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Some remarks on the signed total Italian $k$-domination number of graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 10 شهریور 1404 اصل مقاله (380.94 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30398.2452 | ||
| نویسنده | ||
| Lutz Volkmann* | ||
| RWTH Aachen University, 52056 Aachen, Germany | ||
| چکیده | ||
| Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. Volkmann \cite{vo21} defined the signed total Italian $k$-dominating function (STIkDF) on a graph $G$ as a function $f:V(G)\rightarrow\{-1,1,2\}$ satisfying the conditions that $\sum_{x\in N(v)}f(x)\ge k$ for each vertex $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and every vertex $u$ for which $f(u)=-1$ is adjacent to at least one vertex $v$ for which $f(v)=2$ or adjacent to two vertices $w$ and $z$ with $f(w)=f(z)=1$. The weight of an STIkDF $f$ is $w(f)=\sum_{v\in V(G)}f(v)$. The signed total Italian $k$-domination number $\gamma_{stI}^k(G)$ of $G$ is the minimum weight of an STIkDF on $G$. In this paper we continue the study of the signed total Italian $k$-domination number. We present new bounds on $\gamma_{stI}^k(G)$, and we determine the signed total Italian $k$-domination number of some complete $p$-partite graphs. Furthermore, we show that the difference $\gamma_{stR}^k(G)-\gamma_{stI}^k(G)$ can be arbitrarily large, where $\gamma_{stR}^k(G)$ is the signed total Roman $k$-domination number. | ||
| کلیدواژهها | ||
| Signed total Italian $k$-dominating function؛ Signed total Italian $k$-domination number؛ signed total Roman domination | ||
| مراجع | ||
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[1] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Roman domination in graphs, Topics in Domination in Graphs (Teresa W. Haynes, Stephen T. Hedetniemi, and Michael A. Henning, eds.), Springer International Publishing, Cham, 2020, pp. 365–409. [2] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, A survey on Roman domination parameters in directed graphs, J. Combin. Math. Combin. Comput. 115 (2020), 141–171.
[3] 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
[4] 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.
[5] E.J. Cockayne, R.M. Dawes, and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980), no. 3, 211–219. https://doi.org/10.1002/net.3230100304
[6] T.W. Haynes, S. Hedetniemi, and P. Slater, Fundamentals of Domination in Graphs, CRC press, 2013.
[7] V. Kulli, On n-total domination number in graphs, Graph Theory, Combinatorics, Algorithms, and Applications, SIAM, Philadelphia, 1991, pp. 319–324.
[8] L. Volkmann, Signed total Roman domination in graphs, J. Comb. Optim. 32 (2016), no. 3, 855–871. https://doi.org/10.1007/s10878-015-9906-6
[9] L. Volkmann, Signed total Roman $k$-domination in graphs, J. Combin. Math. Combin. Comput. 105 (2018), 105–116.
[10] L. Volkmann, Signed total Italian $k$-domination in graphs, Commun. Comb. Optim. 6 (2021), no. 2, 171–183. https://doi.org/10.22049/cco.2020.26919.1164 | ||
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