{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:09:03Z","timestamp":1753880943121,"version":"3.41.2"},"reference-count":15,"publisher":"World Scientific Pub Co Pte Ltd","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:p> For a given graph [Formula: see text], the independence number [Formula: see text] of [Formula: see text] is the size of the maximum independent set of [Formula: see text]. Finding the maximum independent set in a graph is NP-hard. Another version of the independence number is defined as the size of the maximum-induced forest of [Formula: see text], and called the forest number of [Formula: see text], and denoted by [Formula: see text]. Finding [Formula: see text] is also an NP-hard problem. Suppose that [Formula: see text] is a graph, and [Formula: see text] is a family of graphs, a graph [Formula: see text] has a [Formula: see text]-free [Formula: see text]-coloring if there exists a decomposition of [Formula: see text] into sets [Formula: see text], [Formula: see text], so that [Formula: see text] for each [Formula: see text], and each [Formula: see text]. [Formula: see text] is [Formula: see text]-free, if the subgraph of [Formula: see text] induced by [Formula: see text] is [Formula: see text]-free, i.e., it contains no copy of [Formula: see text]. Finding a maximum subset [Formula: see text] of [Formula: see text], such that [Formula: see text] is a [Formula: see text]-free graph is a very hard problem as well. In this paper, we study the generalized version of the independence number of a graph. Also, we give some bounds about the size of the maximum [Formula: see text]-free subset of graphs as another purpose of this paper. <\/jats:p>","DOI":"10.1142\/s1793830922501324","type":"journal-article","created":{"date-parts":[[2022,8,12]],"date-time":"2022-08-12T01:41:38Z","timestamp":1660268498000},"source":"Crossref","is-referenced-by-count":0,"title":["Some bounds on the size of maximum G-free sets in graphs"],"prefix":"10.1142","volume":"15","author":[{"given":"Yaser","family":"Rowshan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran"}]}],"member":"219","published-online":{"date-parts":[[2023,3,29]]},"reference":[{"key":"S1793830922501324BIB001","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0012-365X(97)84217-3","volume":"165","author":"Achlioptas D.","year":"1997","journal-title":"Discrete Math."},{"issue":"3","key":"S1793830922501324BIB002","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1002\/jgt.1028","volume":"38","author":"Alon N.","year":"2001","journal-title":"J. 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