Graham Farr’s research while affiliated with Monash University (Australia) and other places

We review the work of Dominic Welsh (1938-2023), tracing his remarkable influence through his theorems, expository writing, students, and interactions. He was particularly adept at bringing different fields together and fostering the development of mathematics and mathematicians. His contributions ranged widely across discrete mathematics over four main career phases: discrete probability, matroids and graphs, computational complexity, and Tutte-Whitney polynomials. We give particular emphasis to his work in matroid theory and Tutte-Whitney polynomials.

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.

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 polynomial is said to factorise if it has a non-constant factor of lower degree. We study the 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 factorise.

A graph is planar if it has a drawing in which no two edges cross. The Hanani-Tutte Theorem states that a graph is planar if it has a drawing D such that any two edges in D cross an even number of times. A graph G is a non-separating planar graph if it has a drawing D such that (1) edges do not cross in D, and (2) for any cycle C and any two vertices u and v that are not in C, u and v are on the same side of C in D. Non-separating planar graphs are closed under taking minors and hence have a finite forbidden minor characterisation. In this paper, we prove a Hanani-Tutte type theorem for non-separating planar graphs. We use this theorem to prove a stronger version of the strong Hanani-Tutte Theorem for planar graphs, namely that a graph is planar if it has a drawing in which any two disjoint edges cross an even number of times or it has a chordless cycle that enables a suitable decomposition of the graph.

A graph G is a non-separating planar graph if there is a drawing D of G on the plane such that (1) no two edges cross each other in D and (2) for any cycle C in D, any two vertices not in C are on the same side of C in D. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a wheel or it is a graph obtained from the disjoint union of two triangles by adding three vertex-disjoint paths between the two triangles. Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges. Thus, maximal linkless graphs can have significantly fewer edges than maximum linkless graphs; Sachs exhibited linkless graphs with n vertices and $4n-10$ edges (the maximum possible) in 1983.

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.

A graph G is a non-separating planar graph if there is a drawing D of G on the plane such that (1) no two edges cross each other in D and (2) for any cycle C in D, any two vertices not in C are on the same side of C in D. Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs. In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor. Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a subgraph of a wheel or it can be obtained by subdividing some of the side-edges of the 1-skeleton of a triangular prism (two disjoint triangles linked by a perfect matching). Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges which provides an answer to a question asked by Horst Sachs about the number of edges of linkless graphs in 1983.

The chromatic polynomial gives the number of proper colourings of a graph in terms of the number of available colours. In general, calculating chromatic polynomials is #P-hard. Two graphs are chromatically equivalent if they have the same chromatic polynomial. At present, determining if two graphs are chromatically equivalent involves computation and comparison of their chromatic polynomials, or similar computational effort. In this paper we investigate a new approach, certificates of chromatic equivalence, first proposed by Morgan and Farr. These give proofs of chromatic equivalence, without directly computing the polynomials. The lengths of these proofs may provide insight into the computational complexity of chromatic equivalence and related problems including chromatic factorisation and chromatic uniqueness. For example, if the lengths of shortest certificates of chromatic equivalence are bounded above by a polynomial in the size of the graphs, then chromatic equivalence belongs to NP. After establishing some links of this type between certificate length and computational complexity, we give some theoretical and computational results on certificate length. We prove that, if the number of different chromatic polynomials falls well short of the number of different graphs, then for all sufficiently large n there are pairs of chromatically equivalent graphs on n vertices with certificate of chromatic equivalence of length $\Omega(n^2/\log n)$. We give a linear upper bound on shortest certificate length for trees. We designed and implemented a program for finding short certificates of equivalence using a minimal set of certificate steps. This program was used to find the shortest certificates of equivalence for all pairs of chromatically equivalent graphs of order $n\leq 7$.

Bill Tutte was born on May 14, 1917 in Newmarket, England. In 1935, he began studying at Trinity College, Cambridge reading natural sciences specializing in chemistry. After completing a master’s degree in chemistry in 1940, he was recruited to work at Bletchley Park as one of an elite group of codebreakers that included Alan Turing. While there, Tutte performed “one of the greatest intellectual feats of the Second World War.” Returning to Cambridge in 1945, he completed a Ph.D. in mathematics in 1948. Thereafter, he worked in Canada, first in Toronto and then as a founding member of the Department of Combinatorics and Optimization at the University of Waterloo. His contributions to graph theory alone mark him as arguably the twentieth century’s leading researcher in that subject. He also made groundbreaking contributions to matroid theory including proving the first excluded-minor theorems for matroids, one of which generalized Kuratowski’s Theorem. He extended Menger’s Theorem to matroids and laid the foundations for structural matroid theory. In addition, he introduced the Tutte polynomial for graphs and extended it and some relatives to matroids. This paper will highlight some of his many contributions focusing particularly on those to matroid theory.

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.