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Beth Bjorkman

Publications and source records attributed to Beth Bjorkman.

15 recordsLinked to original sources
arXiv2026

Cost-Benefit Analysis for PMU Placement in Power Grids

Power domination is a graph-theoretic model for the observance of a power grid using phasor measurement units (PMUs). There are many costs associated with the installation of a PMU, but also costs associated with not observing the entire power grid. In this work, we propose and study a power domination cost function, which balances these two costs. Given a graph $G$, a set of sensor locations $S$, and a parameter $\beta$ (which is the ratio of the cost of a PMU to the cost of non-observance of any given vertex), we define the cost function \[ \mathrm{C}(G;S,\beta)=|S|+\beta\cdot (|V(G)|-|\mathrm{Obs}(G;S)|) \] where $|\mathrm{Obs}(G;S)|$ is the number of vertices observed by sensors placed at $S\subseteq V(G)$ in the power domination process. We explore the values of $k$ for which there is a set $S$ of size $k$ that minimizes this cost function, and explore which values of $\beta$ guarantee that it is optimal to observe the entire power grid to minimize cost. We also introduce notions of marginal cost and marginal observance, providing tools to analyze how many PMUs one should install on a given power grid.

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arXiv2025

Leaky Forcing: Extending Zero Forcing Results to a Fault-Tolerant Setting

We study a recent variation of zero forcing called leaky forcing. Zero forcing is a propagation process on a network whereby some nodes are initially blue with all others white. Blue vertices can "force" a white neighbor to become blue if all other neighbors are blue. The goal is to find the minimum number of initially blue vertices to eventually force all vertices blue after exhaustively applying the forcing rule above. Leaky forcing is a fault-tolerant variation of zero forcing where certain vertices (not necessarily initially blue) cannot force. The goal in this context is to find the minimum number of initially blue vertices needed that can eventually force all vertices to be blue, regardless of which small number of vertices can't force. This work extends results from zero forcing in terms of leaky forcing. In particular, we provide a complete determination of leaky forcing numbers for all unicyclic graphs and upper bounds for generalized Petersen graphs. We also provide bounds for the effect of both edge removal and vertex removal on the $\ell$-leaky forcing number. Finally, we completely characterize connected graphs that have the minimum and maximum possible $1$-leaky forcing number (i.e., when $Z_{1}(G) = 2$ and when $Z_{1}(G) = |V(G)|-1$).

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arXiv2025

On Fragile Power Domination

Power domination is a graph theoretic model which captures how phasor measurement units (PMUs) can be used to monitor a power grid. Fragile power domination takes into account the fact that PMUs may break or otherwise fail. In this model, each sensor fails independently with probability $q\in [0,1]$ and the surviving sensors monitor the grid according to classical power domination. We study the expected number of observed nodes under the fragile power domination model. We give a characterization for when two networks and initial sensor placements will behave the same according to this expectation. We also show how to control the behavior of this expectation by adding structure to a network.

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arXiv2023

Power domination with random sensor failure

The power domination problem seeks to determine the minimum number of phasor measurement units (PMUs) needed to monitor an electric power network. We introduce random sensor failure before the power domination process occurs and call this the fragile power domination process. For a given graph, PMU placement, and probability of PMU failure $q$, we study the expected number of observed vertices at the termination of the fragile power domination process. This expected value is a polynomial in $q$, which we relate to fault-tolerant and PMU-defect-robust power domination. We also study the probability of that the entire graph becomes observed and give results for some graph families.

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arXiv2023

An Introduction to PMU-Defect-Robust Power Domination: Bounds, Bipartites, and Block Graphs

Sensors called phasor measurement units (PMUs) are used to monitor the electric power network. The power domination problem seeks to minimize the number of PMUs needed to monitor the network. We extend the power domination problem and consider the minimum number of sensors and appropriate placement to ensure monitoring when $k$ sensors are allowed to fail with multiple sensors allowed to be placed in one location. That is, what is the minimum multiset of the vertices, $S$, such that for every $F\subseteq S$ with $|F|=k$, $S\setminus F$ is a power dominating set. Such a set of PMUs is called a $k$-PMU-defect-robust power domination set. This paper generalizes the work done by Pai, Chang and Wang in 2010 on fault-tolerant power domination, which did not allow for multiple sensors to be placed at the same vertex. We provide general bounds and determine the $k$-PMU-defect-robust power domination number of some graph families.

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arXiv2023

Triangle-degree and triangle-distinct graphs

Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of $v$ in $G$ is the number of triangles that contain $v$. While every graph has at least two vertices with the same degree, there are graphs in which every vertex has a distinct triangle-degree. In this paper, we construct an infinite family of graphs with this property. We also study the vertex degrees and size of graphs with this property.

