On The Total Roman Domination in Trees
A on a graph is a function : ( ) → {0, 1, 2} satisfying the following conditions: (i) every vertex for which ) = 0 is adjacent to at least one vertex for which ) = 2 and (ii) the subgraph of induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dom...
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| Veröffentlicht in: | Discussiones Mathematicae. Graph Theory 2019-01, Vol.39 (2), p.519-532 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | A
on a graph
is a function
:
(
) → {0, 1, 2} satisfying the following conditions: (i) every vertex
for which
) = 0 is adjacent to at least one vertex
for which
) = 2 and (ii) the subgraph of
induced by the set of all vertices of positive weight has no isolated vertex. The weight of a total Roman dominating function
is the value
(
)) = Σ
f (
). The
) is the minimum weight of a total Roman dominating function of
. Ahangar
in [H.A. Ahangar, M.A. Henning, V. Samodivkin and I.G. Yero,
, Appl. Anal. Discrete Math. 10 (2016) 501–517] recently showed that for any graph
without isolated vertices, 2
) ≤
) ≤ 3
), where
) is the domination number of
, and they raised the problem of characterizing the graphs
achieving these upper and lower bounds. In this paper, we provide a constructive characterization of these trees. |
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| ISSN: | 1234-3099 2083-5892 |
| DOI: | 10.7151/dmgt.2099 |