In 2017, Hedetniemi asked the question: “For which graphs \( G \) does the indexed family \( \{N_G(v) \mid v \in V(G)\} \) of open neighborhoods have a system of distinct representatives?” In [1], we answered that question. Now, we move on to other special set families in graphs and examine whether they do or do not have a system of distinct representatives.

We give necessary and sufficient conditions for two matroids on the same ground set to be the upper and lower matroids of a \( \Delta \)-matroid.

Let \( D \) be a digraph on \( n \) vertices. A cycle \( C \) in \( D \) is said to be 1-extendable if there exists a cycle \( C’ \) in \( D \) such that the vertex set of \( C’ \) contains the vertex set of \( C \) and \( C’ \) contains exactly one additional vertex. A digraph is 1-cycle-extendable if every non-Hamiltonian cycle is 1-extendable.

A cycle \( C \) in \( D \) is said to be 2-extendable if there exists a cycle \( C’ \) in \( D \) such that the vertex set of \( C’ \) contains the vertex set of \( C \) and \( C’ \) contains exactly two additional vertices. A digraph is 2-cycle-extendable if every cycle on at most \( n-2 \) vertices is 2-extendable.

A digraph is 1,2-cycle-extendable if every non-Hamiltonian cycle is either 1-extendable or 2-extendable. It has been previously shown that not all strong tournaments (orientations of a complete undirected graph) are 1-extendable, but are 2-extendable. The structure of all non 1-extendable tournaments is shown as a type of block Kronecker product of 1-extendable subtournaments.

For a toroidal graph \( G = (V, E) \) embedded in the torus, let \( \mathcal{F}(G) \) denote the set of faces of \( G \). Then, \( G \) is called a \( C_{n} \)-face-magic torus graph if there exists a bijection \( f: V(G) \rightarrow \{1, 2, \ldots, |V(G)|\} \) such that for any \( F \in \mathcal{F}(G) \) with \( F \cong C_{n} \), the sum of all the vertex labelings along \( C_{n} \) is a constant \( S \).

Let \( x_{v} = f(v) \) for all \( v \in V(G) \). We call \( \{x_{v} : v \in V(G)\} \) a \( C_{n} \)-face magic torus labeling on \( G \).

We say that a \( C_{4} \)-face-magic torus labeling \( \{x_{i,j} \} \) on \( C_{2n} \times C_{2n} \) is antipodal balanced if \( x_{i,j} + x_{i+n,j+n} = \frac{1}{2}S \) for all \( (i, j) \in V(C_{2n} \times C_{2n}) \).

We determine all antipodal balanced \( C_{4} \)-face-magic torus labelings on \( C_{4} \times C_{4} \) up to symmetries on a torus.

Steinhaus graphs are a small (there are \( 2^{n-1} \) of them on \( n \) vertices) but interesting family of graphs. They have been studied for over forty years, and it has been shown that almost all graphs have certain properties if and only if almost all Steinhaus graphs have these properties.

In this paper, we find and count all the complements of Steinhaus graphs that are claw-free.

Edge-Nim is a combinatorial game played on finite regular graphs with positive, integrally weighted edges. Two players alternately move from an initialized vertex to an adjacent vertex, decreasing the weight of the incident edge to a strictly non-negative integer as they travel across it. The game ends when no incident edge has a nonzero weight and a player is unable to move, in which case that player loses.

We characterize the winner of Edge-Nim on the complete bipartite graphs \( K_{2,n} \) for all positive integers \( n \), giving the solution and complete strategy for the player able to win.

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem.

In this paper, we investigate the pansophy of two classes of graphs that contain a vertex that we define as the superuser. The superuser of a graph is a vertex that is adjacent to every other vertex. We close with future research directions.

A grid on a cell of a game board attacks all neighboring cells. The domination number counts the minimum number of grids such that each cell of a board is occupied or attacked by a grid.

For square boards (chess boards), the domination number has been determined in a series of papers. Here, we start to consider grids on hexagon boards \( B_n \) as parts of the Euclidean tessellation by congruent regular hexagons, where \( B_1 \) is one hexagon, \( B_2 \) consists of the three hexagons around one vertex, and \( B_n \) for \( n \geq 3 \) consists of \( B_{n-2} \) together with all hexagons having at least one hexagon in common with \( B_{n-2} \).

An upper bound is presented for the grid domination number, and exact values are determined by computer for small \( n \).

Given a graph \( G \), we are interested in finding disjoint paths for a given set of distinct pairs of vertices. In 2017, we formally defined a new parameter, the pansophy of \( G \), in the context of the disjoint path problem.

In this paper, we construct a method to determine the pansophy of any complete bipartite graph, and then generalize the method to compute the pansophy of any complete multipartite graph. We close with future research directions.

The \emph{Reconstruction Number} of a graph \( G \), denoted \( RN(G) \), is the minimum number \( k \) such that there exist \( k \) vertex-deleted subgraphs of \( G \) which determine \( G \) up to isomorphism. More precisely, \( RN(G) = k \) if and only if there are vertex-deleted subgraphs \( G_1, G_2, \ldots, G_k \), such that if \( H \) is any graph with vertex-deleted subgraphs \( H_1, H_2, \ldots, H_k \), and \( G_i \cong H_i \) for \( i = 1, 2, \ldots, k \), then \( G \cong H \).

A \emph{unicyclic graph} is a connected graph with exactly one cycle. In this paper, we find reconstruction numbers for various types of unicyclic graphs. With one exception, all unicyclic graphs considered have \( RN(G) = 3 \).