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arXiv2023

Introduction to Robust Power Domination

Sensors called phasor measurement units (PMUs) are used to monitor the electric power network. The power domination problem seeks to minimize the number of PMUs needed to monitor the network. We extend the power domination problem and consider the minimum number of sensors and appropriate placement to ensure monitoring when $k$ sensors are allowed to fail with multiple sensors allowed to be placed in one location. That is, what is the minimum multiset of the vertices, $S$, such that for every $F\subseteq S$ with $|F|=k$, $S\setminus F$ is a power dominating set. Such a set of PMUs is called a $k$-robust power domination set. This paper generalizes the work done by Pai, Chang and Wang in 2010 on vertex-fault-tolerant power domination, which did not allow for multiple sensors to be placed at the same vertex. We provide general bounds and determine the $k$-robust power domination number of some graph families.

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arXiv2023

The Power Domination Toolbox

Phasor Measurement Units (PMUs) are placed at strategic vertices in an electrical power network to monitor the flow of power. Determining the minimum number and optimal placement of PMUs is modeled by the graph theoretic process called Power Domination. This paper describes the Power Domination Toolbox (PDT), which efficiently identifies a minimum number of PMU locations that monitor the entire network. The PDT leverages graph theoretic literature to reduce the complexity of determining optimal PMU placements by: reducing the order of the graph (contraction), leveraging zero forcing forts, sorting the remaining solution space, and parallel computing. The PDT is a drop-in replacement of the current state-of-the-art exhaustive search algorithm in Python and maintains compatibility with SageMath. The PDT can identify minimum PMU placements for graphs with hundreds of vertices on personal computers and can analyze larger graphs on high performance computers. The PDT affords users the ability to investigate power domination on graphs previously considered infeasible due to the number of vertices resulting in a prohibitively long run-time.

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arXiv2022

Power domination reconfiguration

The study of token addition and removal and token jumping reconfiguration graphs for power domination is initiated. Some results established here can be extended by applying the methods used for power domination to reconfiguration graphs for other parameters such as domination and zero forcing, so these results are first established in a universal framework.

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arXiv2020

Non-Asymptotic Connectivity of Random Graphs and Their Unions

Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When graphs are time-varying, it is common to assume that their unions are connected over time, yet, to the best of our knowledge, there are not results that determine the number of finite-size random graphs needed to attain a connected union. Therefore, this paper bounds the probability that individual random graphs are connected and bounds the same probability for connectedness of unions of random graphs. The random graph model used is a generalization of the classic Erdos-Renyi model which allows some edges never to appear. Numerical results are presented to illustrate the analytical developments made.

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arXiv2019

Infectious power domination of hypergraphs

The power domination problem seeks to find the placement of the minimum number of sensors needed to monitor an electric power network. We generalize the power domination problem to hypergraphs using the infection rule from Bergen et al: given an initial set of observed vertices, $S_0$, a set $A\subseteq S_0$ may infect an edge $e$ if $A\subseteq e$ and for any unobserved vertex $v$, if $A\cup \{v\}$ is contained in an edge, then $v\in e$. We combine a domination step with this infection rule to create \emph{infectious power domination}. We compare this new parameter to the previous generalization by Chang and Roussel. We provide general bounds and determine the impact of some hypergraph operations.

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arXiv2018

Cubic Preferences and the Character Admissibility Problem

In multiple-question referendum elections, the separability problem occurs when a voter's preferences on some questions or proposals depend on the predicted outcomes of others. The notion of separability formalizes the study of interdependence in multidimensional preferences, and the character admissibility problem deals with the construction of voter preferences with given separability structures. In this paper, we develop a graph theoretic approach to the character admissibilty problem, using Hamiltonian paths to generate voter preferences. We apply this method specifically to the hypercube graph, defining the class of cubic preferences. We then explore how the algebraic structure of the group of symmetries of the hypercube impacts the separability structures exhibited by cubic preferences. We prove that the characters of cubic preferences satisfy set theoretic properties distinct from those produced by previous methods, and we define two functions to construct cubic preferences. Our results have potential applications to experimental work involving election simulation.

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arXiv2017

$k$-Foldability of Words

We extend results regarding a combinatorial model introduced by Black, Drellich, and Tymoczko (2017+) which generalizes the folding of the RNA molecule in biology. Consider a word on alphabet $\{A_1, \overline{A}_1, \ldots, A_m, \overline{A}_m\}$ in which $\overline{A}_i$ is called the complement of $A_i$. A word $w$ is foldable if can be wrapped around a rooted plane tree $T$, starting at the root and working counterclockwise such that one letter labels each half edge and the two letters labeling the same edge are complements. The tree $T$ is called $w$-valid. We define a bijection between edge-colored plane trees and words folded onto trees. This bijection is used to characterize and enumerate words for which there is only one valid tree. We follow up with a characterization of words for which there exist exactly two valid trees. In addition, we examine the set $\mathcal{R}(n,m)$ consisting of all integers $k$ for which there exists a word of length $2n$ with exactly $k$ valid trees. Black, Drellich, and Tymoczko showed that for the $n$th Catalan number $C_n$, $\{C_n,C_{n-1}\}\subset \mathcal{R}(n,1)$ but $k\not\in\mathcal{R}(n,1)$ for $C_{n-1}<k<C_n$. We describe a superset of $\mathcal{R}(n,1)$ in terms of the Catalan numbers by which we establish more missing intervals. We also prove $\mathcal{R}(n,1)$ contains all non-negative integer less than $n+1$.

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arXiv2017

Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.

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