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A central question is which parts of such graphs are essential with respect to leaf-observable information, and which parts can be removed without changing this information. Resolving this question can lead to principled simplification methods for phylogenetic networks, such as the recent normalization approach of Francis et al. In this paper, we study this question from three related perspectives: clusters displayed by a DAG\n                    <jats:italic>G<\/jats:italic>\n                    , least common ancestors (LCAs) of subsets of its leaf set, and visibility, a path-based property of vertices. We first introduce an LCA-based simplification procedure called\n                    <jats:italic>i<\/jats:italic>\n                    <jats:italic>-regularization<\/jats:italic>\n                    . For a DAG\n                    <jats:italic>G<\/jats:italic>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$i\\ge 1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the DAG\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textsc {Reg}_i(G)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:msub>\n                              <mml:mstyle>\n                                <mml:mi>E<\/mml:mi>\n                                <mml:mi>G<\/mml:mi>\n                              <\/mml:mstyle>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    retains precisely those vertices that occur as unique LCAs of leaf subsets of size at most\n                    <jats:italic>i<\/jats:italic>\n                    , removes the remaining non-leaf vertices by a graph-editing operation\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ominus $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>\u2296<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and then deletes shortcuts. We show that\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textsc {Reg}_i(G)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:msub>\n                              <mml:mstyle>\n                                <mml:mi>E<\/mml:mi>\n                                <mml:mi>G<\/mml:mi>\n                              <\/mml:mstyle>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    admits a Hasse-diagram characterization in terms of the corresponding lca-clusters. We then compare LCA-based regularization with normalization. Using the same\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ominus $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mo>\u2296<\/mml:mo>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -operator, we describe the cover construction underlying normalization, identify visible vertices that are nevertheless removed, and characterize when regularization and normalization coincide. Together, these results provide a unified framework for cluster-based, LCA-based, and visibility-based simplifications of DAGs and phylogenetic networks.\n                  <\/jats:p>","DOI":"10.1007\/s11538-026-01783-4","type":"journal-article","created":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T14:10:44Z","timestamp":1792591844000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Regularizing and Normalizing DAGs and Phylogenetic Networks"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1620-5508","authenticated-orcid":false,"given":"Marc","family":"Hellmuth","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9664-1918","authenticated-orcid":false,"given":"Anna","family":"Lindeberg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9371-6435","authenticated-orcid":false,"given":"Vincent","family":"Moulton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,21]]},"reference":[{"issue":"1","key":"1783_CR1","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1142\/S0218126698000043","volume":"8","author":"WMP van der Aalst","year":"1998","unstructured":"van der Aalst WMP (1998) The application of petri nets to workflow management. 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