{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T00:35:40Z","timestamp":1748392540295,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T00:00:00Z","timestamp":1644278400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Algorithmica"],"published-print":{"date-parts":[[2022,6]]},"abstract":"Abstract<\/jats:title>Given a graphG<\/jats:italic>, and terminal verticess<\/jats:italic>andt<\/jats:italic>, theTracking Paths<\/jats:sc>problem asks to compute a set of minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each$$s$$<\/jats:tex-math>s<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-$$t$$<\/jats:tex-math>t<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>path is unique.Tracking Paths<\/jats:sc>isNP<\/jats:sc>-hard in both directed and undirected graphs in general. In this paper we give a collection of polynomial time algorithms for some restricted versions ofTracking Paths<\/jats:sc>. We prove thatTracking Paths<\/jats:sc>is polynomial time solvable for undirected chordal graphs and tournament graphs. We also show thatTracking Paths<\/jats:sc>isNP<\/jats:sc>-hard in graphs with bounded maximum degree$$\\Delta \\ge 6$$<\/jats:tex-math>\u0394<\/mml:mi>\u2265<\/mml:mo>6<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and give a$$2(\\Delta +1)$$<\/jats:tex-math>2<\/mml:mn>(<\/mml:mo>\u0394<\/mml:mi>+<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graphG<\/jats:italic>, a tracking set$$T\\subseteq V(G)$$<\/jats:tex-math>T<\/mml:mi>\u2286<\/mml:mo>V<\/mml:mi>(<\/mml:mo>G<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and a sequence of trackers$$\\pi $$<\/jats:tex-math>\u03c0<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, returns the unique$$s$$<\/jats:tex-math>s<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-$$t$$<\/jats:tex-math>t<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>path inG<\/jats:italic>that corresponds to$$\\pi $$<\/jats:tex-math>\u03c0<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, if one exists. Finally we analyze the version of tracking$$s$$<\/jats:tex-math>s<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-$$t$$<\/jats:tex-math>t<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.<\/jats:p>","DOI":"10.1007\/s00453-022-00931-1","type":"journal-article","created":{"date-parts":[[2022,2,8]],"date-time":"2022-02-08T09:06:40Z","timestamp":1644311200000},"page":"1548-1570","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Polynomial Time Algorithms for Tracking Path Problems"],"prefix":"10.1007","volume":"84","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1648-288X","authenticated-orcid":false,"given":"Pratibha","family":"Choudhary","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,8]]},"reference":[{"issue":"3","key":"931_CR1","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1137\/S0895480196305124","volume":"12","author":"V Bafna","year":"1999","unstructured":"Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. 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