Ryan Pepper

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 k-Domination Number of a Graph

The k-domination number of a graph is the cardinality of a smallest set of vertices such that every vertex not in the set is adjacent to at least k vertices of the set. We prove two bounds on the k-domination number of a graph, inspired by two conjectures of the computer program Graffiti.pc. In particular, we show that for any graph with minimum degree at least 2k 1, the k-domination number is at most the matching number.

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.

Note: A note on dominating sets and average distance

We show that the total domination number of a simple connected graph is greater than the average distance of the graph minus one-half, and that this inequality is best possible. In addition, we show that the domination number of the graph is greater than two-thirds of the average distance minus one-third, and that this inequality is best possible. Although the latter inequality is a corollary to a result of P. Dankelmann, we present a short and direct proof.

Lower bounds for the domination number

In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.

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

The k-residue of a graph, introduced by Jelen in a 1999 paper, is a lower bound on the k-independence number for every positive integer k. This generalized earlier work by Favaron, Maheo, and Sacle, by Griggs and Kleitman, and also by Triesch, who all showed that the independence number of a graph is at least as large as its Havel-Hakimi residue, defined by Fajtlowicz. We show here that, for every positive integer k, the k-residue of disjoint unions is at most the sum of the k-residues of the connected components considered separately, and give applications of this lemma. Our main application is an improvement on Jelens bound for connected graphs which have a maximum degree cut-vertex. We demonstrate constructively that, in some cases, our extensions give better approximations to the k-independence number than all known lower bounds-including bounds of Hopkins and Staton, Caro and Tuza, Favaron, Caro and Hansberg, as well as Jelens k-residue bound itself. Additionally, we apply this disjoint union lemma to prove a theorem for function graphs (those graphs formed by connecting vertices from a graph and its copy according to a given function) while simultaneously giving, in this context, different classes of non-trivial examples for which our new results improve on the k-residue, further motivating our first application of the lemma.

Regular independent sets

The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs. In this article, we generalize and extend these results to find lower bounds for the regular k -independence number for trees, forests, planar graphs, k -trees and k -degenerate graphs.

A Characterization of Graphs Where the Independence Number Equals the Radius

In a classical 1986 paper by Erdös, Saks and Saós every graph of radius r has an induced path of order at least 2r − 1. This result implies that the independence number of such graphs is at least r. In this paper, we use a result of S. Fajtlowicz about radius-critical graphs to characterize graphs where the independence number is equal to the radius, for all possible values of the radius except 2, 3, and 4. We briefly discuss these remaining cases as well.