Extreme-Support Total Monophonic Graphs

Abstract

For a connected graph $$G=(V, E)$$ of order at least two, a total monophonic set of a graph $$G$$ is a monophonic set $$S$$ such that the subgraph $$G[S]$$ induced by $$S$$ has no isolated vertices. The minimum cardinality of a total monophonic set of $$G$$ is the total monophonic number of $$G$$ and is denoted by $${m}_{t}(G)$$ . The number of extreme vertices and support vertices of $$G$$ is its extreme-support order $$es(G)$$ . A graph $$G$$ is an extreme-support total monophonic graph if $${m}_{t}\left(G\right)=es(G)$$ . Some interesting results on the extreme-support total monophonic graphs $$G$$ are studied. Graphs $$G$$ with $${m}_{t}\left(G\right)=3=es(G)$$ are characterized. It is shown that for any three positive integers $$r, d$$ and $$k\ge 6$$ with $$r<d$$ , there exists an extreme-support total monophonic graph $$G$$ with monophonic radius $$r$$ , monophonic diameter $$d$$ and total monophonic number $$k$$ . Also, for any three positive integers $$d, k$$ and $$p$$ with $$2\le d\le p-5$$ and $$5\le k\le p-2$$ and $$p-d-k\ge 0$$ , there exists an extreme-support total monophonic graph $$G$$ of order $$p$$ with monophonic diameter $$d$$ and $${m}_{t}\left(G\right)=k$$ .