{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:42:20Z","timestamp":1760060540506,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T00:00:00Z","timestamp":1756771200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A set S of vertices in a graph G is a total dominating set of G if every vertex of G is totally dominated by at least one vertex of S. For a total dominating set S of a graph G and a vertex v of G, let \u03c3S(v) denote the number of vertices in S that totally dominate v. A total dominating set S in a graph G is a proper total dominating set if \u03c3S(u)\u2260\u03c3S(v) for every two adjacent vertices u and v of G. While proper total dominating sets in trees have been previously studied, the primary goal here is to extend this study to classes of trees with a symmetric structure or that possess subtrees with a symmetric structure. Those trees belonging to several of the most-studied classes of trees that possess a proper total dominating set are determined. Graphical structures of proper total dominating sets in these trees are investigated. The minimum cardinality of a proper total dominating set in a graph G is the proper total domination number of G. Characterizations are obtained for all trees T with a small proper total domination number. Other results and problems are also presented on proper total dominating sets in trees in general.<\/jats:p>","DOI":"10.3390\/sym17091429","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T08:23:38Z","timestamp":1756801418000},"page":"1429","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Proper Total Domination in Trees"],"prefix":"10.3390","volume":"17","author":[{"given":"Sawyer","family":"Osborn","sequence":"first","affiliation":[{"name":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA"}]},{"given":"Ping","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,2]]},"reference":[{"key":"ref_1","first-page":"258","article-title":"Sur le couplage maximum d\u2019un graphe","volume":"247","author":"Berge","year":"1958","journal-title":"Comptes Rendus Acad. Sci. Paris"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ore, O. (1962). Theory of Graphs, American Mathematical Society Colloquium Publications.","DOI":"10.1090\/coll\/038"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1002\/net.3230070305","article-title":"Towards a theory of domination in graphs","volume":"7","author":"Cockayne","year":"1977","journal-title":"Networks"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Fundamentals of Domination in Graphs, Marcel Dekker.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Slater, P.J. (1998). Domination in Graphs: Advanced Topics, Marcel Dekker.","DOI":"10.1002\/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Haynes, T.W., Hedetniemi, S.T., and Henning, M.A. (2023). Domination in Graphs: Core Concepts, Springer.","DOI":"10.1007\/978-3-031-09496-5"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1002\/net.3230100304","article-title":"Total domination in graphs","volume":"10","author":"Cockayne","year":"1977","journal-title":"Networks"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Henning, M.A., and Yeo, A. (2013). Total Domination in Graphs, Springer.","DOI":"10.1007\/978-1-4614-6525-6"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Broe, P. (2022). Irregular Orbital Domination in Graphs. [Ph.D. Dissertation, Western Michigan University].","DOI":"10.1080\/23799927.2021.2014977"},{"key":"ref_10","unstructured":"Gera, R., Hedetniemin, S., and Larson, C. (2016). Highly irregular. Graph Theory-Favorite Conjectures and Open Problems, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chartrand, G., Haynes, T.W., Henning, M.A., and Zhang, P. (2019). From Domination to Coloring: Stephen Hedetniemi\u2019s Graph Theory and Beyond, Springer.","DOI":"10.1007\/978-3-030-31110-0"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"58","DOI":"10.47443\/ejm.2024.024","article-title":"Proper total domination in graphs","volume":"7","author":"Chatterjee","year":"2024","journal-title":"Electron. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"40","DOI":"10.47443\/cm.2024.063","article-title":"Vertex colorings resulting from proper total domination of graphs","volume":"10","author":"Osborn","year":"2024","journal-title":"Contrib. Math."},{"key":"ref_14","unstructured":"Talanda-Fisher, M. (2021). Domination Functions in Graphs. [Ph.D. Dissertation, Western Michigan University]."},{"key":"ref_15","unstructured":"Osborn, S., and Zhang, P. (J. Combin. Math. Combin. Comput., 2025). From princes on chessboards to proper total Domination in graphs, J. Combin. Math. Combin. Comput., to appear."},{"key":"ref_16","first-page":"3","article-title":"On orbital domination numbers of graphs","volume":"37","author":"Chartrand","year":"2001","journal-title":"J. Combin. Math. Combin. Comput."},{"key":"ref_17","first-page":"193","article-title":"Universal domination sequences of graphs","volume":"54","author":"Hayes","year":"1998","journal-title":"Util. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cabrera Mart\u00ednez, A., and Rodr\u00edguez-Vel\u00e1zquez, J.A. (2020). Total domination in rooted product graphs. Symmetry, 12.","DOI":"10.3390\/sym12111929"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Klobu\u010dar, A., and Klobu\u010dar Bari\u0161i\u0107, A. (2024). Total and double total domination on octagonal grid. Axioms, 13.","DOI":"10.3390\/axioms13110792"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chartrand, G., and Zhang, P. (2020). Chromatic Graph Theory, Chapman & Hall\/CRC Press. [2nd ed.].","DOI":"10.1201\/9780429438868"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Soifer, A. (2024). The New Mathematical Coloring Book, Springer. [2nd ed.].","DOI":"10.1007\/978-1-0716-3597-1"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1869","DOI":"10.1007\/s00373-020-02193-6","article-title":"Decomposing degenerate graphs into locally irregular subgraphs","volume":"36","author":"Bensmail","year":"2020","journal-title":"Graphs Combin."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1429\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:37:43Z","timestamp":1760035063000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1429"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,2]]},"references-count":22,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091429"],"URL":"https:\/\/doi.org\/10.3390\/sym17091429","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,9,2]]}}}