Alane M. de Lima, Murilo V. G. da Silva, A. L. Vignatti
. Let G be an undirected graph with non-negative edge weights and let S be a subset of its shortest paths such that, for every pair ( u, v ) of distinct vertices, S contains exactly one shortest path between u and v . In this paper we define a range space associated with S and prove that its VC dimension is 2. As a consequence, we show a bound for the number of shortest paths trees required to be sampled in order to solve a relaxed version of the All-pairs Shortest Paths problem (APSP) in G . In this version of the problem we are interested in computing all shortest paths with a certain “importance” at least ε . Given any 0 < ε , δ < 1, we propose a O ( m + n log n + (diam V ( G ) ) 2 ) sampling algorithm that outputs with probability 1 − δ the (exact) distance and the shortest path between every pair of vertices ( u, v ) that appears as subpath of at least a proportion ε of all shortest paths in the set S , where diam V ( G ) is the vertex-diameter of G . The bound that we obtain for the sample size depends only on ε and δ , and do not depend on the size of the graph.
． 设G是一个边权为非负的无向图，设S是其最短路径的子集，使得对于每一对(u, v)不同的顶点，S包含u和v之间的最短路径。定义了一个与S相关的值域空间，并证明了它的VC维为2。因此，我们给出了为了解决G中的全对最短路径问题(APSP)的一个宽松版本而需要采样的最短路径树的数量界限。在这个版本的问题中，我们感兴趣的是计算具有一定“重要性”至少为ε的所有最短路径。给出任何0 <δε,< 1,我们提出一个O (m + n O (log n) +(直径V (G)) 2)抽样算法输出的概率1−δ(精确的)距离和每一对顶点之间的最短路径(u, V)出现的子路径至少ε比例的最短路径设置年代,在直径V (G)的vertex-diameter G。我们得到的样本大小的界只取决于ε和δ，而不取决于图的大小。
{"title":"A Range Space with Constant VC Dimension for All-pairs Shortest Paths in Graphs","authors":"Alane M. de Lima, Murilo V. G. da Silva, A. L. Vignatti","doi":"10.7155/jgaa.00636","DOIUrl":"https://doi.org/10.7155/jgaa.00636","url":null,"abstract":". Let G be an undirected graph with non-negative edge weights and let S be a subset of its shortest paths such that, for every pair ( u, v ) of distinct vertices, S contains exactly one shortest path between u and v . In this paper we define a range space associated with S and prove that its VC dimension is 2. As a consequence, we show a bound for the number of shortest paths trees required to be sampled in order to solve a relaxed version of the All-pairs Shortest Paths problem (APSP) in G . In this version of the problem we are interested in computing all shortest paths with a certain “importance” at least ε . Given any 0 < ε , δ < 1, we propose a O ( m + n log n + (diam V ( G ) ) 2 ) sampling algorithm that outputs with probability 1 − δ the (exact) distance and the shortest path between every pair of vertices ( u, v ) that appears as subpath of at least a proportion ε of all shortest paths in the set S , where diam V ( G ) is the vertex-diameter of G . The bound that we obtain for the sample size depends only on ε and δ , and do not depend on the size of the graph.","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71225272","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":"Planar Confluent Orthogonal Drawings of 4-Modal Digraphs","authors":"Sabine Cornelsen, Gregor Diatzko","doi":"10.7155/jgaa.00632","DOIUrl":"https://doi.org/10.7155/jgaa.00632","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136299752","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}
Alfredo García, Javier Tejel, Birgit Vogtenhuber, Alexandra Weinberger
{"title":"Empty Triangles in Generalized Twisted Drawings of $K_n$","authors":"Alfredo García, Javier Tejel, Birgit Vogtenhuber, Alexandra Weinberger","doi":"10.7155/jgaa.00637","DOIUrl":"https://doi.org/10.7155/jgaa.00637","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135450723","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":"Minimum Linear Arrangement of Generalized Sierpinski Graphs","authors":"Sundara Rajan R, Berin Greeni A, Leo Joshwa P","doi":"10.7155/jgaa.00644","DOIUrl":"https://doi.org/10.7155/jgaa.00644","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135611750","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}
. The angular resolution of a straight-line drawing of a graph is the smallest angle formed by any two edges incident to a vertex. The angular resolution of a graph is the supremum of the angular resolutions of all straight-line drawings of the graph. We show that testing whether a graph has angular resolution at least π/ (2 k ) is complete for ∃ R , the existential theory of the reals, for every fixed k ≥ 2 . This remains true if the graph is planar and a plane embedding of the graph is fixed.
