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Ganesamoorthy K., Lakshmi Priya S. Forcing total outer connected monophonic number of a graph. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2022, vol. 22, iss. 3, pp. 278-286. DOI: 10.18500/1816-9791-2022-22-3-278-286, EDN: IMTPKR

# Forcing total outer connected monophonic number of a graph

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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