Minimum Cardinality Research Articles

Vertex Cover Hop Dominating Sets in Graphs

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Pewarnaan Titik Ketakteraturan Lokal Graf Pensil

The graph used in this paper is a simple and connected graph consisting of a set of vertices and a set of edge . A function is called vertex irregulaarity labelling and . If for every , where and then, f is called local irregularity vertex coloring. The chromatic number of the local irregular vertex coloring of graph is denoted by , which is the minimum cardinality of the largest label of all local irregular vertex coloring. In this article we will learn about coloring points of local irregularities in pencil graphs and finding the exact value of the chromatic number. Keywords: local irregularity; vertex coloring; pencil graphs

Leaf Domination of Graphs

Let G be connected graph. A dominating set D in G is said to be a leaf dominating set if <D> must have at least one leaf vertex. The minimum cardinality of a leaf dominating set is said to be the leaf domination number, denoted by Yld (G). Any minimal dominating set with minimum cardinality is called the Yld set. In this paper, we determine the leaf domination number and calculate the leaf domination number for standard graphs.

On the Local Metric Dimension of Graphs

Let [Formula: see text] be a graph. A set [Formula: see text] is a local resolving set of [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] for any [Formula: see text]. The local metric dimension [Formula: see text] of [Formula: see text] is the minimum cardinality of all the local resolving sets of [Formula: see text]. In this paper, we characterize the graphs with [Formula: see text]. Next, we obtain the Nordhaus–Gaddum-type results for local metric dimension. Finally, the local metric dimension of several graph classes is given.

Weakly connected resolving sets in graphs

Harary and Melter introduced the concept of resolving sets in graphs. Slater stated the term locating set and the minimum resolving set as reference sets. Chartrand mentioned the locating set as a resolving set and the minimum cardinality of a resolving set as metric dimension or simply dimension in later stage. Several types of resolving sets like connected resolving sets, independent resolving sets, acyclic resolving sets, etc. have been discussed in the past. In this work, weakly connected resolving set is introduced and we studied its bounds.

Star structure fault tolerance of Bicube networks

Processor and communication link failures are inevitable in a large multiprocessor system, and so the fault tolerance of its underlying interconnection network has become a key scientific issue. Connectivity is an important parameter to characterize network fault tolerance, and there are many novel variants of classical connectivity to measure the fault tolerance of interconnection networks. However, these new strategies only consider a single faulty vertex. Structure connectivity and substructure connectivity make up for this deficiency, which underline the fault situation with certain specific structures. H-structure-connectivity κ ( G ; H ) (resp. H-substructure-connectivity κ s ( G ; H ) ) of G is the minimum cardinality of H-structure-cuts (resp. H-substructure-cuts). For the n-dimensional Bicube network B Q n , we establish the structure and substructure connectivity of Bicube networks, i.e. κ ( B Q n ; K 1 , 1 ) = κ s ( B Q n ; K 1 , 1 ) = n for odd n ≥ 5 ; κ ( B Q n ; K 1 , 1 ) = κ s ( B Q n ; K 1 , 1 ) = n − 1 for even n ≥ 4 and κ ( B Q n ; K 1 , r ) = κ s ( B Q n ; K 1 , r ) = ⌈ n 2 ⌉ for n ≥ 6 and 2 ≤ r ≤ n − 1 .

The equidistant dimension of graphs: NP-completeness and the case of lexicographic product graphs

<abstract><p>Let $ V(G) $ be the vertex set of a simple and connected graph $ G $. A subset $ S\subseteq V(G) $ is a distance-equalizer set of $ G $ if, for every pair of vertices $ u, v\in V(G)\setminus S $, there exists a vertex in $ S $ that is equidistant to $ u $ and $ v $. The minimum cardinality among the distance-equalizer sets of $ G $ is the equidistant dimension of $ G $, denoted by $ \xi(G) $. In this paper, we studied the problem of finding $ \xi(G\circ H) $, where $ G\circ H $ denotes the lexicographic product of two graphs $ G $ and $ H $. The aim was to express $ \xi(G\circ H) $ in terms of parameters of $ G $ and $ H $. In particular, we considered the cases in which $ G $ has a domination number equal to one, as well as the cases where $ G $ is a path or a cycle, among others. Furthermore, we showed that $ \xi(G)\le \xi(G\circ H)\le \xi(G)|V(H)| $ for every connected graph $ G $ and every graph $ H $ and we discussed the extreme cases. We also showed that the general problem of finding the equidistant dimension of a graph is NP-hard.</p></abstract>

