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 ∆ ≥ k, then F k (G) ≤ (∆−k+1)n ∆−k+1+min {δ,k} . This simplifies to, for the zero forcing number case of k = 1, Z(G) = F 1 (G) ≤ ∆n ∆+1 . Moreover, when ∆ ≥ 2 and the graph is k-connected, we prove that F k (G) ≤ (∆−2)n+2 ∆+k−2 , which is an improvement when k ≤ 2, and specializes to, for the zero forcing number case, Z(G) = F 1 (G) ≤ (∆−2)n+2 ∆−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.

We investigate the zero-forcing number for triangle-free graphs. We improve upon the trivial bound, δ ≤ Z(G) where δ is the minimum degree, in the triangle-free case. In particular, we show that 2δ − 2 ≤ Z(G) for graphs with girth of at least 5, and this can be further improved when G has a small cut set. Lastly, we make a conjecture that the lower bound for Z(G) increases as a function of the girth, g, and δ.

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.

A total forcing set in a graph G is a forcing set (zero forcing set) in G which induces a subgraph without isolated vertices. Total forcing sets were introduced and first studied by Davila [11]. The total forcing number of G, denoted F t (G) is the minimum cardinality of a total forcing set in G. We study basic properties of F t (G), relate F t (G) to various domination parameters, and establish N P -completeness of the associated decision problem for F t (G). Our main contribution is to prove that if G is a connected graph of order n ≥ 3 with maximum degree ∆, then F t (G) ≤ ( ∆ ∆+1 )n, with equality if and only if G is a complete graph K ∆+1 , or a star K 1,∆ .