Bojan Nikolić, Marko Djukanović, Milana Grbić, Dragan Matić
{"title":"Theoretical studies of the k-strong Roman domination problem","authors":"Bojan Nikolić, Marko Djukanović, Milana Grbić, Dragan Matić","doi":"10.1016/j.kjs.2024.100283","DOIUrl":null,"url":null,"abstract":"<div><p>The concept of Roman domination has been a subject of intrigue for more than two decades, with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the <span><math><mi>k</mi></math></span>-<em>strong Roman domination</em>. In this problem variant, defenders within a network are tasked with safeguarding any <span><math><mi>k</mi></math></span> vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.</p></div>","PeriodicalId":17848,"journal":{"name":"Kuwait Journal of Science","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2307410824001081/pdfft?md5=11535318026ccae769d19361f71f2a09&pid=1-s2.0-S2307410824001081-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kuwait Journal of Science","FirstCategoryId":"103","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2307410824001081","RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
Abstract
The concept of Roman domination has been a subject of intrigue for more than two decades, with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically motivated generalization of this problem, known as the -strong Roman domination. In this problem variant, defenders within a network are tasked with safeguarding any vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.
期刊介绍:
Kuwait Journal of Science (KJS) is indexed and abstracted by major publishing houses such as Chemical Abstract, Science Citation Index, Current contents, Mathematics Abstract, Micribiological Abstracts etc. KJS publishes peer-review articles in various fields of Science including Mathematics, Computer Science, Physics, Statistics, Biology, Chemistry and Earth & Environmental Sciences. In addition, it also aims to bring the results of scientific research carried out under a variety of intellectual traditions and organizations to the attention of specialized scholarly readership. As such, the publisher expects the submission of original manuscripts which contain analysis and solutions about important theoretical, empirical and normative issues.