Relating the annihilation number and the 2-domination number for unicyclic graphs

The 2-domination number ?2(G) of a graph G is the minimum cardinality of a set S ? V(G) such that every vertex from V(G)\S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that ?2(G) ? a(G)+1 holds for every non-trivial connected graph G. The conjecture was earlier confirmed for graphs of minimum degree 3, trees, block graphs and some bipartite cacti. However, a class of cacti were found as counterexample graphs recently by Yue et al. [9] to the above conjecture. In this paper, we consider the above conjecture from the positive side and prove that this conjecture holds for all unicyclic graphs.

Algorithmic Aspects of Vertex-edge Domination in Some Graphs

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set \(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

Predicting the evolution of pH and total soluble solids during coffee fermentation using near-infrared spectroscopy coupled with chemometrics

Currently, coffee fermentation is visually operated, which results in incomplete or excessive processes and coffees with undesirable characteristics. In front of it, pH and total soluble solids (TSS) have been shown to be good fermentation indicators, although this requires rapid, accurate, and chemical-free measurement techniques such as NIR spectroscopy. However, the complexity of the NIR spectra requires optimization steps in which variable selection techniques simplify profiles and subsequent models. This work tests a new covering array feature selection (CAFS) approach on NIR spectra to optimize prediction models in coffee samples during fermentation. Spectral profiles in the range 1100–2100 nm were extracted from coffee beans (Typica, Caturra, and Catimor varieties) raw and during fermentation (4, 8, 12, 16, 20, and 24 h). Partial least-squares regressions (PLSR) were performed using full spectra using a five-fold cross-validation strategy for training and validation. The relevant wavelengths were then selected using the β coefficients, the important projection of variables (VIP), and the CAFS method. Finally, optimized models were performed using the relevant wavelengths and compared among these using their statistical metrics. The models performed using the selected variables (22–47) of CAFS showed the best performance in predicting pH (R2 = 0.825–0.903, RMSE = 0.096–0.158, RPD = 6.33–10.38) and TSS (R2 = 0.865–0.922, RMSE = 0.688–1.059, RPD = 0.94–1.45) compared to the other methods. These findings suggest that simple and efficient models could be performed and implemented in routine analysis due to the maximum coverage and minimum cardinality of CAFS.

Burning, edge burning & chromatic burning classification of some graph family

Graph ‘G’ is a Simple and undirected graph, which has a lowest number of color that is required to color the edge is called chromatic index. It is denoted by the symbol χ1(G). In order to burn a graph, a minimum source of edges is required which is called edge burning index or edge burning number and it is denoted by the symbol b1(G). A minimum cardinality of color graph set and the source of the burning index set are called edge chromatic burning and let us assume that it is represented by the symbol χb1(G). In this paper we are mainly concentrating on identifying the chromatic burning and edge chromatic burning graphs and their chromatic burning index using a specific graph family. They are Petersen graph, Helm graph and n- Sunlet graph.

A Study on Variants of Status Unequal Coloring in Graphs and Its Properties

Let be a simple connected graph with vertex set and edge set . The status of a vertex is defined as ∑q≠pd(p, q). A subset of is called a status unequal dominating set (stu‐dominating set) of ; for every , there exists p in such that p and q are adjacent and st(p) ≠ st(q). Among the most admired, extensively explored, and extensively discussed topics in the field of graph theory is graph coloring. This article recommends a new research study on the topic of incorporating stu‐dominating set with coloring. Diverse coloring variants can be found in literature, some of which serve as inspiration is chromatic number, dominator coloring, color class domination, and colorful dominating sets. A novel study on the topic of incorporating stu‐dominating set with coloring is suggested in this article. The stu‐chromatic number, stu‐dominator coloring, stu‐color class domination, and colorful stu‐domination have thus been explored and investigated. For each parameter introduced, the minimum cardinality of certain standard classes of graphs, bounds, and characterization results was explored.