{"title":"The Complexity of Angular Resolution","authors":"M. Schaefer","doi":"10.7155/jgaa.00634","DOIUrl":"https://doi.org/10.7155/jgaa.00634","url":null,"abstract":". The angular resolution of a straight-line drawing of a graph is the smallest angle formed by any two edges incident to a vertex. The angular resolution of a graph is the supremum of the angular resolutions of all straight-line drawings of the graph. We show that testing whether a graph has angular resolution at least π/ (2 k ) is complete for ∃ R , the existential theory of the reals, for every fixed k ≥ 2 . This remains true if the graph is planar and a plane embedding of the graph is fixed.","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71225261","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}
Rahnuma Islam Nishat, Venkatesh Srinivasan, Sue Whitesides
{"title":"1-Complex $s,t$ Hamiltonian Paths: Structure and Reconfiguration in Rectangular Grids","authors":"Rahnuma Islam Nishat, Venkatesh Srinivasan, Sue Whitesides","doi":"10.7155/jgaa.00624","DOIUrl":"https://doi.org/10.7155/jgaa.00624","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135685951","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}
Steven Chaplick, Krzysztof Fleszar, Fabian Lipp, Alexander Ravsky, Oleg Verbitsky, Alexander Wolff
It is well known that any graph admits a crossing-free straight-line drawing in $mathbb{R}^3$ and that any planar graph admits the same even in $mathbb{R}^2$. For a graph $G$ and $d in {2,3}$, let $rho^1_d(G)$ denote the minimum number of lines in $mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results.
{"title":"The Complexity of Drawing Graphs on Few Lines and Few Planes","authors":"Steven Chaplick, Krzysztof Fleszar, Fabian Lipp, Alexander Ravsky, Oleg Verbitsky, Alexander Wolff","doi":"10.7155/jgaa.00630","DOIUrl":"https://doi.org/10.7155/jgaa.00630","url":null,"abstract":"It is well known that any graph admits a crossing-free straight-line drawing in $mathbb{R}^3$ and that any planar graph admits the same even in $mathbb{R}^2$. For a graph $G$ and $d in {2,3}$, let $rho^1_d(G)$ denote the minimum number of lines in $mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. ","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783904","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}
Ioannis Katsikarelis, Michael Lampis, Vangelis Th. Paschos
In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization.
{"title":"Improved (In-)Approximability Bounds for d-Scattered Set","authors":"Ioannis Katsikarelis, Michael Lampis, Vangelis Th. Paschos","doi":"10.7155/jgaa.00621","DOIUrl":"https://doi.org/10.7155/jgaa.00621","url":null,"abstract":"In the $d$-Scattered Set problem we are asked to select at least $k$ vertices of a given graph, so that the distance between any pair is at least $d$. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. ","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135181370","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}
Vasileios Geladaris, Panagiotis Lionakis, Ioannis G. Tollis
. Computing a minimum Feedback Arc Set (FAS) is important for visualizing directed graphs in hierarchical style. It is the first step of both known frameworks for hierarchical graph drawing of directed graphs and it is NP-hard. We present a new heuristic algorithm for computing a minimum FAS in directed graphs. The new technique produces solutions that, for graph drawing datasets, are better than the ones produced by the best previously known heuristics, often reducing the FAS size by more than 50%. The heuristic is based on computing the PageRank score of the nodes of the directed line graph of the input directed graph. Although the time required by our heuristic is heavily influenced by the size of the produced line graph, our experimental results show that it runs very fast even for very large graphs used in graph drawing. We compare results produced by our heuristic to known exact results for specific graphs used in a previous study and discuss the interesting trade-off. Finally, our experimental results on large web-graphs show that our technique found smaller FAS than it was known before for some web-graphs from a data set used in a recent study.
{"title":"Effective Computation of a Feedback Arc Set Using PageRank","authors":"Vasileios Geladaris, Panagiotis Lionakis, Ioannis G. Tollis","doi":"10.7155/jgaa.00641","DOIUrl":"https://doi.org/10.7155/jgaa.00641","url":null,"abstract":". Computing a minimum Feedback Arc Set (FAS) is important for visualizing directed graphs in hierarchical style. It is the first step of both known frameworks for hierarchical graph drawing of directed graphs and it is NP-hard. We present a new heuristic algorithm for computing a minimum FAS in directed graphs. The new technique produces solutions that, for graph drawing datasets, are better than the ones produced by the best previously known heuristics, often reducing the FAS size by more than 50%. The heuristic is based on computing the PageRank score of the nodes of the directed line graph of the input directed graph. Although the time required by our heuristic is heavily influenced by the size of the produced line graph, our experimental results show that it runs very fast even for very large graphs used in graph drawing. We compare results produced by our heuristic to known exact results for specific graphs used in a previous study and discuss the interesting trade-off. Finally, our experimental results on large web-graphs show that our technique found smaller FAS than it was known before for some web-graphs from a data set used in a recent study.","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135450045","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":"Efficient Point-to-Point Resistance Distance Queries in Large Graphs","authors":"C. Gotsman, K. Hormann","doi":"10.7155/jgaa.00612","DOIUrl":"https://doi.org/10.7155/jgaa.00612","url":null,"abstract":"","PeriodicalId":35667,"journal":{"name":"Journal of Graph Algorithms and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71225317","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}
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