# Total Roman Domination Edge-Supercritical and Edge-Removal-Supercritical Graphs

 dc.contributor.author Ogden, Shannon dc.date.accessioned 2020-06-08T21:12:52Z dc.date.available 2020-06-08T21:12:52Z dc.date.copyright 2020 en_US dc.date.issued 2020-06-08 dc.description.abstract A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} such that every vertex v with f(v) = 0 is adjacent to some vertex u with f(u) = 2, and the subgraph of G induced by the set of vertices w such that f(w) > 0 has no isolated vertices. The weight of f is Σv∈V (G)f(v). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function on G. A graph G is γtR-edge-supercritical if γtR(G + e) = γtR(G) − 2 for every e ∈ E(G) = ∅, and γtR-edge-stable if γtR(G + e) = γtR(G) for every e ∈ E(G) = ∅. For an edge e ∈ E(G) incident with a degree 1 vertex, we deﬁne γtR(G − e) = ∞. A graph G is γtR-ER-critical if γtR(G − e) > γtR(G) for every e ∈ E(G), γtR-ER-supercritical if γtR(G − e) ≥ γtR(G) + 2 for every e ∈ E(G), and γtR-ER-stable if γtR(G − e) = γtR(G) for every e ∈ E(G). We characterize γtR-ER-critical and γtR-ER-supercritical graphs. In addition, we investigate connected γtR-edge-supercritical graphs and exhibit inﬁnite classes of such graphs. We present a connection between γtR-ER-supercritical and γtR-edge-stable graphs, and similarly between γtR-edge-supercritical and γtR-ER-stable graphs. en_US dc.description.reviewstatus Reviewed en_US dc.description.scholarlevel Undergraduate en_US dc.description.sponsorship Jamie Cassels Undergraduate Research Award (JCURA) en_US dc.identifier.uri http://hdl.handle.net/1828/11815 dc.language.iso en en_US dc.subject total domination; total Roman domination; total Roman domination edge-critical graphs; total Roman domination edge-supercritical graphs; total Roman domination edge-stable graphs; total Roman domination edge-removal-critical graphs; total Roman domination edge-removal-supercritical graphs; total Roman domination edge-removal-stable graphs en_US dc.title Total Roman Domination Edge-Supercritical and Edge-Removal-Supercritical Graphs en_US dc.type Poster en_US

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