Iain Beaton

activity

20172026

most citedOn the Unimodality of Domination Polynomials

1 citations · 2 across the 14 of their papers we have counts for

collaborators

16 papers

math.CO2026

Obstructions to $k$ -colouring $H$ -free graphs

Iain Beaton, Ben Cameron, Adam van Omme

A graph is $H$ -free if it has no induced subgraph isomorphic to $H$ . In 2020, Chudnovsky, Goedgebeur, Schaudt, and Zhong characterized all graphs $H$ such that there are only finit…

math.CO2026

A dichotomy for the number of vertex-critical ( $P_{5}$ , $H$ )-free graphs when $H$ is bipartite

Iain Beaton, Ben Cameron

A graph $G$ is $k$ -vertex-critical if $χ (G) = k$ , but $χ (H) < k$ for every induced subgraph $H$ of $G$ . A graph $G$ is $(H_{1}, H_{2}, \dots, H_{m})$ -free if does not contain $H_{i}$ as an induce…

math.CO2026

Vertex-critical graphs in subfamilies of $(P_{4} + ℓ P_{1})$ -free graphs

Iain Beaton, Ben Cameron

A graph $G$ is $k$ -vertex-critical if $χ (G) = k$ but $χ (G - v) < k$ for all $v \in V (G)$ . In this paper we make progress on the open problem of the finiteness of $k$ -vertex-critical $(P_4…

math.CO2025

Critical $(P_{5}, W_{4})$ -Free Graphs

Wen Xia, Jorik Jooken, Jan Goedgebeur +3

A graph $G$ is $k$ -vertex-critical if $χ (G) = k$ but $χ (G - v) < k$ for all $v \in V (G)$ . A graph is $(H_{1}, H_{2})$ -free if it contains no induced subgraph isomorphic to $H_{1}$ nor $H_{2}$ .…

math.CO2024

Reconfiguration graphs for minimal domination sets

Iain Beaton

A dominating set $S$ in a graph is a subset of vertices such that every vertex is either in $S$ or adjacent to a vertex in $S$ . A minimal dominating set $M$ is a dominating set suc…

math.CO2024

On the unimodality of nearly well-dominated trees

Iain Beaton, Sam Schoonhoven

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the numbe…

math.CO2024

Vertex-critical graphs in co-gem-free graphs

Iain Beaton, Ben Cameron

A graph $G$ is $k$ -vertex-critical if $χ (G) = k$ but $χ (G - v) < k$ for all $v \in V (G)$ and $(G, H)$ -free if it contains no induced subgraph isomorphic to $G$ or $H$ . We show that there a…

math.CO2022

A Tight Upper Bound on the Average Order of Dominating Sets of a Graph

Iain Beaton, Ben Cameron

In this paper we study the the average order of dominating sets in a graph, $avd (G)$ . Like other average graph parameters, the extremal graphs are of interest. Beato…

math.CO2020

On the Real Roots of Domination Polynomials

Iain Beaton, Jason I. Brown

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$ . The domination polynomia…

math.CO2020★ 1 cited

On the Unimodality of Domination Polynomials

Iain Beaton, Jason I. Brown

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the numbe…

math.CO2020

The Average Order of Dominating Sets of a Graph

Iain Beaton, Jason I. Brown

This papers focuses on the average order of dominating sets of a graph. We find the extremal graphs for the maximum and minimum value over all graphs on $n$ vertices, while for tre…

math.CO2020

Chromatic polynomials of 2-edge coloured graphs

I. Beaton, D. Cox, C. Duffy +1

Using the definition of colouring of $2$ -edge-coloured graphs derived from $2$ -edge-coloured graph homomorphism, we extend the definition of chromatic polynomial to $2$ -edge-colour…

math.CO2020

On the largest real root of the independence polynomial of a unicyclic graph

Iain Beaton, Ben Cameron

The independence polynomial of a graph $G$ , denoted $I (G, x)$ , is the generating polynomial for the number of independent sets of each size. The roots of $I (G, x)$ are called the \te…

math.CO2019

Optimal Domination Polynomials

I. Beaton, J. I. Brown, D. Cox

Let $G$ be a graph on $n$ vertices and $m$ edges and $D (G, x)$ the domination polynomial of $G$ . In this paper we completely characterize the values of $n$ and $m$ for which optimal…

math.CO2018

Independence Equivalence Classes of Paths and Cycles

Iain Beaton, Jason I. Brown, Ben Cameron

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if t…

math.CO2017★ 1 cited

The Domination Equivalence Classes of Paths

Iain Beaton, Jason I. Brown

A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$ . %The domination number $…

Obstructions to k-colouring H-free graphs

A dichotomy for the number of vertex-critical (P5​, H)-free graphs when H is bipartite

Vertex-critical graphs in subfamilies of (P4​+ℓP1​)-free graphs

Critical (P5​,W4​)-Free Graphs

Reconfiguration graphs for minimal domination sets

On the unimodality of nearly well-dominated trees

Vertex-critical graphs in co-gem-free graphs

A Tight Upper Bound on the Average Order of Dominating Sets of a Graph

On the Real Roots of Domination Polynomials

On the Unimodality of Domination Polynomials

The Average Order of Dominating Sets of a Graph

Chromatic polynomials of 2-edge coloured graphs

On the largest real root of the independence polynomial of a unicyclic graph

Optimal Domination Polynomials

Independence Equivalence Classes of Paths and Cycles

The Domination Equivalence Classes of Paths

Obstructions to $k$ -colouring $H$ -free graphs

A dichotomy for the number of vertex-critical ( $P_{5}$ , $H$ )-free graphs when $H$ is bipartite

Vertex-critical graphs in subfamilies of $(P_{4} + ℓ P_{1})$ -free graphs

Critical $(P_{5}, W_{4})$ -Free Graphs