3 citations · 7 across the 16 of their papers we have counts for
18 papers
Transmission Zero Forcing
Adam H. Berliner, Chassidy Bozeman, Karen L. Collins +3
We initiate the study of transmission zero forcing, a variant of the well-studied zero forcing graph parameter. In this variant, a subset of vertices is assigned an initial unit we…
Discrete-time treatment number of binary trees
Karen L. Collins, Margaret-Ellen Messinger, Ann N. Trenk
The discrete-time treatment number of a graph , denoted by , was introduced in arXiv:2408.0531(3) and arises from a deterministic process in which each vertex is assigned…
Color -switches and neighborhood -balanced graphs with colors
Karen L. Collins, Jonelle Hook, Cayla McBee +1
This paper examines vertex colorings of graphs with constraints on the distribution of colors in vertex neighborhoods. We introduce color 2-switches and color degree matrices. The…
Discrete-time treatment number
N. E. Clarke, K. L. Collins, M. E. Messinger +2
We introduce the discrete-time treatment number of a graph, in which each vertex is in exactly one of three states at any given time-step: compromised, vulnerable, or treated. Our…
Split graphs and Block Representations
Karen L. Collins, Ann N. Trenk, Rebecca Whitman
In this paper, we study split graphs and related classes of graphs from the perspective of their sequence of vertex degrees and an associated lattice under majorization. Following…
Great-circle Tree Thrackles
Karen Collins, Cleo Roberts
A thrackle is a graph drawing in which every pair of edges meets exactly once. The Thrackle Conjecture (established by John Conway) states that the number of edges of a thrackle ca…
A Note on the Immersion Number of Generalized Mycielski Graphs
Karen L. Collins, Megan E. Heenehan, Jessica McDonald
The immersion number of a graph , denoted im, is the largest such that has a -immersion. In this note we are interested in determining the immersion number of…
Product Throttling
Sarah E. Anderson, Karen L. Collins, Daniela Ferrero +4
Throttling addresses the question of minimizing the sum or the product of the resources used to accomplish a task and the time needed to complete that task for various graph search…
Product throttling for power domination
Sarah E. Anderson, Karen L. Collins, Daniela Ferrero +4
The product power throttling number of a graph is defined to study product throttling for power domination. The domination number of a graph is an upper bound for its product power…
Clique immersion in graph products
Karen L. Collins, Megan E. Heenehan, Jessica McDonald
Let be graphs and represent a particular graph product of and . We define to be the largest such that has a -immersion and ask: given $im(G)…
The distinguishing number and distinguishing chromatic number for posets
Karen L. Collins, Ann N. Trenk
In this paper we introduce the concepts of the distinguishing number and the distinguishing chromatic number of a poset. For a distributive lattice and its set of join-ir…
An upper bound on the Wiener Index of a k-connected graph
Zhongyuan Che, Karen L. Collins
The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index…
Treewidth Bounds for Planar Graphs Using Three-Sided Brambles
Karen L. Collins, Brett C. Smith
Square grids play a pivotal role in Robertson and Seymour's work on graph minors as planar obstructions to small treewidth. We introduce a three-sided bramble in a plane graph call…
Finding Balance: Split Graphs and Related Classes
Karen L. Collins, Ann N. Trenk
A graph is a split graph if its vertex set can be partitioned into a clique and a stable set. A split graph is unbalanced if there exist two such partitions that are distinct. Chen…
Split Graphs and Nordhaus-Gaddum Graphs
Christine Cheng, Karen L. Collins, Ann Trenk
A graph G is an NG-graph if χ(G) + χ(G complement) = |V(G)| + 1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also expl…
Constructing graphs with no immersion of large complete graphs
Karen L. Collins, Megan E. Heenehan
In 1989, Lescure and Meyniel proved, for , that every -chromatic graph contains an immersion of , and in 2003 Abu-Khzam and Langston conjectured that this holds for…
Nordhaus-Gaddum Theorem for the Distinguishing Chromatic Number
Karen L. Collins, Ann Trenk
Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. F…
A Note on Planar and Dismantlable Lattices
Karen L. Collins
It is shown that any finite, rank-connected, dismantlable lattice is lexicographically shellable (hence Cohen-Macaulay). A ranked, interval-connected lattice is shown to be rank-co…