On the Kernel and Related Problems in Interval Digraphs

Given a digraph $G$, a set $X\subseteq V(G)$ is said to be absorbing set (resp. dominating set) if every vertex in the graph is either in $X$ or is an in-neighbour (resp. out-neighbour) of a vertex in $X$. A set $S\subseteq V(G)$ is said to be an independent set if no two vertices in $S$ are adjacent in $G$. A kernel (resp. solution) of $G$ is an independent and absorbing (resp. dominating) set in $G$. We explore the algorithmic complexity of these problems in the well known class of interval digraphs. A digraph $G$ is an interval digraph if a pair of intervals $(S_u,T_u)$ can be assigned to each vertex $u$ of $G$ such that $(u,v)\in E(G)$ if and only if $S_u\cap T_v\neq\emptyset$. Many different subclasses of interval digraphs have been defined and studied in the literature by restricting the kinds of pairs of intervals that can be assigned to the vertices. We observe that several of these classes, like interval catch digraphs, interval nest digraphs, adjusted interval digraphs and chronological interval digraphs, are subclasses of the more general class of reflexive interval digraphs -- which arise when we require that the two intervals assigned to a vertex have to intersect. We show that all the problems mentioned above are efficiently solvable, in most of the cases even linear-time solvable, in the class of reflexive interval digraphs, but are APX-hard on even the very restricted class of interval digraphs called point-point digraphs, where the two intervals assigned to each vertex are required to be degenerate, i.e. they consist of a single point each. The results we obtain improve and generalize several existing algorithms and structural results for subclasses of reflexive interval digraphs.Comment: 26 pages, 3 figure

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Interval Orders with Restrictions on the Interval Lengths

This thesis examines several classes of interval orders arising from restrictions on the permissible interval lengths. We first provide an accessible proof of the characterization theorem for the class of interval orders representable with lengths between 1 and k for each k in {1,2,...}. We then consider the interval orders representable with lengths exactly 1 and k for k in {0,1,...}. We characterize the class of interval orders representable with lengths 0 and 1, both structurally and algorithmically. To study the other classes in this family, we consider a related problem, in which each interval has a prescribed length. We derive a necessary and sufficient condition for an interval order to have a representation with a given set of prescribed lengths. Using this result, we provide a necessary condition for an interval order to have a representation with lengths 1 and 2

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Investigating Abstract Algebra Students' Representational Fluency and Example-Based Intuitions

The quotient group concept is a difficult for many students getting started in abstract algebra (Dubinsky et al., 1994; Melhuish, Lew, Hicks, and Kandasamy, 2020). The first study in this thesis explores an undergraduate, a first-year graduate, and second-year graduate students' representational fluency as they work on a "collapsing structure", quotient, task across multiple registers: Cayley tables, group presentations, Cayley digraphs to Schreier coset digraphs, and formal-symbolic mappings. The second study characterizes the (partial) make-up of two graduate learners' example-based intuitions related to orbit-stabilizer relationships induced by group actions. The (partial) make-up of a learner's intuition as a quantifiable object was defined in this thesis as a point viewed in R17, 12 variable values collected with a new prototype instrument, The Non-Creative versus Creative Forms of Intuition Survey (NCCFIS), 2 values for confidence in truth value, and 3 additional variables: error to non-error type, unique versus common, and network thinking. The revised Fuzzy C-Means Clustering Algorithm (FCM) by Bezdek et al. (1981) was used to classify the (partial) make-up of learners' reported intuitions into fuzzy sets based on attribute similarity