Fractional strong matching preclusion of some Cartesian product graphs

The fractional strong matching preclusion number of a graph is the minimum number of edges and vertices whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional strong matching preclusion number for the Cartesian product of a graph and a cycle. As an application, the fractional strong matching preclusion number for torus networks is also obtained.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

Connected domination game played on Cartesian products

The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

γ-Graphs of Trees

For a graph G = ( V , E ) , the γ -graph of G, denoted G ( γ ) = ( V ( γ ) , E ( γ ) ) , is the graph whose vertex set is the collection of minimum dominating sets, or γ -sets of G, and two γ -sets are adjacent in G ( γ ) if they differ by a single vertex and the two different vertices are adjacent in G. In this paper, we consider γ -graphs of trees. We develop an algorithm for determining the γ -graph of a tree, characterize which trees are γ -graphs of trees, and further comment on the structure of γ -graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.