula: see text] is both a distance-[Formula: see text] locating set and a distance-[Formula: see text] dominating set of [Formula: see text]. For any connected graph [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text]. We characterize connected graphs [Formula: see text] satisfying [Formula: see text] equals [Formula: see text] and [Formula: see text], respectively. We examine the relationship among [Formula: see text], [Formula: see text] and [Formula: see text]; along the way, we show that [Formula: see text] if [Formula: see text]. We also show that there exist graphs [Formula: see text] and [Formula: see text] with [Formula: see text] such that [Formula: see text] can be arbitrarily large. Moreover, we examine some graph classes.

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.

On the binary locating-domination number of regular and strongly-regular graphs

On the binary locating-domination number of regular and strongly-regular graphs

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).

The adjacency dimension of graphs

It is known that the problem of computing the adjacency dimension of a graph is NP-hard. This suggests finding the adjacency dimension for special classes of graphs or obtaining good bounds on this invariant. In this work we obtain general bounds on the adjacency dimension of a graph G in terms of known parameters of G . We discuss the tightness of these bounds and, for some particular classes of graphs, we obtain closed formulae. In particular, we show the close relationships that exist between the adjacency dimension and other parameters, like the domination number, the location-domination number, the 2 -domination number, the independent 2 -domination number, the vertex cover number, the independence number and the super domination number.

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.

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.

BILANGAN DOMINASI LOKASI PADA LINE PAN GRAPH DAN MIDDLE PAN GRAPH

Let G is a simple, connected, and undirected graph. A set D is a subset of V. If every node of V adjacent to at least one node D, then D is a dominating set of the graph G . One of the topics from dominating set is locating dominating set. Locating dominating set is dominating set on condition if every two vertices u,v elements of V-D satisfy the intersection of N(v) and D not equal to the intersection of N(u) and D with u not equal to v. The locating domination number of a graph G is the minimum cardinality of a locating dominating set in a graph G . In this study discussed the locating domination number on line pan graph (L(Tn,1)) and middle pan graph (M(Tn,1)). Locating domination number was obtained by finding dominating set from some graph. Then, does the dominating set meet the condition of locating dominating set? If it meets locating dominating set condition, then we can find the locating domination number of the graphs. In the last procedure, we get pattern location domination number of line pan graph and middle pan graph. The results of the study obtained the locating domination number of (L(Tn,1)) and (M(Tn,1)) consecutive are floor function of (2n/5)+1 and floor function of ((2n+1)/3)+1.

Analisis Locating Dominating Set pada Graf Khusus dan Hasil Operasi Comb Sisi

Assume that G = (V;E) is an undirected and connected graph with vertex set V and edge set E. D is called a dominating set of the vertex in G such that for each vertex v 2 V one of: v 2 D or a neighbor u of v in D with u 2 D. While locating dominating set of G is a dominating set D of G when satisfy this condition: for every two vertices u; v 2 (V 􀀀 D);N(u) \ DN(v) \ D. The minimum cardinality of a locating dominating set of G is the location domination number L(G). In this paper, locating dominating set and location domination number of special graph and edge comb product operation result will be determined. Location domination number theorem on triangular book graph Btn and edge comb product operation result that is Cm D Btn and Sm D Btn are the results from this experiment.

Covering Codes of a Graph Associated with a Finite Vector Space

UDC 512.5 In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [Commun. Algebra, 44, 3918 – 3926 (2016)], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.

Covering codes of a graph associated to a finite vector space

UDC 512.5 In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das [Commun. Algebra, <strong>44</strong>, 3918 – 3926 (2016)], such that we can uniquely identify any vertex by examining the vertices that cover it. We use locating-dominating sets and identifying codes, which are closely related concepts for this purpose. We find the location-domination number and the identifying number of the graph and study the exchange property for locating-dominating sets and identifying codes.

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.

A note on the complexity of locating-total domination in graphs

A total dominating set of a graph G=(V,E) with no isolated vertex is a set D⊆V(G) such that every vertex of G is adjacent to a vertex in D. A total dominating set D of G is a locating-total dominating set if for every pair of distinct vertices u and v in V−D, N(u)∩D≠N(v)∩D. Let γLt(G) be the minimum cardinality of a locating-total dominating set of G. In this paper, we show that the decision problem for locating-total domination number is NP-complete for several classes of graphs. In particular, we answer a problem posed in Miller et al. (2017) [13]. We also define classes of graphs for which the complexities of domination number, locating domination number and locating-total domination number are different.

Location domination number of line graph

A set S which dominate and uniquely locate all vertices in the set V(G) – S is a locating dominating set. In this paper location domination number of line graph of Pn, Cn, Sn, Bm, n, F1, n and Wn have been found