Equality of graphoidal and acyclic graphoidal covering number of a graph

On 2-graphoidal Covering Number of a Graph

A 2-graphoidal cover of a graph G is a collection ψ of paths (not necessarily open) in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most two paths in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a 2-graphoidal cover of G is called the 2-graphoidal covering number of G and is denoted by η2(G) or η2. Here, we...

The Monophonic Graphoidal Covering Number of a Graph

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 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. The minimum card...

Induced Acyclic Graphoidal Covers in a Graph

An induced acyclic graphoidal cover of a graph G is a collection ψ of open paths in G such that every path in ψ has atleast two vertices, every vertex of G is an internal vertex of at most one path in ψ, every edge of G is in exactly one path in ψ and every member of ψ is an induced path. The minimum cardinality of an induced acyclic graphoidal cover of G is called the induced acyclic graphoida...

Simple acyclic graphoidal covers in a graph

Detour Monophonic Graphoidal Covering Number of Corona Product Graph of Some Standard Graphs with the Wheel

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 longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A  detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex  of at most on...