Forcing edge detour monophonic number of a graph

Document Type : Research Paper

Authors

1 Department of Mathematics, University College of Engineering Nagercoil, Nagercoil-629 004, India

2 Department of Mathematics, Coimbatore Institute of Technology, Coimbatore - 641 014, India

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 $0\leq a<b$ and $b\geq 2$‎, ‎there exists a connected graph $G$ such that $f_{edm}(G)=a$ and $edm(G)=b$‎.

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References

[1] F. Harary, Graph Theory, Addison-Wesley, 1969.
[2] P. Titus and K. Ganesamoorthy, On the Detour Monophonic Number of a Graph, Ars Combin., 129 (2016) 33–42.
[3] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math.
Combin. Comput., 84 (2013) 179-188.
[4] A.vP. Santhakumaran, P. Titus, K. Ganesamoorthy and P. Balakrishnan, Edge Detour Monophonic Number of a
Graph, Proyecciones, 32 (2013) 183–198.