Searcharxiv⌕ Search

arXiv subjects

Kathie Cameron

Publications and source records attributed to Kathie Cameron.

17 recordsLinked to original sources
arXiv2026

A positive instance of Scott's Conjecture on induced subdivisions

For a graph $G$, $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ denotes the size of the largest clique in $G$. A hereditary class of graphs is called $\chi$-bounded if there is a function $f$ such that for each graph $G$ in the class, $\chi(G) \le f(\omega(G))$. Scott (1997) conjectured that for every graph $H$, the class of graphs which do not contain any subdivision of $H$ as an induced subgraph is $\chi$-bounded. He proved his conjecture when $H$ is a tree and when $H$ is the complete graph on four vertices, $K_4$. Esperet and Trotignon (2019) proved that the conjecture holds when $H$ is $K_4$ with one edge subdivided once. Scott's conjecture was disproved by Pawlik et al. (2014). Chalopin et al. (2016) gave more counterexamples including the graph obtained from $K_4$ by subdividing each edge of a 4-cycle once. We prove that the conjecture holds when $H$ consists of a complete bipartite graph with and additional vertex which has exactly two neighbours, on the same side of the bipartition. As a special case, this proves Scott's conjecture when $H$ is obtained from $K_4$ by subdividing two disjoint edges.

math.CO↗
arXiv2025

Kempe changes in $H$-free graphs

Given a $k$-colouring of a graph $G$ and two of the colours, a $Kempe$ $chain$ is a connected component of the subgraph of $G$ induced by the vertices coloured with one of these two colours. A $Kempe$ $swap$ changes one colouring into another by interchanging the colours of the vertices in a Kempe chain. Two colourings are $Kempe$ $equivalent$ if each can be obtained from the other by a series of Kempe swaps; the set of Kempe equivalent colourings is called a $Kempe$ $class$. For a graph $G$, let $\chi(G)$ denote its chromatic number and let $\mathcal{C}_{k}(G)$ denote the set of all $k$-colourings of $G$. We say $G$ is $Kempe$ $connected$ if for all $k\ge \chi(G)$, $\mathcal{C}_{k}(G)$ forms a Kempe class. For a graph $H$, graph $G$ is called $H$-$free$ if no induced subgraph of $G$ is isomorphic to $H$. We prove that every $H$-free graph is Kempe connected if and only if $H$ is an induced subgraph of the path on four vertices, $P_4$. The graph 2$K_2$ consists of four vertices and two edges which are not adjacent. We prove that for all $p\ge 0$, there is a $k$-colourable 2$K_2$-free graph $G$ such that $\mathcal{C}_{k+p}(G)$ does not form a Kempe class.

math.CO↗
arXiv2025

On graphs without four-vertex induced subgraphs

Given a family F of graphs, a graph G is F-free if it does not contain any graph in F as an induced subgraph. The problem of determining the complexity of colouring (claw, 4K1)- free graphs is a well-known open problem. In this paper we solve the colouring problem for a subclass of (claw, 4K1)-free graphs. We design a polynomial-time algorithm to colour (claw, 4K1, bridge, C4-twin)-free graphs. This algorithm is derived from a structural theorem on (claw, 4K1, bridge, C4-twin)-free graphs.

