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.
Highlights
By a graph G = (V, E) we mean a finite, undirected connected graph without loops or multiple edges
For any two vertices u and v in a connected graph G, the monophonic distance dm(u, v) from u to v is defined as the length of a longest u−v monophonic path in G
A monophonic graphoidal cover of a graph G is a collection ψm of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm
Summary
For the graph G given, ψm = {(v1, v2, v3, v4, v5, v6, v7), (v3, v10, v1, v8, v7, v9, v5)} is a minimum monophonic graphoidal cover of G and so ηm(G) = 2. For any monophonic graphoidal cover ψm of a graph G, let tψm denote the number of vertices of G which are not internal vertices of any path in ψm.
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