THE MONOPHONIC GRAPHOIDAL COVERING NUMBER OF A GRAPH
P. Titus, Sushama Kumari · International Journal of Pure and Apllied Mathematics · 2014
A chord of a path P is an edge joining two non-adjacent vertices of P .A path P is called a monophonic path if it is a chordless path.A monophonic graphoidal cover of a graph G is a collection ψ m of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in ψ m and every edge of G is in exactly one monophonic path in ψ m .The minimum cardinality of a monophonic graphoidal cover of G is called the monophonic graphoidal covering number of G and is denoted by η m .We determine bounds for it and characterize graphs which realize these bounds.Also, for any positive integer n with q -p + 2 ≤ n ≤ q -1, there exists a tree T such that the monophonic graphoidal covering number is n.