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It is known that <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\textrm{psn}\\,}}(G) \\in O(c^{\\Delta })$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mtext>psn<\/mml:mtext>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>G<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>c<\/mml:mi>\n                        <mml:mi>\u0394<\/mml:mi>\n                      <\/mml:msup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for every planar graph <jats:italic>G<\/jats:italic> of maximum degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Delta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0394<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This upper bound has been improved to <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\Delta ^5)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u0394<\/mml:mi>\n                      <mml:mn>5<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if <jats:italic>G<\/jats:italic> has treewidth three, and to <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\Delta )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>\u0394<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> if <jats:italic>G<\/jats:italic> has treewidth two. In this paper we prove <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\textrm{psn}\\,}}(G) \\le \\max \\{4,\\Delta \\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mtext>psn<\/mml:mtext>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>G<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mo>max<\/mml:mo>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>\u0394<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> when <jats:italic>G<\/jats:italic> is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(\\Delta ^2)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u0394<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> slopes suffice for nested pseudotrees.\n<\/jats:p>","DOI":"10.1007\/s00453-024-01230-7","type":"journal-article","created":{"date-parts":[[2024,5,9]],"date-time":"2024-05-09T14:01:56Z","timestamp":1715263316000},"page":"2413-2447","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees"],"prefix":"10.1007","volume":"86","author":[{"given":"Steven","family":"Chaplick","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giordano","family":"Da Lozzo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emilio","family":"Di Giacomo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giuseppe","family":"Liotta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabrizio","family":"Montecchiani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,9]]},"reference":[{"issue":"6","key":"1230_CR1","doi-asserted-by":"publisher","first-page":"2527","DOI":"10.1007\/s00453-018-00542-9","volume":"81","author":"P Angelini","year":"2019","unstructured":"Angelini, P., Bekos, M.A., Liotta, G., Montecchiani, F.: Universal slope sets for 1-bend planar drawings. 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