In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very s...

We enumerate prime graphs with respect to the Cartesian multiplication of graphs by expressing the species of connected graphs as the exponential composition of the species of prime graphs, where the exponential composition of species is constructed based on the arithmetic product of species of Maia and Méndez and the quotient species.

A set of k spanning trees rooted at the same vertex r in a graph G are called independent spanning trees (ISTs) if for any vertex x 6= r, the k paths from v to r, one path in each tree, are internally disjoint. The design of ISTs on graphs has applications to fault-tolerant broadcasting and secure message distribution in networks. It was conjectured that for any k-connected graph there exist k ...

System level diagnosis that can identify the faulty units in the system was introduced by Preparata, Metze, and Chien [1]. In this area, the fundamental problem is to decide the diagnosability of given networks. We study the diagnosability of networks represented by the cartesian product. Our result is the optimal one with respect to the restriction of degrees of vertices of graphs that represe...

9 We study the inheritance of path-pairability in the Cartesian product of 10 graphs and prove additive and multiplicative inheritance patterns of path11 pairability, depending on the number of vertices in the Cartesian product. 12 We present path-pairable graph families that improve the known upper 13 bound on the minimal maximum degree of a path-pairable graph. Further 14 results and open que...

There are known exact results of the crossing numbers of the Cartesian product of all graphs of order at most four with paths, cycles and stars. Moreover, for the path Pn of length n, the crossing numbers of Cartesian products G Pn for all connected graphs G on five vertices and for forty graphs G on six vertices are known. In this paper, we extend these results by determining the crossing numb...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

The distinguishing chromatic number χD (G) of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G. We study the distinguishing chromatic number of Cartesian products of graphs by focusing on how much it can exceed the trivial lower bound of the chromatic number χ(·). Our main result is that for every graph G, th...

In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the k-decomposition for the reciprocal sum-degree distance. Finally,we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs.