Extreme-Support Total Monophonic Graphs

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 mt(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 mt(G)= es(G). Some interesting results on the extreme-support total monophonic graphs G are studied. Graphs G with mt(G) = 3 = es(G) are characterized. It is shown that for any three positive integers r, d and k ≥ 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 ≤ d ≤ p − 5 and 5 ≤ k ≤ p − 2 and p − d − k ≥ 0, there exists an extreme-support total monophonic graph G of order p with monophonic diameter d and mt(G) = k.