Abstract
This work is aimed to continue studying the packing sets of digraphs via the perspective of partitioning the vertex set of a digraph into packing sets (which can be interpreted as a type of vertex coloring of digraphs) and focused on finding the minimum cardinality among all packing partitions for a given digraph D, called the packing partition number of D. Some lower and upper bounds on this parameter are proven, and their exact values for directed trees are given in this paper. In the case of directed trees, the proof results in a polynomial-time algorithm for finding a packing partition of minimum cardinality. We also consider this parameter in digraph products. In particular, a complete solution to this case is presented when dealing with the rooted products.
Highlights
Introduction and PreliminariesGraph partitions are universally extended procedures for equilibrating a property of a graph into smaller pieces of it
We directly proceed to discuss the main goal of our exposition, which is aimed to develop some combinatorial properties of partitioning a directed graph into packing sets, which, can be interpreted as a type of vertex coloring of the digraph
A −→p (D)-function will be a PP-function of D with f V(D) of minimum cardinality
Summary
Graph partitions are universally extended procedures for equilibrating a property of a graph into smaller pieces of it. For all four standard products of digraphs D and F, the vertex set is V(D) × V(F) Their arc sets are defined as follows. Our main focus is given to the vertex partitioning of a digraph D into packing sets. The set of all frequencies assigned to the whole set of radio stations clearly forms a partition of the set of radio stations by putting together in a same set of such partition those radio stations with the same frequency In this digraph model, and requiring a system with the smallest possible number of assigned frequencies, we can readily observe that −→p (D−1) equals the minimum number of frequencies, which must be assigned to the stations in order to satisfy the above-mentioned conditions (here D−1 is the digraph obtained by reversing the direction of every arc of D). A −→p (D)-function will be a PP-function of D with f V(D) of minimum cardinality
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