On the Relationships between Zero Forcing Numbers and Certain Graph Coverings

Abstract

Abstract:The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number. We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.

Grundy dominating sequences and zero forcing sets

“…Motivated by the above two concepts, we introduce in this paper the closely related concept of the Z-Grundy domination number γ Z gr (G) of a graph G. We are going to prove in Section 2 that to determine γ Z gr (G) is equivalent to compute Z(G), where Z(G) is the extensively studied zero forcing number [4] which is in turn closely related to the concept of power domination [6,19]. The complexity of the decision problem whether Z(G) is at least some constant was shown to be NP-complete by [1] and the zero forcing number has been determined for many classes of graphs [4,5,22,25]. This connection on the one hand enables us to deduce γ Z gr (G) for many classes of graphs, and to determine the zero forcing number of Sierpiński graphs and lexicographic products of paths and cycles on the other hand.…”

k-Geodetic propagation in graphs

“…For G with order n ≥ 2 and Clearly all the vertices of V (P m P n )\S belongs to some 2-geodesic between the vertices in S. But all the vertices in S is adjacent to at least two non propagated vertices of V (P m P n )\S, hence can not initiate the propagation process. From [10] the propagation number for grids, i.e p(P m P n ) = m. Thus the m(n−1) 2 from S together with the m boundary vertices can initiate the propagation process and propagate the remaining non propagated vertices of V (P m P n )\S simultaneously. Thus g 2 p(P m P n ) ≤ m(n−1) 2 + m = m(n+1)…”

Spectral bounds for the zero forcing number of a graph

“…This parameter has been extensively studied in over half a century, largely due to its connection to inverse eigenvalue problems for graphs and its applications to other problems. Up to now, there have been lots of research work on bounding the zero forcing number of a graph in terms of its various parameters, such as connected domination number [1], perfect domination number [9], degree sequence [7,10], girth [11] and chromatic number [12], etc. We will not list them all here, but we will focus primarily on those related to the spectral bounds for the zero forcing number.…”