Total and forcing total edge-to-vertex monophonic number of a graph

For a connected graph (G = left( V,Eright) ), a set (Ssubseteq E(G)) is called a total edge-to-vertex monophonic set of a connected graph G if the subgraph induced by S has no isolated edges. The total edge-to-vertex monophonic number (m_{tev}(G)) of G is the minimum cardinality of its total edge-to-vertex monophonic set of G. The total edge-to-vertex monophonic number of certain classes of graphs is determined and some of its general properties are studied. Connected graphs of size (q ge 3 ) with total edge-to-vertex monophonic number q is characterized. It is shown that for positive integers (r_{m},d_{m}) and (lge 4) with (r_{m}< d_{m} le 2 r_{m}), there exists a connected graph G with (textit{rad}_ {m} G = r_{m}), (textit{diam}_ {m} G = d_{m}) and (m_{tev}(G) = l) and also shown that for every integers a and b with (2 le a le b), there exists a connected graph G such that ( m_{ev}left( Gright) = b) and (m_{tev}(G) = a + b). A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of S, denoted by (f_{tev}(S)) is the cardinality of a minimum forcing subset of S. The forcing total edge-to-vertex monophonic number of G, denoted by (f_{tev}(G) = textit{min}{f_{tev}(S)}), where the minimum is taken over all total edge-to-vertex monophonic set S in G. The forcing total edge-to-vertex monophonic number of certain classes of graphs are determined and some of its general properties are studied. It is shown that for every integers a and b with (0 le a le b) and (b ge 2), there exists a connected graph G such that (f_{tev}(G) = a) and ( m _{tev}(G) = b), where ( f _{tev}(G)) is the forcing total edge-to-vertex monophonic number of G.