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Weak signed Roman $k$-domatic number of a digraph | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 9، دوره 11، شماره 2، شهریور 2026، صفحه 449-464 اصل مقاله (428.81 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29074.1834 | ||
| نویسنده | ||
| Lutz Volkmann* | ||
| RWTH Aachen University, 52056 Aachen, Germany | ||
| چکیده | ||
| Let $D$ be a digraph with vertex set $V(D)$, and let $k\ge 1$ be an integer. A weak signed Roman $k$-dominating function on a digraph $D$ is a function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that $\sum_{u\in N^-[v]}f(u)\ge k$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct weak signed Roman $k$-dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le k$ for each $v\in V(D)$, is called a weak signed Roman $k$-dominating family (of functions) on $D$. The maximum number of functions in a weak signed Roman $k$-dominating family on $D$ is the weak signed Roman $k$-domatic number of $D$, denoted by $d_{wsR}^k(D)$. In this paper we initiate the study of the weak signed Roman $k$-domatic number in digraphs, and we present sharp bounds for $d_{wsR}^k(D)$. In addition, we determine the weak signed Roman $k$-domatic number of some digraphs. | ||
| کلیدواژهها | ||
| digraphs؛ weak signed Roman $k$-dominating function؛ weak signed Roman $k$-domination number؛ weak signed Roman $k$-domatic number | ||
| مراجع | ||
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[1] H.A. Ahangar, M.A. Henning, C. Löwenstein, Y. Zhao, and V. Samodivkin, Signed Roman domination in graphs, J. Comb. Optim. 27 (2014), no. 2, 241–255. https://doi.org/10.1007/s10878-012-9500-0
[2] M. Chellali, N. Jafari Rad, S. M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination, Structures of Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer International Publishing, Cham, 2021, pp. 273–307. [3] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Roman domination in graphs, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer International Publishing, Cham, 2020, pp. 365–409.
[4] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Varieties of Roman domination II, AKCE Int. J. Graphs Comb. 17 (2020), no. 3, 966–984. https://doi.org/10.1016/j.akcej.2019.12.001
[5] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, The Roman domatic problem in graphs and digraphs: A survey, Discuss. Math. Graph Theory 42 (2022), no. 3, 861–891. https://doi.org/10.7151/dmgt.2313
[6] E.J. Cockayne and S.T. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977), no. 3, 247–261. https://doi.org/10.1002/net.3230070305 [7] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, CRC press, 2013.
[8] M.A. Henning and L. Volkmann, Signed Roman k-domination in graphs, Graphs Combin. 32 (2016), no. 1, 175–190. https://doi.org/10.1007/s00373-015-1536-3 [9] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Am. Math. Mon. 63 (1956), no. 3, 175–177.
[10] S.M. Sheikholeslami and L. Volkmann, The signed Roman domatic number of a graph, Ann. Math. Inform. 40 (2012), 105–112.
[11] S.M. Sheikholeslami and L. Volkmann, Signed Roman domination in digraphs, J. Comb. Optim. 30 (2015), no. 3, 456–467. https://doi.org/10.1007/s10878-013-9648-2
[12] P.J. Slater and E.L. Trees, The signed Roman k-domatic number of a graph, J. Combin. Math. Combin. Comput. 40 (2002), 171–181.
[13] L. Volkmann, The signed Roman k-domatic number of a graph, Discrete Appl. Math. 180 (2015), 150–157. https://doi.org/10.1016/j.dam.2014.07.030 [14] L. Volkmann, The signed Roman k-domatic number of digraphs, Australas. J. Combin. 64 (2016), no. 3, 444–457.
[15] L. Volkmann, Signed Roman k-domination in digraphs, Graphs Combin. 32 (2016), no. 3, 1217–1227. https://doi.org/10.1007/s00373-015-1641-3
[16] L. Volkmann, Weak signed Roman domination in graphs, Commun. Comb. Optim. 5 (2020), no. 2, 111–123. https://doi.org/10.22049/cco.2019.26598.1123
[17] L. Volkmann, Weak signed Roman domination in digraphs, Tamkang J. Math. 52 (2021), no. 4, 497–508. https://doi.org/10.5556/j.tkjm.52.2021.3523
[18] L. Volkmann, Weak signed Roman k-domination in graphs, Commun. Comb. Optim. 6 (2021), no. 1, 1–15. https://doi.org/10.22049/cco.2020.26734.1137
[19] L. Volkmann, Weak signed Roman k-domatic number of a graph, Commun. Comb. Optim. 7 (2022), no. 1, 17–27. https://doi.org/10.22049/cco.2021.26998.1178
[20] L. Volkmann, Weak signed Roman k-domination in digraphs, Opuscula Math. 44 (2024), no. 2, 285–296. https://doi.org/10.7494/OpMath.2024.44.2.285 | ||
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