We provide tight bounds on the diameter of γ-graphs, which are reconfiguration graphs of the minimum dominating sets of a graph G. In particular, we prove that for any tree T of order n ≥ 3, the diameter of its γ-graph is at most n/2 in the single vertex replacement adjacency model, whereas in the slide adjacency model, it is at most 2(n − 1)/3. Our proof is constructive, leading to a simple linear-time algorithm for determining the optimal sequence of “moves” between two minimum dominating sets of a tree.
{"title":"Reconfiguring Minimum Dominating Sets in Trees","authors":"M. Lemańska, P. Żyliński","doi":"10.7155/jgaa.00517","DOIUrl":"https://doi.org/10.7155/jgaa.00517","url":null,"abstract":"We provide tight bounds on the diameter of γ-graphs, which are reconfiguration graphs of the minimum dominating sets of a graph G. In particular, we prove that for any tree T of order n ≥ 3, the diameter of its γ-graph is at most n/2 in the single vertex replacement adjacency model, whereas in the slide adjacency model, it is at most 2(n − 1)/3. Our proof is constructive, leading to a simple linear-time algorithm for determining the optimal sequence of “moves” between two minimum dominating sets of a tree.","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71225077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several dierent ways. This allows us obtain several linear time multi-sweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3-sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2-sweep MNS certifying recognition algorithm.
{"title":"Maximal Neighborhood Search and Rigid Interval Graphs","authors":"Peng Li, Yaokun Wu","doi":"10.7155/jgaa.00293","DOIUrl":"https://doi.org/10.7155/jgaa.00293","url":null,"abstract":"A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several dierent ways. This allows us obtain several linear time multi-sweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3-sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2-sweep MNS certifying recognition algorithm.","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71224847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Special Issue with Selected Papers from the 18th International Symposium on Graph Drawing, GD 2010 : Guest Editors' Foreword","authors":"U. Brandes, Sabine Cornelsen","doi":"10.7155/JGAA.00248","DOIUrl":"https://doi.org/10.7155/JGAA.00248","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71225245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Image segmentation with specific constraints has found applications in several areas such as biomedical image analysis and data mining. In this paper, we study the problem of simultaneous detection of both borders of a doughnut-shaped and smooth objects in 2-D medical images. Image objects of that shape are often studied in medical applications. We present an O(IJU(U-L)logJUlog(U-L)) time algorithm, where the size of the input 2-D image is I x J, M is the smoothness parameter with 1
具有特定约束的图像分割在生物医学图像分析和数据挖掘等多个领域都有应用。本文研究了二维医学图像中甜甜圈形状和光滑物体边界的同时检测问题。这种形状的图像对象经常在医学应用中进行研究。我们提出了一种O(IJU(U-L)logJUlog(U-L))时间算法，其中输入二维图像的大小为I x J, M为平滑参数1
{"title":"Simultaneous Border Segmentation of Doughnut-Shaped Objects in Medical Images.","authors":"Xiaodong Wu, Michael Merickel","doi":"10.7155/jgaa.00143","DOIUrl":"https://doi.org/10.7155/jgaa.00143","url":null,"abstract":"<p><p>Image segmentation with specific constraints has found applications in several areas such as biomedical image analysis and data mining. In this paper, we study the problem of simultaneous detection of both borders of a doughnut-shaped and smooth objects in 2-D medical images. Image objects of that shape are often studied in medical applications. We present an O(IJU(U-L)logJUlog(U-L)) time algorithm, where the size of the input 2-D image is I x J, M is the smoothness parameter with 1 </= M </= J, and L and U are the thickness parameters specifying the thickness between two border contours of a doughnut-shaped object. Previous approaches for solving this segmentation problem are computationally expensive and/or need a lot of user interference. Our algorithm improves the straightforward dynamic programming algorithm by a factor of O(J(U-L)M2UlogJUlog(U-L)). We explore some interesting observations, which make possible to apply the divide-and-conquer strategy combined with dynamic programming. Our algorithm is also based on computing optimal paths in an implicitly represented graph.</p>","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2768307/pdf/nihms75919.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"28093448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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