The Total Restrained Monophonic Number of a Graph
نویسنده
چکیده
For a connected graph G = (V,E) of order at least two, a total restrained monophonic set S of a graph G is a restrained monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total restrained monophonic set of G is the total restrained monophonic number of G and is denoted by mtr(G). A total restrained monophonic set of cardinality mtr(G) is called a mtr-set of G. We determine bounds for it and characterize graphs which realize these bounds. It is shown that if p,d and k are positive integers such that 2 ≤ d ≤ p−2, 3 ≤ k ≤ p and p−d − k+2 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and mtr(G) = k. AMS subject classification: 05C12.
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