arXiv subjects

Kerri Morgan

Publications and source records attributed to Kerri Morgan.

11 recordsLinked to original sources

arXiv2024

Graph polynomials: some questions on the edge

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.

math.CO↗

arXiv2022

Bounds On The Inducibility Of Double Loop Graphs

In the area of extremal graph theory, there exists a problem that investigates the maximum induced density of a $k$-vertex graph $H$ in any $n$-vertex graph $G$. This is known as the problem of \emph{inducibility} that was first introduced by Pippenger and Golumbic in 1975. In this paper, we give a new upper bound for the inducibility for a family of \emph{Double Loop Graphs} of order $k$. The upper bound obtained for order $k=5$ is within a factor of 0.964506 of the exact inducibility, and the upper bound obtained for $k=6$ is within a factor of 3 of the best known lower bound.

math.CO↗

arXiv2021

Supernodes

In this paper, we present two new concepts related to subgraph counting where the focus is not on the number of subgraphs that are isomorphic to some fixed graph $H$, but on the frequency with which a vertex or an edge belongs to such subgraphs. In particular, we are interested in the case where $H$ is a complete graph. These new concepts are termed vertex participation and edge participation respectively. We combine these concepts with that of the rich-club to identify what we call a Super rich-club and rich edge-club. We show that the concept of vertex participation is a generalisation of the rich-club. We present experimental results on randomised Erdös Rényi and Watts-Strogatz small-world networks. We further demonstrate both concepts on a complex brain network and compare our results to the rich-club of the brain.

cs.DM↗

arXiv2021

Instance Space Analysis for the Car Sequencing Problem

We investigate an important research question for solving the car sequencing problem, that is, which characteristics make an instance hard to solve? To do so, we carry out an instance space analysis for the car sequencing problem, by extracting a vector of problem features to characterize an instance. In order to visualize the instance space, the feature vectors are projected onto a two-dimensional space using dimensionality reduction techniques. The resulting two-dimensional visualizations provide new insights into the characteristics of the instances used for testing and how these characteristics influence the behaviours of an optimization algorithm. This analysis guides us in constructing a new set of benchmark instances with a range of instance properties. We demonstrate that these new instances are more diverse than the previous benchmarks, including some instances that are significantly more difficult to solve. We introduce two new algorithms for solving the car sequencing problem and compare them with four existing methods from the literature. Our new algorithms are shown to perform competitively for this problem but no single algorithm can outperform all others over all instances. This observation motivates us to build an algorithm selection model based on machine learning, to identify the niche in the instance space that an algorithm is expected to perform well on. Our analysis helps to understand problem hardness and select an appropriate algorithm for solving a given car sequencing problem instance.

math.OC↗

arXiv2020

Observement as Universal Measurement

Measurement theory is the cornerstone of science, but no equivalent theory underpins the huge volumes of non-numerical data now being generated. In this study, we show that replacing numbers with alternative mathematical models, such as strings and graphs, generalises traditional measurement to provide rigorous, formal systems (`observement') for recording and interpreting non-numerical data. Moreover, we show that these representations are already widely used and identify general classes of interpretive methodologies implicit in representations based on character strings and graphs (networks). This implies that a generalised concept of measurement has the potential to reveal new insights as well as deep connections between different fields of research.

cs.OH↗

arXiv2020

Further Results on an Abstract Model for Branching and its Application to Mixed-Integer Programming

A key ingredient in branch and bound (B&B) solvers for mixed-integer programming (MIP) is the selection of branching variables since poor or arbitrary selection can affect the size of the resulting search trees by orders of magnitude. A recent article by Le Bodic and Nemhauser [Mathematical Programming, (2017)] investigated variable selection rules by developing a theoretical model of B&B trees from which they developed some new, effective scoring functions for MIP solvers. In their work, Le Bodic and Nemhauser left several open theoretical problems, solutions to which could guide the future design of variable selection rules. In this article, we first solve many of these open theoretical problems. We then implement an improved version of the model-based branching rules in SCIP 6.0, a state-of-the-art academic MIP solver, in which we observe an 11% geometric average time and node reduction on instances of the MIPLIB 2017 Benchmark Set that require large B&B trees.

math.OC↗

arXiv2020

A survey of repositories in graph theory

Since the pioneering work of R. M. Foster in the 1930s, many graph repositories have been created to support research in graph theory. This survey reviews many of these graph repositories and summarises the scope and contents of each repository. We identify opportunities for the development of repositories that can be queried in more flexible ways.

math.CO↗

arXiv2020

Tutte Invariants for Alternating Dimaps

An alternating dimap is an orientably embedded Eulerian directed graph where the edges incident with each vertex are directed inwards and outwards alternately. Three reduction operations for alternating dimaps were investigated by Farr. A minor of an alternating dimap can be obtained by reducing some of its edges using the reduction operations. Unlike classical minor operations, these reduction operations do not commute in general. A Tutte invariant for alternating dimaps is a function $ P $ defined on every alternating dimap and taking values in a field such that $ P $ is invariant under isomorphism and obeys a linear recurrence relation involving reduction operations. It is well known that if a graph $ G $ is planar, then the Tutte polynomial $ T $ satisfies $ T(G;x,y)=T(G^{*};y,x) $. We note an analogous relation for the extended Tutte invariants for alternating dimaps introduced by Farr. We then characterise the Tutte invariant for alternating dimaps of genus zero under several conditions. As a result of the non-commutativity of the reduction operations, the recursions based on them cannot always be satisfied. We investigate the properties of alternating dimaps of genus zero that are required in order to obtain a well defined Tutte invariant. Some excluded minor characterisations for these alternating dimaps are also given.

math.CO↗

arXiv2019

Factorisation of Greedoid Polynomials of Rooted Digraphs

Gordon and McMahon defined a two-variable greedoid polynomial $ f(G;t,z) $ for any greedoid $ G $. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by $ 1 + z $ under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial $ f(D) $ of a rooted digraph $ D $ of order $ n $ GM-factorises if $ f(D) = f(G) \cdot f(H) $ such that $ G $ and $ H $ are rooted digraphs of order at most $ n $ and $ f(G),f(H) \ne 1 $. We study the GM-factorability of greedoid polynomials of rooted digraphs, particularly those that are not divisible by $ 1 + z $. We give some examples and an infinite family of rooted digraphs that are not direct sums but their greedoid polynomials GM-factorise.

math.CO↗

arXiv2016

Algebraic properties of chromatic roots

A \emph{chromatic root} is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments. We conjecture that, for every algebraic integer $α$, there is a natural number $n$ such that $α+n$ is a chromatic root. This is proved for quadratic integers, an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

math.CO↗

arXiv2013

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.

cs.CG↗