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

Abstract

For a connected graph $$G = \left( V,E\right) $$ , a set $$S\subseteq E(G)$$ 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 $$l\ge 4$$ 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 $$ m_{ev}\left( G\right) = b$$ 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.