In this paper the
concept of semi-total monophonic domination number of a graph is introduced. A
set of vertices of a graph is called a total monophonic set if is a monophonic set and its induced subgraph
has no isolated vertices. The minimum cardinality of all total monophonic sets
of is called the total monophonic number
and is denoted by. A set of
vertices in is called a monophonic dominating set if is both a monophonic set and a dominating
set. The minimum cardinality of a monophonic dominating set of is its monophonic domination number and is
denoted by . A monophonic dominating set of size is said to be a set. A set
of vertices in a graph with no
isolated vertices is said to be a semi-total monophonic set of if it is a monophonic set of and every vertex in is within distance 2 of another vertex of .
The semi-total monophonic AMS Subject classification: 05C12 number, denoted by , is the minimum
cardinality of a semitotal monophonic dominating set of .
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