An open-locating-dominating set of a graph models a detection system for a facility with a possible “intruder” or a multiprocessor network with a possible malfunctioning processor. A “sensor” or “detector” is assumed to be installed at a subset of vertices where each can detect an intruder or a malfunctioning processor in its neighborhood, but not at its own location. We consider a fault-tolerant variant of an open-locating-dominating set called an error-correcting open-locating-dominating set, which can correct a false-positive or a false-negative signal from a detector. In particular, we prove the problem of finding a minimum error-correcting open-locating-dominating set in an arbitrary graph is NP-complete. Additionally, we characterize the existence criteria for an error-correcting open-locating-dominating set in an arbitrary graph. We also consider extremal graphs that require every vertex to be a detector and minimum error-correcting open-locating-dominating sets in infinite grids.

Let \( G = (V, E) \) be a graph with vertex set \( V \) and edge set \( E \). A set \( S \subset V \) is a dominating set if every vertex in \( V – S \) is adjacent to at least one vertex in \( S \), an independent set if no two vertices in \( S \) are adjacent, and a total dominating set if every vertex in \( V \) is adjacent to at least one vertex in \( S \). The domatic number \( \text{dom}(G) \), idomatic number \( \text{idom}(G) \), and total domatic number \( \text{tdom}(G) \), of a graph \( G \) equal the maximum order \( k \) of a partition \( \pi = \{V_1, V_2, \ldots, V_k\} \) of \( V \) into {dominating sets, independent dominating sets, total dominating sets}, respectively. A queens graph \( Q_n \) is a graph defined on the \( n^2 \) squares of an \( n \)-by-\( n \) chessboard, such that two squares are adjacent if and only if a queen on one square can move to the other square in one move, that is, the two squares lie on a common row, column, or diagonal. In this note, we determine the value of these three numbers for \( Q_n \) for the first several values of \( n \). In addition, we introduce the concepts of graphs being \( \gamma \)-domatic, \( i \)-domatic, \( \alpha \)-domatic, \( \Gamma \)-domatic, \( \gamma_t \)-domatic, and \( \Gamma_t \)-domatic.

A Magic Venn Diagram is a magic figure where regions of a Venn diagram are labeled such that the sums of the regional labels of each set are the same. We have developed a backtracking search to count the number of Magic Venn Diagrams. The algorithm could determine the number of Magic Venn Diagrams for all Venn diagrams with four sets. This paper presents the algorithm with its applied heuristics and lists the computational results.

A famous open problem in the field of rendezvous search is to ascertain the rendezvous value of the symmetric rendezvous problem on the line, wherein both agents begin two units apart. We provide a new, Bayesian framework to both create new strategies for the agents to follow and to provide a new interpretation of previously posited strategies.
Additionally, we have developed a method that modifies any strategy, even those with potentially infinite expected meeting time, into a new strategy that is guaranteed to have a finite expected meeting time. This process, combined with using our Bayesian framework to create new strategies, yields an upper bound that is within one percent of the current best upper bound for the symmetric rendezvous value.

The Astronaut Problem is an open problem in the field of rendezvous search. The premise is that two astronauts randomly land on a planet and want to find one another. Research explores what strategies accomplish this in the least expected time.
To investigate this problem, we create a discrete model which takes place on the edges of the Platonic solids. Some baseline assumptions of the model are:

1. The agents can see all of the faces around them.
2. The agents travel along the edges from vertex to vertex and cannot jump.
3. The agents move at a rate of one edge length per unit time.

The 3-dimensional nature of our model makes it different from previous work. We explore multi-step strategies, which are strategies where both agents move randomly for one step, and then follow a pre-determined sequence.

For the cube and octahedron, we are able to prove optimality of the “Left Strategy,” in which the agents move in a random direction for the first step and then turn left. In an effort to find lower expected times, we explore mixed strategies. Mixed strategies incorporate an asymmetric case which, under certain conditions, can result in lower expected times.

Most of the calculations were done using first-step decompositions for Markov chains.

Our research focuses on the winning probability of a novel problem posted on a question-and-answer website. There are \( n \) people in a line at positions \( 1, 2, \ldots, n \). For each round, we randomly select a person at position \( i \), where \( i \) is odd, to leave the line, and shift each person at a position \( j \) such that \( j > i \) to position \( j – 1 \). We continue to select people until there is only one person left, who then becomes the winner.

We are interested in which initial position has the greatest chance to survive, that is, the highest probability to be the last one remaining. Specifically, we have derived recursions to solve for exact values and the formula of the winning probabilities.

We have also considered variations of the problem, where people are grouped into triples, quadruples, etc., and the first person in each group is at the risk of being selected. We will also present various sequences we have discovered while solving for the winning probabilities of the different variations, as well as other possible extensions and related findings concerning this problem.

Our research studies the expected survival time in a novel problem posted on a question-and-answer website. There are \( n \) people in a line at positions \( 1, 2, \ldots, n \). For each round, we randomly select a person at position \( k \), where \( k \) is odd, to leave the line, and shift the person at each position \( i > k \) to position \( i-1 \). We continue to select people until there is only one person left, who then becomes the winner.

We are interested in which initial position has the largest expected number of turns to stay in the line before being selected, which we refer to as “expected survival time.” In this paper, we use a recursive approach to solve for exact values of the expected survival time. We have proved the exact formula of the expected survival time of the first and the last position as well. We will also present our work on the expected survival time of the other positions, \( 2, 3, 4, \ldots \) from an asymptotic perspective.

A set \( S \subseteq V \) is a dominating set of \( G \) if every vertex in \( V – S \) is adjacent to at least one vertex in \( S \). The domination number \( \gamma(G) \) of \( G \) equals the minimum cardinality of a dominating set \( S \) in \( G \); we say that such a set \( S \) is a \( \gamma \)-set.

A generalization of this is partial domination, which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman. In \emph{partial domination}, a set \( S \) is a \( p \)-dominating set if it dominates a proportion \( p \) of the vertices in \( V \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality of a \( p \)-dominating set in \( G \).

In this paper, we investigate further properties of partial dominating sets, particularly ones related to graph products and locating partial dominating sets. We also introduce the concept of a \( p \)-influencing set as the union of all \( p \)-dominating sets for a fixed \( p \) and investigate some of its properties.

For bipartite graphs \( F \) and \( H \) and a positive integer \( s \), the \( s \)-bipartite Ramsey number \( BR_s(F,H) \) of \( F \) and \( H \) is the smallest integer \( t \) with \( t \geq s \) such that every red-blue coloring of \( K_{s,t} \) results in a red \( F \) or a blue \( H \). We evaluate this number for all positive integers \( s \) when \( F \) and \( H \) are both stars, are both matchings, or one is a star and the other is a matching, as well as when \( F = H \) is an arbitrary double star.