{"title":"Recognizing LBFS trees of bipartite graphs","authors":"Robert Scheffler","doi":"10.1016/j.ipl.2024.106483","DOIUrl":null,"url":null,"abstract":"<div><p>The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is <span><math><mi>NP</mi></math></span>-hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless <span><math><mi>P</mi><mo>=</mo><mrow><mi>NP</mi></mrow></math></span>) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.</p></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"186 ","pages":"Article 106483"},"PeriodicalIF":0.7000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0020019024000139/pdfft?md5=0a705320becd861a4100ad392710d19e&pid=1-s2.0-S0020019024000139-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information Processing Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020019024000139","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is -hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless ) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.
期刊介绍:
Information Processing Letters invites submission of original research articles that focus on fundamental aspects of information processing and computing. This naturally includes work in the broadly understood field of theoretical computer science; although papers in all areas of scientific inquiry will be given consideration, provided that they describe research contributions credibly motivated by applications to computing and involve rigorous methodology. High quality experimental papers that address topics of sufficiently broad interest may also be considered.
Since its inception in 1971, Information Processing Letters has served as a forum for timely dissemination of short, concise and focused research contributions. Continuing with this tradition, and to expedite the reviewing process, manuscripts are generally limited in length to nine pages when they appear in print.