Forcing total outer connected monophonic number of a graph

Abstract

. 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 (cid:54) 𝑎 < 𝑏 and 𝑏 (cid:62) 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.