{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,26]],"date-time":"2026-09-26T12:02:38Z","timestamp":1790424158851,"version":"4.1.0"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>For each integer $k\\ge 2$, we determine a sharp bound on ${mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$, where $I$ is an independent set and $G[F_k]$ is a forest in which each component has at most $k$ vertices. For each $k$ we construct an infinite family of examples showing our result is the best possible. Our results imply that every planar graph $G$ of girth at least 9 (resp., 8, 7) has a partition of $V(G)$ into an independent set $I$ and a set $F$ such that $G[F]$ is a forest with each component of order at most 3 (resp., 4, 6). Hendrey, Norin, and Wood asked for the largest function $g(a,b)$ such that if ${mad}(G)&lt;g(a,b)$, then $V(G)$ has a partition into sets $A$ and $B$ such that ${mad}(G[A])&lt;a$ and ${mad}(G[B])&lt;b$. They specifically asked for the value of $g(1,b)$, i.e., the case when $A$ is an independent set. Previously, the only values known were $g(1,4\/3)$ and $g(1,2)$. We find $g(1,b)$ whenever $4\/3&lt; b&lt;2$.<\/jats:p>","DOI":"10.1137\/21m1392280","type":"journal-article","created":{"date-parts":[[2021,8,10]],"date-time":"2021-08-10T14:43:51Z","timestamp":1628606631000},"page":"1769-1791","source":"Crossref","is-referenced-by-count":3,"title":["Vertex Partitions into an Independent Set and a Forest with Each Component Small"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3592-6105","authenticated-orcid":true,"given":"Daniel W.","family":"Cranston","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthew P.","family":"Yancey","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,8,10]]},"reference":[{"key":"atypb1","unstructured":"Open problems for the Barbados graph theory workshop\n                      2019, L. Cook and S. Spirkl, eds.https:\/\/sites.google.com\/site\/sophiespirkl\/openproblems2019."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.20467"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2013.07.014"},{"key":"atypb4","first-page":"1004","volume":"52","author":"Borodin O. V.","year":"2011","journal-title":"Sibirsk. Mat. Zh."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2013.10.002"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2020.111921"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/19M1299712"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.37236\/6815"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.21659"},{"key":"atypb10","unstructured":"W. Nadara and M. Smulewicz,\n                      Decreasing the Maximum Average Degree by Deleting an Independent Set or a d-Degenerate Subgraph\n                      , preprint,https:\/\/arxiv.org\/abs\/1909.10701(2019)."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1392280","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:06:55Z","timestamp":1787321215000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1392280"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/21M1392280"],"URL":"https:\/\/doi.org\/10.1137\/21m1392280","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}