Signed Total Roman Domination in Digraphs

Abstract

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman dominating function (STRDF) on a digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑xN(v)f(x) ≥ 1 for each vV (D), where N(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight of an STRDF f is w(f) = ∑vV (D)f(v). The signed total Roman domination number γstR(D) of D is the minimum weight of an STRDF on D. In this paper we initiate the study of the signed total Roman domination number of digraphs, and we present different bounds on γstR(D). In addition, we determine the signed total Roman domination number of some classes of digraphs. Some of our results are extensions of known properties of the signed total Roman domination number γstR(G) of graphs G.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] S. Arumugam, K. Jacop and L. Volkmann, Total and connected domination in digraphs, Australas. J. Combin. 39 (2007) 283–292.

  • [2] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).

  • [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Editors, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998).

  • [4] M.A. Henning, Signed total domination in graphs, Discrete Math. 278 (2004) 109–125. doi:10.1016/j.disc.2003.06.002

  • [5] E. Shan and T.C.E. Cheng, Remarks on the minus (signed) total domination in graphs, Discrete Math. 308 (2008) 3373–3380. doi:10.1016/j.disc.2007.06.015

  • [6] S.M. Sheikholeslami, Signed total domination numbers of directed graphs, Util.Math. 85 (2011) 213–218.

  • [7] S.M. Sheikholeslami and L. Volkmann, The Roman domination number of a digraph Acta Univ. Apulensis Math. Inform. 27 (2011) 77–96.

  • [8] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016) 855–871. doi 10.1007/s10878-015-9906-6

  • [9] B. Zelinka, Signed total domination numbers of a graph, Czechoslovak Math. J. 51 (2001) 225–229. doi:10.1023/A:1013782511179

OPEN ACCESS

Journal + Issues

Search