Abstract
Let be a simple graph with vertex set and edge set . In a graph , a subset of edges denoted by is referred to as an edge-dominating set of if every edge that is not in is incident to at least one member of . A set is the locating edge-dominating set if for every two edges , the sets and are nonempty and different. The edge domination number of is the minimum cardinality of all edge-dominating sets of . The purpose of this study is to determine the locating edge domination number of certain types of claw-free cubic graphs.
1. Introduction
The domination number is a key metric in graph theory, which plays a crucial role in analyzing and comprehending a graph structure [1]. The initial investigations into the concept of the domination number can be credited to scholars such as Berge [2]. He examined numerous facets of this notion and its applications in graph theory. This metric finds utility in diverse domains, encompassing network architecture [3], facility placement [4], and network efficiency and security challenges [5]. Determining the domination number of a graph is frequently a pivotal stage in resolving optimization problems within graph theory [6]. Edge-dominating sets play a crucial role in a wide range of applications, including network architecture, where the primary objective is to achieve effective connectivity while minimizing the number of edges required. Mitchell and Hedetniemi are the ones who initially proposed the concept of an edge-dominating set [7].
The extension of domination properties to fuzzy graphs is also possible. The following are important properties related to domination in fuzzy extensions on graphs: fuzzy domination, fuzzy total domination, fuzzy edge domination, fuzzy locating domination, and fuzzy connected domination ([8, 9]). In a recent study by Sarwar et al., the researchers examined certain formulas determining the lower and upper bounds of dominating and double-dominating energy. The authors also introduced the concept of double-dominating energy of -polar fuzzy digraphs [10]. A decision model utilizing -polar fuzzy preference relations was proposed to address multicriteria decision-making challenges. Studying rough approximations of graphs and hypergraphs [11], distance measures in rough environments [12], and decision-making based on a color spectrum in rough environments provides an extension of domination features [13]. The computation of domination numbers has been conducted for certain categories of graphs, including sparse graphs [14], grid graphs [15], planar graphs [16], regular graphs [17], bipartite graphs [18], social networks, random geometric graphs, pseudofractal scale-free web graphs [19], wheel graphs [20], neural networks [21], and others [22].
Motivated by the above articles, the present study undertakes the task of calculating the edge domination in claw-free cubic graphs. Claw-free cubic graphs have several applications in different fields; in network design, they can be used as models for network design problems, such as designing communication or transportation networks, and their properties make them suitable for creating efficient and reliable network topologies. In VLSI (very large-scale integration) design, claw-free cubic graphs can be used to represent the layout of integrated circuits, and the absence of claws simplifies the layout design process and leads to more regular and efficient chip layouts. In algorithm development, algorithms for various problems can be tested and developed on claw-free cubic graphs due to their well-defined properties. In computational biology, they have been used in modeling biological networks and evolutionary relationships and provide a structured way to represent and analyze complex biological data. In grid computing and mesh topologies, they can serve as models for designing grid or mesh topologies, and these are important for interconnecting computing nodes efficiently. In scheduling and timetabling, claw-free cubic graphs have been applied to scheduling and timetabling problems, where the goal is to allocate resources or time slots efficiently. In planar graphs, these graphs are a special case of planar graphs, which have applications in geographic information systems, cartography, and layout problems in computer-aided design.
This paper is organized in the following way. To demonstrate our primary findings, we will begin by reviewing some fundamental concepts, definitions, and notations relevant to graph theory and combinatorics in Section 2. In Section 3, we will demonstrate the significance of our findings by referring to the concepts presented in Section 2. The paper’s final section has some concluding thoughts regarding the entire study.
2. Preliminaries
For undetermined notations and terminologies, we refer the readers to read the book by Haynes et al. [23].
Let be a simple graph with vertex set and edge set . The neighborhood or open neighborhood of a vertex (or edge ) of the graph is represented by the variable (or, resp., ), and it is the set of all neighbors of the vertex (or edge ), respectively. In a graph , a subset of edges denoted by is referred to as an edge-dominating set of if every edge that is not in is incident to at least one member of . A set is the locating edge-dominating set if for every two edges , the sets and are nonempty and different. The edge domination number of is the minimum cardinality of all edge-dominating sets of [20]. Slater was the first person to present and investigate the idea of positioning, often known as locating dominating set [24–26].
