{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T23:15:33Z","timestamp":1723072533812},"reference-count":50,"publisher":"Association for Computing Machinery (ACM)","issue":"3","funder":[{"name":"NSF TRIPODS","award":["CCF-1740850, CCF-1813165, CCF-1909790, and W911NF1910294"]},{"name":"ERC","award":["633509"]},{"name":"ISF","award":["1028\/16"]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[2022,6,30]]},"abstract":"\n We consider the problem of counting the number of copies of a fixed graph\n H<\/jats:italic>\n within an input graph\n G<\/jats:italic>\n . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input\n G<\/jats:italic>\n has\n bounded degeneracy<\/jats:italic>\n . This is a rich family of graphs, containing all graphs without a fixed minor (e.g., planar graphs), as well as graphs generated by various random processes (e.g., preferential attachment graphs). We say that\n H<\/jats:italic>\n is\n easy<\/jats:italic>\n if there is a linear-time algorithm for counting the number of copies of\n H<\/jats:italic>\n in an input\n G<\/jats:italic>\n of bounded degeneracy. A seminal result of Chiba and Nishizeki from \u201985 states that every\n H<\/jats:italic>\n on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all\n H<\/jats:italic>\n on 5 vertices and further proved that for every\n \n \\( k \\gt 5 \\)<\/jats:tex-math>\n <\/jats:inline-formula>\n there is a\n k<\/jats:italic>\n -vertex\n H<\/jats:italic>\n which is not easy. They left open the natural problem of characterizing all easy graphs\n H<\/jats:italic>\n .\n <\/jats:p>\n \n Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph\n H<\/jats:italic>\n to be easy. Here, we show that this sufficient condition is also necessary, thus fully answering the Bera\u2013Pashanasangi\u2013Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms.\n <\/jats:p>","DOI":"10.1145\/3520240","type":"journal-article","created":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T11:29:32Z","timestamp":1648639772000},"page":"1-21","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Counting Subgraphs in Degenerate Graphs"],"prefix":"10.1145","volume":"69","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-5699-2948","authenticated-orcid":false,"given":"Suman K.","family":"Bera","sequence":"first","affiliation":[{"name":"University of California, Santa Cruz, CA"}]},{"given":"Lior","family":"Gishboliner","sequence":"additional","affiliation":[{"name":"School of Mathematics, Tel Aviv University, Tel Aviv, Israel"}]},{"given":"Yevgeny","family":"Levanzov","sequence":"additional","affiliation":[{"name":"School of Mathematics, Tel Aviv University, Tel Aviv, Israel"}]},{"given":"C.","family":"Seshadhri","sequence":"additional","affiliation":[{"name":"University of California, Santa Cruz, CA"}]},{"given":"Asaf","family":"Shapira","sequence":"additional","affiliation":[{"name":"School of Mathematics, Tel Aviv University, Tel Aviv, Israel"}]}],"member":"320","published-online":{"date-parts":[[2022,6,29]]},"reference":[{"key":"e_1_3_3_2_2","first-page":"434","volume-title":"Proceedings of the 2014 IEEE 55th Annual Symposium on Foundations of Computer Science","author":"Abboud A.","unstructured":"A. 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