A double Italian dominating (DID) function of a graph $G=(V,E)$ is a function $f: V(G)\to\{0,1,2,3\}$ having the property that for every vertex $v\in V$, $\sum_{u\in N_G[v]}f(u)\geq 3$, if $f(v)\in \{0,1\}$. A restrained double Italian dominating (RDID) function is a DID function $f$ such that the subgraph induced by the vertices with label $0$ has no isolated vertex. A total restrained double Italian dominating (TRDID) function is an RDID function $f$ such that the set $\{v\in V: f(v)> 0\}$ induces a subgraph with no isolated vertex.\\ We initiate the study of TRDID function of any graph $G$. The TRDID and RDID functions of the middle of any graph $G$ are investigated, and then, the sharp bounds for these parameters are established. Finally, for a graph $H$, we provide the minimum value of TRDID and RDID functions for corona graphs, $H \circ K_1$, $H \circ K_2$ and middle of them.
Mojdeh, D. A., & Abdallah, M. (2024). On the total restrained double Italian domination. Journal of Algebra and Related Topics, 12(1), 105-126. doi: 10.22124/jart.2023.24056.1507
MLA
D. A. Mojdeh; M. Abdallah. "On the total restrained double Italian domination". Journal of Algebra and Related Topics, 12, 1, 2024, 105-126. doi: 10.22124/jart.2023.24056.1507
HARVARD
Mojdeh, D. A., Abdallah, M. (2024). 'On the total restrained double Italian domination', Journal of Algebra and Related Topics, 12(1), pp. 105-126. doi: 10.22124/jart.2023.24056.1507
VANCOUVER
Mojdeh, D. A., Abdallah, M. On the total restrained double Italian domination. Journal of Algebra and Related Topics, 2024; 12(1): 105-126. doi: 10.22124/jart.2023.24056.1507
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