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 \( RN(G) = 3 \).

In this work, we introduce the Interval Permutation Segment (IP-SEG) model that naturally generalizes the geometric intersection models of interval and permutation graphs.
We study properties of two graph classes that arise from the IP-SEG model and present a family of forbidden subgraphs for these classes. In addition, we present polynomial algorithms for the following problems on these classes, when the model is given as part of the input.