{"title":"Algorithmic and complexity aspects of problems related to total restrained domination for graphs","authors":"Yu Yang, Cai-Xia Wang, Shou-Jun Xu","doi":"10.1007/s10878-023-01090-x","DOIUrl":null,"url":null,"abstract":"<p>Let <i>G</i> be a graph with vertex set <i>V</i> and a subset <span>\\(D\\subseteq V\\)</span>. <i>D</i> is a <i>total dominating set</i> of <i>G</i> if every vertex in <i>V</i> is adjacent to a vertex in <i>D</i>. <i>D</i> is a <i>restrained dominating set</i> of <i>G</i> if every vertex in <span>\\(V\\setminus D\\)</span> is adjacent to a vertex in <i>D</i> and another vertex in <span>\\(V\\setminus D\\)</span>. <i>D</i> is a <i>total restrained dominating set</i> if <i>D</i> is both a total dominating set and a restrained dominating set. The minimum cardinality of total dominating sets (resp. restrained dominating sets, total restrained dominating sets) of <i>G</i> is called the <i>total domination number</i> (resp. <i>restrained domination number</i>, <i>total restrained domination number</i>) of <i>G</i>, denoted by <span>\\(\\gamma _{t}(G)\\)</span> (resp. <span>\\(\\gamma _{r}(G)\\)</span>, <span>\\(\\gamma _{tr}(G)\\)</span>). The MINIMUM TOTAL RESTRAINED DOMINATION (MTRD) problem for a graph <i>G</i> is to find a total restrained dominating set of minimum cardinality of <i>G</i>. The TOTAL RESTRAINED DOMINATION DECISION (TRDD) problem is the decision version of the MTRD problem. In this paper, firstly, we show that the TRDD problem is NP-complete for undirected path graphs, circle graphs, S-CB graphs and C-CB graphs, respectively, and that, for a S-CB graph or C-CB graph with <i>n</i> vertices, the MTRD problem cannot be approximated within a factor of <span>\\((1-\\epsilon )\\textrm{ln} n\\)</span> for any <span>\\(\\epsilon >0\\)</span> unless <span>\\(NP\\subseteq DTIME(n^{O(\\textrm{loglog}n)})\\)</span>. Secondly, for a graph <i>G</i>, we prove that the problem of deciding whether <span>\\(\\gamma _{r}(G) =\\gamma _{tr}(G)\\)</span> is NP-hard even when <i>G</i> is restricted to planar graphs with maximum degree at most 4, and that the problem of deciding whether <span>\\(\\gamma _{t}(G) =\\gamma _{tr}(G)\\)</span> is NP-hard even when <i>G</i> is restricted to planar bipartite graphs with maximum degree at most 5. Thirdly, we show that the MTRD problem is APX-complete for bipartite graphs with maximum degree at most 4. Finally, we design a linear-time algorithm for solving the MTRD problem for generalized series–parallel graphs.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01090-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a graph with vertex set V and a subset \(D\subseteq V\). D is a total dominating set of G if every vertex in V is adjacent to a vertex in D. D is a restrained dominating set of G if every vertex in \(V\setminus D\) is adjacent to a vertex in D and another vertex in \(V\setminus D\). D is a total restrained dominating set if D is both a total dominating set and a restrained dominating set. The minimum cardinality of total dominating sets (resp. restrained dominating sets, total restrained dominating sets) of G is called the total domination number (resp. restrained domination number, total restrained domination number) of G, denoted by \(\gamma _{t}(G)\) (resp. \(\gamma _{r}(G)\), \(\gamma _{tr}(G)\)). The MINIMUM TOTAL RESTRAINED DOMINATION (MTRD) problem for a graph G is to find a total restrained dominating set of minimum cardinality of G. The TOTAL RESTRAINED DOMINATION DECISION (TRDD) problem is the decision version of the MTRD problem. In this paper, firstly, we show that the TRDD problem is NP-complete for undirected path graphs, circle graphs, S-CB graphs and C-CB graphs, respectively, and that, for a S-CB graph or C-CB graph with n vertices, the MTRD problem cannot be approximated within a factor of \((1-\epsilon )\textrm{ln} n\) for any \(\epsilon >0\) unless \(NP\subseteq DTIME(n^{O(\textrm{loglog}n)})\). Secondly, for a graph G, we prove that the problem of deciding whether \(\gamma _{r}(G) =\gamma _{tr}(G)\) is NP-hard even when G is restricted to planar graphs with maximum degree at most 4, and that the problem of deciding whether \(\gamma _{t}(G) =\gamma _{tr}(G)\) is NP-hard even when G is restricted to planar bipartite graphs with maximum degree at most 5. Thirdly, we show that the MTRD problem is APX-complete for bipartite graphs with maximum degree at most 4. Finally, we design a linear-time algorithm for solving the MTRD problem for generalized series–parallel graphs.
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.