On the edge metric dimension of some classes of cacti

<abstract><p>The cactus graph has many practical applications, particularly in radio communication systems. Let $ G = (V, E) $ be a finite, undirected, and simple connected graph, then the edge metric dimension of $ G $ is the minimum cardinality of the edge metric generator for $ G $ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices $ \mathcal{G}_e = \{g_1, g_2, ..., g_k \} $ of a connected graph $ G $, for any edge $ e\in E $, we referred to the $ k $-vector (ordered $ k $-tuple), $ r(e|\mathcal{G}_e) = (d(e, g_1), d(e, g_2), ..., d(e, g_k)) $ as the edge metric representation of $ e $ with respect to $ G_e $. In this regard, $ \mathcal{G}_e $ is an edge metric generator for $ G $ if, and only if, for every pair of distinct edges $ e_1, e_2 \in E $ implies $ r (e_1 |\mathcal{G}_e) \neq r (e_2 |\mathcal{G}_e) $. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $ \mathfrak{C}(n, c, r) $ and $ \mathfrak{C}(n, m, c, r) $ in terms of the number of cycles $ (c) $ and the number of paths $ (r) $.</p></abstract>

The true twin classes-based investigation for connected local dimensions of connected graphs

<abstract><p>Let $ G $ be a connected graph of order $ n $. The representation of a vertex $ v $ of $ G $ with respect to an ordered set $ W = \{w_1, w_2, ..., w_k\} $ is the $ k $-vector $ r(v|W) = (d(v, w_1), d(v, w_2), ..., d(v, w_k)) $, where $ d(v, w_i) $ represents the distance between vertices $ v $ and $ w_i $ for $ 1\leq i\leq k $. An ordered set $ W $ is called a connected local resolving set of $ G $ if distinct adjacent vertices have distinct representations with respect to $ W $, and the subgraph $ \langle W\rangle $ induced by $ W $ is connected. A connected local resolving set of $ G $ of minimum cardinality is a connected local basis of $ G $, and this cardinality is the connected local dimension $ \mathop{\text{cld}}(G) $ of $ G $. Two vertices $ u $ and $ v $ of $ G $ are true twins if $ N[u] = N[v] $. In this paper, we establish a fundamental property of a connected local basis of a connected graph $ G $. We analyze the connected local dimension of a connected graph without a singleton true twin class and explore cases involving singleton true twin classes. Our investigation reveals that a graph of order $ n $ contains at most two non-singleton true twin classes when $ \mathop{\text{cld}}(G) = n-2 $. Essentially, our work contributes to the characterization of graphs with a connected local dimension of $ n-2 $.</p></abstract>

A Central Local Metric Dimension of Generalized Fan Graph, Generalized Broken Fan Graph, and Cm ⊙ K¯m

The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let 𝐺 be a connected graph with order 𝑛 and vertex set is 𝑉(𝐺). A subset 𝑊={𝑥1,𝑥2,…,𝑥𝑘}⊆𝑉(𝐺) is a local metric set of graph 𝐺 if the metric code of every two adjacent vertices 𝑢,𝑣 in 𝐺 are 𝑟(𝑢|𝑊)≠𝑟(𝑣|𝑊), where 𝑟(𝑢|𝑊)=(𝑑(𝑢,𝑥1),𝑑(𝑢,𝑥2),…,𝑑(𝑢,𝑥𝑘)) and 𝑟(𝑣|𝑊)=(𝑑(𝑣,𝑥1),𝑑(𝑣,𝑥2),…,𝑑(𝑣,𝑥𝑘)). A vertex 𝑥∈𝑉(𝐺) is a central vertex in 𝐺 if 𝑥 have the the shorthest distance to the another all vertices in 𝐺. If 𝑊 consist of all central vertices in 𝐺, then 𝑊 is called a central local metric set of 𝐺. The minimal cardinality of 𝑊 is called a central basis local set of 𝐺 and its cardinality is called central local metric dimension of 𝐺 or denoted by 𝑙𝑚𝑑𝑠(𝐺). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ⊙ K¯n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ⊙ K¯n are equal with the central local metric dimension of 𝐶𝑚.

Roman [1,2]-domination of graphs

A [1,2]-set of a graph G is a set S of vertices of G if every vertex not in S has one or two neighbors in S. The [1,2]-domination number γ[1,2](G) of G equals the minimum cardinality of a [1,2]-set of G. A Roman [1,2]-dominating function (R[1,2]DF) on a graph G is a function f from the vertex set V of G to the set {0,1,2} such that any vertex assigned 0 under f has one or two neighbors assigned 2. The weight of an R[1,2]DF f is the sum ∑x∈Vf(x). The Roman [1,2]-domination number γR[1,2](G) of G equals the minimum weight of an R[1,2]DF on G. In this paper, we prove that the decision problem on the Roman [1,2]-domination is NP-complete for bipartite and chordal graphs. Moreover, we give some bounds on the Roman [1,2]-domination number. In particular, we show that for any nontrivial tree T, γR[1,2](T)≥γ[1,2](T)+1 and characterize all trees obtaining equality in this bound.

