University of IsfahanTransactions on Combinatorics2251-865710420211201Forcing edge detour monophonic number of a graph2012112562210.22108/toc.2021.119182.1670ENP.TitusDepartment of Mathematics, University College of Engineering Nagercoil, Nagercoil-629 004, IndiaK.GanesamoorthyDepartment of Mathematics, Coimbatore Institute of Technology, Coimbatore - 641 014, IndiaJournal Article20190915For a connected graph $G=(V,E)$ of order at least two, an <em>edge detour monophonic set</em> 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 <em>edge detour monophonic number</em> 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 <em>forcing edge detour monophonic subset</em> 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 <em>minimum forcing edge detour monophonic subset</em> of $S$. The <em>forcing edge detour monophonic number</em> $f_{edm}(S)$ in $G$ is the cardinality of a minimum forcing edge detour monophonic subset of $S$. The <em>forcing edge detour monophonic number</em> 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 <em>a</em>, <em>b</em> of positive integers with $0\leq a<b$ and $b\geq 2$, there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$.https://toc.ui.ac.ir/article_25622_c063c926a1ab86fa5bf537a4b45903ac.pdf