Total Monophonic sets and related parameters in graphs

Abstract

For a connected graph G = (V;E) of order at least two, a newlinechord of a path P is an edge joining two non-adjacent vertices of newlineP. A path P is called a monophonic path if it is a chordless path. newlineA set S of vertices of G is a monophonic set of G if each vertex newlinev of G lies on a x and#8722; y monophonic path for some elements x and newliney in S. The minimum cardinality of a monophonic set of G is the newlinemonophonic number of G, denoted by m(G). A connected monophonic newlineset of G is a monophonic set S such that the induced subgraph G[S] newlineis connected. The minimum cardinality of a connected monophonic newlineset of G is the connected monophonic number of G and is denoted newlineby mc(G). For any two vertices u and v in a connected graph G, the newlinemonophonic distance dm(u;v) from u to v is defined as the length of newlinea longest uand#8722;v monophonic path in G and it is denoted by dm(u;v). newlineThe monophonic eccentricity em(v) of a vertex v in G is em(v) = max newline{dm(v;u) : u and#8712; V(G)}. A vertex v of G such that dm(u;v) = em(u) is newlinecalled a monophonic eccentric vertex of u. The monophonic radius, newlineradm(G) of G is radm(G) = min {em(v) : v and#8712; V(G)}. The monophonic newlinediameter, diamm(G) of G is diamm(G) = max {em(v) : v and#8712; V(G)}. newlineA total monophonic set of a graph G is a monophonic set newlineS such that the induced subgraph G[S] has no isolated vertices. newline