{"title":"How vulnerable is an undirected planar graph with respect to max flow","authors":"Lorenzo Balzotti, Paolo G. Franciosa","doi":"10.1002/net.22205","DOIUrl":null,"url":null,"abstract":"We study the problem of computing the vitality of edges and vertices with respect to the <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0001.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0001\" display=\"inline\" location=\"graphic/net22205-math-0001.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">s</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi></mrow>$$ st $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-max flow in undirected planar graphs, where the vitality of an edge/vertex is the <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0002.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0002\" display=\"inline\" location=\"graphic/net22205-math-0002.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">s</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi></mrow>$$ st $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0003.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0003\" display=\"inline\" location=\"graphic/net22205-math-0003.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">s</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi></mrow>$$ st $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0004.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic- data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0004\" display=\"inline\" location=\"graphic/net22205-math-0004.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple;clearspeak:unit\" data-semantic-children=\"0,1\" data-semantic-content=\"2\" data-semantic-role=\"implicit\" data-semantic-speech=\"s t\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">s</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"3\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"3\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">t</mi></mrow>$$ st $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-max flow <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0005.png\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,13\" data-semantic-content=\"14,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"upper O left parenthesis n log log n right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"12\" data-semantic-content=\"1,6\" data-semantic- data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,10\" data-semantic-content=\"11\" data-semantic- data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"3,8\" data-semantic-content=\"9,3\" data-semantic- data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,5\" data-semantic-content=\"7,4\" data-semantic- data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mjx-mi data-semantic-font=\"normal\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\"><mjx-c></mjx-c><mjx-c></mjx-c><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-mrow></mjx-mrow></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0005\" display=\"inline\" location=\"graphic/net22205-math-0005.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,13\" data-semantic-content=\"14,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"upper O left parenthesis n log log n right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">O</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"15\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"12\" data-semantic-content=\"1,6\" data-semantic-parent=\"15\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,10\" data-semantic-content=\"11\" data-semantic-parent=\"13\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"12\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"12\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"3,8\" data-semantic-content=\"9,3\" data-semantic-parent=\"12\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"10\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"4,5\" data-semantic-content=\"7,4\" data-semantic-parent=\"10\" data-semantic-role=\"prefix function\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-font=\"normal\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"prefix function\" data-semantic-type=\"function\">log</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></mrow></mrow></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"13\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ O\\left(n\\mathrm{loglog}n\\right) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in <mjx-container aria-label=\"Menu available. Press control and space , or space\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\" location=\"graphic/net22205-math-0006.png\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"0,4\" data-semantic-content=\"5,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"upper O left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"2\" data-semantic-content=\"1,3\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"4\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"4\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"urn:x-wiley:net:media:net22205:net22205-math-0006\" display=\"inline\" location=\"graphic/net22205-math-0006.png\" overflow=\"scroll\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"0,4\" data-semantic-content=\"5,0\" data-semantic-role=\"simple function\" data-semantic-speech=\"upper O left parenthesis n right parenthesis\" data-semantic-type=\"appl\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\">O</mi><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"appl\" data-semantic-parent=\"6\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"2\" data-semantic-content=\"1,3\" data-semantic-parent=\"6\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"4\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"4\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"4\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow></mrow>$$ O(n) $$</annotation></semantics></math></mjx-assistive-mml></mjx-container> space.","PeriodicalId":54734,"journal":{"name":"Networks","volume":"14 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Networks","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1002/net.22205","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, HARDWARE & ARCHITECTURE","Score":null,"Total":0}
引用次数: 0
Abstract
We study the problem of computing the vitality of edges and vertices with respect to the -max flow in undirected planar graphs, where the vitality of an edge/vertex is the -max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the -max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute -max flow . We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in space.
期刊介绍：
Network problems are pervasive in our modern technological society, as witnessed by our reliance on physical networks that provide power, communication, and transportation. As well, a number of processes can be modeled using logical networks, as in the scheduling of interdependent tasks, the dating of archaeological artifacts, or the compilation of subroutines comprising a large computer program. Networks provide a common framework for posing and studying problems that often have wider applicability than their originating context.
The goal of this journal is to provide a central forum for the distribution of timely information about network problems, their design and mathematical analysis, as well as efficient algorithms for carrying out optimization on networks. The nonstandard modeling of diverse processes using networks and network concepts is also of interest. Consequently, the disciplines that are useful in studying networks are varied, including applied mathematics, operations research, computer science, discrete mathematics, and economics.
Networks publishes material on the analytic modeling of problems using networks, the mathematical analysis of network problems, the design of computationally efficient network algorithms, and innovative case studies of successful network applications. We do not typically publish works that fall in the realm of pure graph theory (without significant algorithmic and modeling contributions) or papers that deal with engineering aspects of network design. Since the audience for this journal is then necessarily broad, articles that impact multiple application areas or that creatively use new or existing methodologies are especially appropriate. We seek to publish original, well-written research papers that make a substantive contribution to the knowledge base. In addition, tutorial and survey articles are welcomed. All manuscripts are carefully refereed.
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