Abstract
For a connected graph $G=(V,E)$ of order at least two, an edge detour monophonic set of $G$ is a set $S$ of vertices such that every edge of $G$ lies on a detour monophonic path joining some pair of vertices in $S$. The edge detour monophonic number of $G$ is the minimum cardinality of its edge detour monophonic sets and is denoted by $edm(G)$. A subset $T$ of $S$ is a forcing edge detour monophonic subset for $S$ if $S$ is the unique edge detour monophonic set of size $edm(G)$ containing $T$. A forcing edge detour monophonic subset for $S$ of minimum cardinality is a minimum forcing edge detour monophonic subset of $S$. The forcing edge detour monophonic number $f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$. The forcing edge detour monophonic number of $G$ is $f_{edm}(G)=min{f_{edm}(S)}$, where the minimum is taken over all edge detour monophonic sets $S$ of size $edm(G)$ in $G$. We determine bounds for it and find the forcing edge detour monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with $0leq a<b$ and $bgeq 2$, there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$.
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