The Connected Edge-To-Vertex Monophonic Number of a Graph

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D. Stalin and J. John

Abstract

A set 𝑀𝑀⊆𝐸𝐸 of edges is called an edge-to-vertex monophonic set if every vertex of 𝐺𝐺 is either incident with an edge of 𝑀𝑀 or lies in the monophonic joining a pair of edges of𝑀𝑀. The edge-to-vertex monophonic number 𝑚𝑚𝑒𝑒𝑒𝑒(𝐺𝐺) of 𝐺𝐺 is the minimum order of its edge-to-vertex monophonic sets and any edge-to-vertex monophonic set of order 𝑚𝑚𝑒𝑒𝑒𝑒 is the edge-to-vertex monophonic basis of 𝐺𝐺 . A connected edge-to-vertex monophonic set 𝑀𝑀 such that the sub graph <𝑀𝑀> induced by 𝑀𝑀 is connected. The minimum cardinality of connected edge-to-vertex monophonic set of 𝐺𝐺 is the connected edge-to-vertex monophonic number of 𝐺𝐺 and is determined by𝑚𝑚𝑐𝑐𝑐 𝑣 (𝐺𝐺). Any edge-to-vertex monophonic set of cardinality 𝑚𝑚𝑐𝑐𝑐 is a connected edge-to-vertex monophonic basis of 𝐺𝐺. some general properties satisfied by this concept are studied . The connected edge-to-vertex monophonic number 2or 𝑞𝑞 or 𝑞𝑞−1 are characterized. It is shown that for every two integers 𝑎𝑎,𝑏𝑏 and 𝑐𝑐 are positive integers with 2≤𝑎𝑎≤𝑏𝑏≤𝑐𝑐, there is a connected graph G such that 𝑚𝑚𝑒𝑒𝑒𝑒(𝐺𝐺)=𝑎𝑎 ,𝑚𝑚𝑐𝑐𝑐 (𝐺𝐺)=𝑏𝑏 and 𝑔𝑔𝑐𝑐𝑐 (𝐺𝐺)=𝑐𝑐.

Volume 11 | Issue 1

Pages: 308-314

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