Randy Davila

Upper bounds on the k -forcing number of a graph

Given a simple undirected graph G and a positive integer k , the k -forcing number of G , denoted F k ( G ) , is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most k non-colored neighbors, then each of its non-colored neighbors becomes colored. When k = 1 , this is equivalent to the zero forcing number, usually denoted with Z ( G ) , a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the k -forcing number. Notable among these, we show that if G is a graph with order n ? 2 and maximum degree Δ ? k , then F k ( G ) ? ( Δ - k + 1 ) n Δ - k + 1 + min { ? , k } . This simplifies to, for the zero forcing number case of k = 1 , Z ( G ) = F 1 ( G ) ? Δ n Δ + 1 . Moreover, when Δ ? 2 and the graph is k -connected, we prove that F k ( G ) ? ( Δ - 2 ) n + 2 Δ + k - 2 , which is an improvement when k ? 2 , and specializes to, for the zero forcing number case, Z ( G ) = F 1 ( G ) ? ( Δ - 2 ) n + 2 Δ - 1 . These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the k -forcing number and the connected k -domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.

Bounds on the Connected Forcing Number of a Graph

In this paper, we study (zero) forcing sets which induce connected subgraphs of a graph. The minimum cardinality of such a set is called the connected forcing number of the graph. We provide sharp upper and lower bounds on the connected forcing number in terms of the minimum degree, maximum degree, girth, and order of the graph.

On the total forcing number of a graph

Let

On the k-residue of disjoint unions of graphs with applications to k-independence

G

Total forcing sets and zero forcing sets in trees

be a simple and finite graph without isolated vertices. In this paper we study forcing sets (zero forcing sets) which induce a subgraph of

The Slater and sub-k-domination number of a graph with applications to domination and k-domination

G

Characterizations of the Connected Forcing Number of a Graph.

without isolated vertices. Such a set is called a total forcing set, introduced and first studied by Davila \cite{Davila}. The minimum cardinality of a total forcing set in

The sub-

G

k

is the total forcing number of

-domination number of a graph with applications to

G