{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T08:27:37Z","timestamp":1759134457320},"reference-count":16,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"name":"ERB, DST, India","award":["SRG\/2022\/000314"],"award-info":[{"award-number":["SRG\/2022\/000314"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:p> Given a finite simple undirected graph G there is a simplicial complex Ind(G), called the independence complex, whose faces correspond to the independent sets of G. This is a well-studied concept because it provides a fertile ground for interactions between commutative algebra, graph theory and algebraic topology. In this paper, we consider a generalization of independence complex. Given [Formula: see text], a subset of the vertex set is called r-independent if the connected components of the induced subgraph have cardinality at most r. The collection of all r-independent subsets of G form a simplicial complex called the r-independence complex and is denoted by Ind<jats:sub>r<\/jats:sub>(G). It is known that when G is a chordal graph the complex Ind<jats:sub>r<\/jats:sub>(G) has the homotopy type of a wedge of spheres. Hence, it is natural to ask which of these complexes are shellable or even vertex decomposable. We prove, using Woodroofe\u2019s chordal hypergraph notion, that these complexes are always shellable when the underlying chordal graph is a tree. Using the notion of vertex splittable ideals we show that for caterpillar graphs the associated r-independence complex is vertex decomposable for all values of r. Further, for any [Formula: see text] we construct chordal graphs on [Formula: see text] vertices such that their r-independence complexes are not sequentially Cohen\u2013Macaulay. <\/jats:p>","DOI":"10.1142\/s0218196723500236","type":"journal-article","created":{"date-parts":[[2023,3,1]],"date-time":"2023-03-01T12:49:50Z","timestamp":1677674990000},"page":"481-498","source":"Crossref","is-referenced-by-count":4,"title":["Chordal graphs, higher independence and vertex decomposable complexes"],"prefix":"10.1142","volume":"33","author":[{"given":"Fred M.","family":"Abdelmalek","sequence":"first","affiliation":[{"name":"University of Toronto, Canada"}]},{"given":"Priyavrat","family":"Deshpande","sequence":"additional","affiliation":[{"name":"Chennai Mathematical Institute, India"}]},{"given":"Shuchita","family":"Goyal","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology Kanpur, India"}]},{"given":"Amit","family":"Roy","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar 752050, India"},{"name":"Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400094, India"}]},{"given":"Anurag","family":"Singh","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology Bhilai, India"}]}],"member":"219","published-online":{"date-parts":[[2023,4,12]]},"reference":[{"issue":"3","key":"S0218196723500236BIB001","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s00493-007-2086-y","volume":"27","author":"Aharoni R.","year":"2007","journal-title":"Combinatorica"},{"issue":"3","key":"S0218196723500236BIB002","doi-asserted-by":"crossref","first-page":"965","DOI":"10.4007\/annals.2007.165.965","volume":"165","author":"Babson E.","year":"2007","journal-title":"Ann. of Math. (2)"},{"key":"S0218196723500236BIB003","doi-asserted-by":"crossref","first-page":"103508","DOI":"10.1016\/j.ejc.2022.103508","volume":"102","author":"Deshpande P.","year":"2022","journal-title":"European J. Combin."},{"issue":"1","key":"S0218196723500236BIB004","first-page":"53","volume":"36","author":"Deshpande P.","year":"2021","journal-title":"J. Ramanujan Math. Soc."},{"issue":"3","key":"S0218196723500236BIB006","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s00493-011-2665-9","volume":"31","author":"Engstr\u00f6m A.","year":"2011","journal-title":"Combinatorica"},{"issue":"8","key":"S0218196723500236BIB007","doi-asserted-by":"crossref","first-page":"2327","DOI":"10.1090\/S0002-9939-07-08841-7","volume":"135","author":"Francisco C. A.","year":"2007","journal-title":"Proc. Amer. Math. Soc."},{"issue":"5","key":"S0218196723500236BIB009","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1016\/j.jcta.2005.08.007","volume":"113","author":"Herzog J.","year":"2006","journal-title":"J. Combin. Theory Ser. A"},{"issue":"2","key":"S0218196723500236BIB010","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/S0097-3165(03)00045-1","volume":"102","author":"Meshulam R.","year":"2003","journal-title":"J. Combin. Theory Ser. A"},{"issue":"1","key":"S0218196723500236BIB011","doi-asserted-by":"crossref","first-page":"43","DOI":"10.7146\/math.scand.a-23295","volume":"118","author":"Moradi S.","year":"2016","journal-title":"Math. Scand."},{"issue":"7","key":"S0218196723500236BIB012","doi-asserted-by":"crossref","first-page":"3943","DOI":"10.2140\/agt.2018.18.3943","volume":"18","author":"Paolini G.","year":"2018","journal-title":"Algebr. Geom. Topol."},{"issue":"3","key":"S0218196723500236BIB013","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/s00493-006-0019-9","volume":"26","author":"Szab\u00f3 T.","year":"2006","journal-title":"Combinatorica"},{"key":"S0218196723500236BIB014","series-title":"Monographs and Research Notes in Mathematics","volume-title":"Monomial Algebras","author":"Villarreal R. H.","year":"2015","edition":"2"},{"issue":"1","key":"S0218196723500236BIB015","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/PL00009450","volume":"22","author":"Wachs M. L.","year":"1999","journal-title":"Discrete Comput. Geom."},{"key":"S0218196723500236BIB016","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1090\/pcms\/013\/09","volume-title":"Geometric Combinatorics","author":"Wachs M. L.","year":"2007"},{"issue":"10","key":"S0218196723500236BIB017","doi-asserted-by":"crossref","first-page":"3235","DOI":"10.1090\/S0002-9939-09-09981-X","volume":"137","author":"Woodroofe R.","year":"2009","journal-title":"Proc. Amer. Math. Soc."},{"issue":"1","key":"S0218196723500236BIB018","doi-asserted-by":"crossref","first-page":"208","DOI":"10.37236\/695","volume":"18","author":"Woodroofe R.","year":"2011","journal-title":"Electron. J. Combin."}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196723500236","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T08:36:07Z","timestamp":1685090167000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196723500236"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,12]]},"references-count":16,"journal-issue":{"issue":"03","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["10.1142\/S0218196723500236"],"URL":"https:\/\/doi.org\/10.1142\/s0218196723500236","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,12]]}}}