As a generalization of line graphs, claw-free graphs were first investigated and analyzed. The complete bipartite graph is a tree usually called the claw graph. A claw-free graph is a graph that does not have a claw as an induced subgraph. The construction of claw-free perfect graphs was the first topic that Chavatal and Sbihi investigated [27]. Later on, Chudnovsky and Seymour show how the claw-free graph can be constructed in the most general way [28]. Cubic graphs are connected graphs having the property that each vertex has a degree of exactly three [29]. A standard problem in structure enumeration is the production of cubic graphs, which can be thought of as a benchmark. Around the close of the 19th century, de Vries and Martinetti provided one of the first comprehensive lists of cubic-connected graphs when he provided a list of all graphs that are cubic with a maximum of 10 vertices [30]. They may be used to represent three-dimensional objects, which will enable you to find a dimension that is missing or investigate the impact that changes made to one or more dimensions have.
Here, we introduce claw-free graphs in a different and generalized manner. A complete graph with the removal of one edge is called diamond [31], and it is denoted by . A string of diamonds is a maximal sequence of diamonds in which a vertex of degree two of is joined by an edge to a vertex of degree two of , and joining the pendent vertex of and to a vertex of (see Figure 1). Furthermore, if is a connected-claw-free simple cubic graph via every vertex in a diamond, then is often referred to as a ring of diamonds (see Figure 2).
At this stage, we are going to look at two infinite families and . Let be a graph that is obtained in the following manner for . Consider two copies of the path , each with the vertex sequences and . For each , join to , to , to , and to to complete the graph’s structure. See for illustration, Figure 3(a). Consider represent the graph formed from for through combining and and deleting and . See, for illustration, Figure 3(b). It has been brought to our attention that both and are cubic graphs of order .
(a)
(b)
3. Main Results
In this section, we will compute the locating edge-dominating set and the locating edge domination number for several kinds of claw-free cubic graphs.
Theorem 1. Let be a string of diamonds (see Figure 1). Then, locating edge-dominating number of is
Proof. The string of diamonds has and vertex and edge sets. The cardinality of the vertex set is , while the cardinality of the edge set is . In order to demonstrate the minimum cardinality required for the locating edge-dominating set, i.e., with . Take edge set of , i.e., . Choose a dominating set from edge set that is . As we observed, the cardinality of locating edge-dominating set is . Subtracting the dominating set from edge set, we get . The result of putting and together is as follows:
From the above expressions, it is simple to see that they are all nonempty and distinct. It indicates that the dominating set dominates all edges in , and it satisfied the condition, i.e., for , the sets and are nonempty and different.
Now, we will prove that the edge set is a minimum locating edge-dominating set. For that, we will take the edge-dominating set , i.e., . To prove the minimum cardinality, we will remove any edge from dominating set. For example, let us remove the edge from the set . Now, consider the new dominating set . Subtracting the new dominating set from the edge set , i.e., . The result of putting and together is as follows:
From the above expressions, it is easy to see that for , the sets and are the same, which is a contradiction itself. So, is the minimum locating edge-dominating set, and the locating edge-dominating number of is .
Theorem 2. Let be a ring of diamonds. Then, locating edge domination number for is .
Proof. The ring of diamonds has and vertex and edge sets. The cardinality of the vertex set is , while the cardinality of the edge set is . In order to demonstrate the minimum cardinality required for the locating edge-dominating set, i.e., for . Take edge set of , i.e., . Choose a dominating set from the edge set that is . As we observed, the cardinality of locating edge-dominating set is . Subtracting the dominating set from edge set, we get . The result of putting and together is as follows:
From the above expressions, it is simple to see that they are all nonempty and distinct. It indicates that the dominating set dominates all edges in , and it satisfied the condition, i.e., for , the sets and are nonempty and different.
