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Discrete Math."],"published-print":{"date-parts":[[2021,1,1]]},"abstract":"<jats:p>We study the Max Partial $H$-Coloring problem: given a graph $G$, find the largest induced subgraph of $G$ that admits a homomorphism into $H$, where $H$ is a fixed pattern graph without loops. Note that when $H$ is a complete graph on $k$ vertices, the problem reduces to finding the largest induced $k$-colorable subgraph, which for $k=2$ is equivalent (by complementation) to Odd Cycle Transversal. We prove that for every fixed pattern graph $H$ without loops, Max Partial $H$-Coloring can be solved in $\\{P_5,F\\}$-free graphs in polynomial time, whenever $F$ is a threshold graph; in $\\{P_5,{bull}\\}$-free graphs in polynomial time; in $P_5$-free graphs in time $n^{\\mathcal{O}(\\omega(G))}$; and in $\\{P_6,{1-subdivided claw}\\}$-free graphs in time $n^{\\mathcal{O}(\\omega(G)^3)}$. Here, $n$ is the number of vertices of the input graph $G$ and $\\omega(G)$ is the maximum size of a clique in $G$. Furthermore, by combining the mentioned algorithms for $P_5$-free and for $\\{P_6,{1-subdivided claw}\\}$-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial $H$-Coloring in these classes of graphs. Finally, we show that even a restricted variant of Max Partial $H$-Coloring is $\\mathsf{NP}$-hard in the considered subclasses of $P_5$-free graphs if we allow loops on $H$.<\/jats:p>","DOI":"10.1137\/20m1367660","type":"journal-article","created":{"date-parts":[[2021,10,14]],"date-time":"2021-10-14T10:34:06Z","timestamp":1634207646000},"page":"2357-2386","source":"Crossref","is-referenced-by-count":3,"title":["Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8920-4944","authenticated-orcid":true,"given":"Maria","family":"Chudnovsky","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jason","family":"King","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Micha\u0142","family":"Pilipczuk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7696-3848","authenticated-orcid":true,"given":"Pawe\u0142","family":"Rza\u0345\u017cewski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sophie","family":"Spirkl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,10,14]]},"reference":[{"key":"atypb1","first-page":"1948","author":"Abrishami T.","year":"2021","journal-title":"SIAM"},{"key":"atypb2","first-page":"3","author":"Alekseev V. 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Thomass\u00e9,\n                      Quasi-Polynomial Time Approximation Schemes for the Maximum Weight Independent Set Problem in $H$-Free Graphs\n                      , preprint, arXiv:1907.04585 [cs.DS], 2019,http:\/\/arxiv.org\/abs\/1907.04585.","DOI":"10.1137\/1.9781611975994.139"},{"key":"atypb12","first-page":"2260","author":"Chudnovsky M.","year":"2020","journal-title":"SIAM"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2008.02.005"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-019-00577-6"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/17M1131301"},{"key":"atypb16","first-page":"1239","author":"Chudnovsky M.","year":"2019","journal-title":"SIAM"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s002249910009"},{"key":"atypb18","first-page":"68","author":"Cournier A.","year":"1994","journal-title":"Algebra and Programming"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"M. 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