Safe sets and in-dominating sets in digraphs

A non-empty subset S of the vertices of a digraph D is a safe set if(i) for every strongly connected component M of D−S, there exists a strongly connected component N of D[S] such that there exists an arc from M to N; and(ii) for every strongly connected component M of D−S and every strongly connected component N of D[S], we have |M|≤|N| whenever there exists an arc from M to N.In the case of acyclic digraphs a set X of vertices is a safe set precisely when X is an in-dominating set, that is, every vertex not in X has at least one arc to X. We prove that, even for acyclic digraphs which are traceable (have a hamiltonian path) it is NP-hard to find a minimum cardinality safe (in-dominating) set. Then we show that the problem is also NP-hard for tournaments and give, for every positive constant c, a polynomial algorithm for finding a minimum cardinality safe set in a tournament on n vertices in which no strong component has size more than clog(n). Under the so called Exponential Time Hypothesis (ETH) this is close to best possible in the following sense: If ETH holds, then, for every ϵ>0 there is no polynomial time algorithm for finding a minimum cardinality safe set for the class of tournaments in which the largest strong component has size at most log1+ϵ(n). We also discuss bounds on the cardinality of safe sets in tournaments.

3-component domination numbers in graphs

Let k be a positive integer and let G=(V(G),E(G)) be a graph. A vertex set D is a k-component dominating set of G if every vertex outside D in G has a neighbor in D and every component of the subgraph G[D] of G induced by D contains at least k vertices. The minimum cardinality of a k-component dominating set of G is the k-component domination number γk(G) of G. It was conjectured that if G is a connected graph of order n≥k+1, and minimum degree at least 2, then γk(G)≤2kn2k+3 except for a finite set of graphs. In this paper, we focus on the parameter γ3(G) of G. We first determine the exact values of 3-component domination numbers of paths and cycles. We then proceed to show that if G is a connected graph of order n with minimum degree at least 2 and maximum degree at most 3, then γ3(G)≤2n3, unless G is one of seven special graphs. This result provides positive support for the conjecture and also generalizes a result by Alvarado et al. (2016) [1].

Transversal total hop domination in graphs

Let [Formula: see text] be a finite simple graph. A set [Formula: see text] is a total hop dominating set of [Formula: see text] if for every [Formula: see text], there exists [Formula: see text] such that [Formula: see text]. Any total hop dominating set of [Formula: see text] of minimum cardinality is a [Formula: see text]-set of [Formula: see text]. A total hop dominating set [Formula: see text] of [Formula: see text] which intersects every [Formula: see text]-set of [Formula: see text] is a transversal total hop dominating set. The minimum cardinality [Formula: see text] of a transversal total hop dominating set in [Formula: see text] is the transversal total hop domination number of [Formula: see text]. In this paper, we initiate the study of transversal total hop domination in graphs. First, we characterize a graph [Formula: see text] of order [Formula: see text] for which [Formula: see text] is [Formula: see text] or [Formula: see text], and also we determine the specific values of [Formula: see text] for some special graphs [Formula: see text]. Next, we solve some realization problems involving [Formula: see text] with other parameters of [Formula: see text]. Finally, we investigate the transversal total hop domination in the complementary prism, corona, and lexicographic product of graphs.

Graphs with distinguishing sets of size k

The size of a resolving set R of a non-trivial connected graph Γ of order n ≥ 2 is the number of edges in the induced subgraph <R>. The minimum cardinality of a resolving set of size k of graph Γ is called the metric dimension of size k, denoted by β(k)(Γ). We study the existence of resolving sets of size k in some families of graphs and investigate their properties. We find bounds on the metric dimension of size k of a graph Γ. We give the necessary condition for the metric dimension of size k and size (k + 1) of a graph Γ, to satisfy the inequality β(k+1)(Γ) − β(k)(Γ) ≤ 1. We will disprove a conjecture on bounds of the metric dimension of size k. For every positive integers k, l, and n such that k + 1 ≤ l ≤ n, we give a realizable result of a graph Γ of order n and l = β(k)(Γ).