Now, we will prove that the edge set is a minimum locating edge-dominating set. For that, we will take edge-dominating set , i.e., . To prove the minimum cardinality, we will remove any edge from dominating set. For example, let us remove the edge from the set . Now, consider the new dominating set . Subtracting the new dominating set from the edge set , i.e., . The result of putting and together is as follows:
From the above expressions, it is easy to see that the set is empty, which is a contradiction itself. So, is the minimum locating edge-dominating set, and the locating edge-dominating number of is .
Theorem 3. Let be a cubic graph. Then, locating edge domination number of is .
Proof. The cubic graph has and vertex and edge sets. The cardinality of the vertex set is , while the cardinality of the edge set is . In order to demonstrate the minimum cardinality required for the locating edge-dominating set, i.e., for . Take edge set of , i.e., . Choose a dominating set from the edge set that is , where is even and . As we observed, the cardinality of locating edge-dominating set is . Subtracting the dominating set from edge set, we get (where is odd). The result of putting and together is as follows:
From the above expressions, it is simple to see that they are all nonempty and distinct. It indicates that the dominating set dominates all edges in , and it satisfied the condition, i.e., for , the sets and are nonempty and different.
Now, we will prove that the edge set is a minimum locating edge-dominating set. For that, we will take edge-dominating set , i.e., , where is even and . To prove the minimum cardinality, we will remove any edge from dominating set. For example, let us remove the edge from the set . Now, consider the new dominating set , where is even and . Subtracting the new dominating set from the edge set , we get (where is odd). The result of putting and together is as follows:
From the above expressions, it is easy to see that for , the sets and are the same and empty, which is a contradiction itself. So, is the minimum locating edge-dominating set, and the locating edge-dominating number of is .
Theorem 4. Let be a cubic graph. Then, locating edge domination number of is .
Proof. The cubic graph has and vertex and edge sets. The cardinality of the vertex set is , while the cardinality of the edge set is . In order to demonstrate the minimum cardinality required for the locating edge-dominating set, i.e., for . Take edge set of , i.e., . Choose a dominating set from the edge set that is , where is even and . As we observed, the cardinality of locating edge-dominating set is . Subtracting the dominating set from edge set, we get , where is odd. The results of putting and together are as follows:
From the above expressions, it is simple to see that they are all nonempty and distinct. It indicates that the dominating set dominates all edges in , and it satisfied the condition, i.e., for , the sets and are nonempty and different.
Now, we will prove that the edge set is a minimum locating edge-dominating set. For that, we will take edge-dominating set , i.e., . To prove the minimum cardinality, we will remove any edge from dominating set. For example, let us remove the edge from the set . Now, consider the new dominating set , where is even and . Subtracting the new dominating set from the edge set , we get , where is odd. The result of putting and together is as follows:
From the above expressions, it is easy to see that for , the sets and are the same, which is a contradiction itself. So, is the minimum locating edge-dominating set, and the locating edge-dominating number of is .
4. Conclusions
Claw-free cubic graphs are graphs with no induced subgraphs that are isomorphic to a claw. In this study, we used fundamental graph theory and combinatorics definitions to ascertain the edge-locating dominating set and the domination number for various classes of claw-free cubic graphs. We have discovered the locating edge-dominating set and the domination number for the cubic graphs and , as well as the string of diamonds and the ring of diamonds . The fact that the edge-dominating number for these graphs is less than almost half of their orders is highly interesting information. Because of their well-defined form and features, graphs are valuable in modeling, analysis, and problem-solving tools that may be applied across various domains. This current work is limited to edge-locating dominating set and domination number of various classes of simple claw-free cubic graphs, and its consequences may not be validated for the directed graphs. For future studies, it is recommended that researchers compute the fuzzy total domination, fuzzy edge domination, and fuzzy locating domination numbers for each of the several classes of claw-free cubic graphs.
Data Availability
All relevant data is given in the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
The first author, Muhammad Shoaib Sardar, conducted this research, conceptualized the idea of the research, investigated the problem, and wrote the original text of the manuscript. The second coauthor, Hamna Choudhry, reviewed, edited, and analyzed this paper. The third and last coauthor, Jia-Bao Liu, supervised this research and also studied the validation of the results.
Acknowledgments
The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to a great deal of improvement of the original manuscript.