Geodesic
Forcing total outer connected monophonic number of a graph
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 3, pp. 278-286.

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For a connected graph $G = (V,E)$ of order at least two, a subset $T$ of a minimum total outer connected monophonic set $S$ of $G$ is a forcing total outer connected monophonic subset for $S$ if $S$ is the unique minimum total outer connected monophonic set containing $T$. A forcing total outer connected monophonic subset for $S$ of minimum cardinality is a minimum forcing total outer connected monophonic subset of $S$. The forcing total outer connected monophonic number $f_{tom}(S)$ in $G$ is the cardinality of a minimum forcing total outer connected monophonic subset of $S$. The forcing total outer connected monophonic number of $G$ is $f_{tom}(G) = \min\{f_{tom}(S)\}$, where the minimum is taken over all minimum total outer connected monophonic sets $S$ in $G$. We determine bounds for it and find the forcing total outer connected monophonic number of a certain class of graphs. It is shown that for every pair $a,b$ of positive integers with $0 \leq a b$ and $b \geq a+4$, there exists a connected graph $G$ such that $f_{tom}(G) = a$ and $cm_{to}(G) = b$, where $cm_{to}(G)$ is the total outer connected monophonic number of a graph. 
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K. Ganesamoorthy; Sh. Lakshmi Priya. Forcing total outer connected monophonic number of a graph. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 3, pp. 278-286. http://geodesic.mathdoc.fr/item/ISU_2022_22_3_a0/

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