Abstract
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 m of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in m and every edge of G is in exactly one monophonic path in m. The minimum cardinality of a monophonic graphoidal cover of G is called the monophonic graphoidal covering number of G and is denoted bym. 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.
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