The Connected Total Monophonic Domination Number of a Graph

摘要

In this paper the concept of connected total monophonic domination number of a graph is introduced. A set of vertices M of a graph G is called a connected total monophonic dominating set if it is a total monophonic dominating set and its induced subgraph (M) is connected. The minimum cardinality of all connected total monophonic dominating sets of M is called the connected total monophonic domination number and is denoted by γ_c_(mt)(G). It is shown that for every pair k,p of integers with 3 < k ≤ p. there exists a connected graph G of order p such that γ_c_(mt)(G) = k. Also, for any positive integers 2 < a < b < c, there exists a connected graph G such that m(G) = a. γ_m(G) = b and γ_(cm_t)(G) = c.