math.CO↗
arXiv2024

Frozen colourings in $2K_2$-free graphs

The \emph{reconfiguration graph of the $k$-colourings} of a graph $G$, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two vertices of $\mathcal{R}_k(G)$ are joined by an edge if the colourings of $G$ they correspond to differ in colour on exactly one vertex. A $k$-colouring of a graph $G$ is called \emph{frozen} if it is an isolated vertex in $\mathcal{R}_k(G)$; in other words, for every vertex $v \in V(G)$, $v$ is adjacent to a vertex of every colour different from its colour. A clique partition is a partition of the vertices of a graph into cliques. A clique partition is called a $k$-clique-partition if it contains at most $k$ cliques. Clearly, a $k$-colouring of a graph $G$ corresponds precisely to a $k$-clique-partition of its complement, $\overline{G}$. A $k$-clique-partition $\mathcal{Q}$ of a graph $H$ is called \emph{frozen} if for every vertex $v \in V(H)$, $v$ has a non-neighbour in each of the cliques of $\mathcal{Q}$ other than the one containing $v$. The cycle on four vertices, $C_4$, is sometimes called the \emph{square}; its complement is called $2K_2$. We give several infinite classes of $2K_2$-free graphs with frozen colourings. We give an operation which transforms a $k$-chromatic graph with a frozen $(k+1)$-colouring into a $(k+1)$-chromatic graph with a frozen $(k+2)$-colouring. Our operation preserves being $2K_2$-free. It follows that for all $k \ge 4$, there is a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-colouring. We prove these results by studying frozen clique partitions in $C_4$-free graphs. We say a graph $G$ is \emph{recolourable} if $R_{\ell}(G)$ is connected for all $\ell$ greater than the chromatic number of $G$. We prove that every 3-chromatic $2K_2$-free graph is recolourable.

math.CO↗
arXiv2024

Recoloring via modular decomposition

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_{k}(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell \geq \chi(G)$+1. We demonstrate how to use the modular decomposition of a graph class to prove that the graphs in the class are recolorable. In particular, we prove that every ($P_5$, diamond)-free graph, every ($P_5$, house, bull)-free graph, and every ($P_5$, $C_5$, co-fork)-free graph is recolorable. A graph is prime if it cannot be decomposed by modular decomposition except into single vertices. For a prime graph $H$, we study the complexity of deciding if $H$ is $k$-colorable and the complexity of deciding if there exists a path between two given $k$-colorings in $R_{k}(H)$. Suppose $\mathcal{G}$ is a hereditary class of graphs. We prove that if every blowup of every prime graph in $\mathcal{G}$ is recolorable, then every graph in $\mathcal{G}$ is recolorable.

math.CO↗
arXiv2024

The sandwich problem for odd-hole-free and even-hole-free graphs

For a property $\mathcal{P}$ of graphs, the $\mathcal{P}$-\textsc{Sandwich-Problem}, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs $(G_1, G_2)$ on the same vertex set $V$, does there exist a graph $G$ such that $V(G)=V$, $E(G_{1})\subseteq E(G) \subseteq E(G_{2})$, and $G$ satisfies $\mathcal{P}$? A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs $\mathcal{C}$ and a graph $G$ we say that $G$ is {\em $\mathcal{C}$-free} if it contains no induced subgraph isomorphic to a member of $\mathcal{C}$. In this paper we prove that if $\mathcal{P}$ is the property of being odd-hole-free or the property of being even-hole-free, then the $\mathcal{P}$-\textsc{Sandwich-Problem} is NP-hard.

math.CO↗
arXiv2023

Recoloring some hereditary graph classes

The reconfiguration graph of the $k$-colorings, denoted $R_k(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R_k(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R_{\ell}(G)$ is connected for all $\ell\geq \chi(G)$+1. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph $G$ which is not recolorable. We show that every (triangle, $H$)-free graph is recolorable if and only if every (paw, $H$)-free graph is recolorable. Every graph in the class of $(2K_2,\ H)$-free graphs, where $H$ is a 4-vertex graph except $P_4$ or $P_3$+$P_1$, is recolorable if $H$ is either a triangle, paw, claw, or diamond. Furthermore, we prove that every ($P_5$, $C_5$, house, co-banner)-free graph is recolorable.

math.CO↗
arXiv2023

Reconfiguration of vertex colouring and forbidden induced subgraphs

The reconfiguration graph of the $k$-colourings, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are adjacent in $\mathcal{R}_k(G)$ if they differ in colour on exactly one vertex. In this paper, we investigate the connectivity and diameter of $\mathcal{R}_{k+1}(G)$ for a $k$-colourable graph $G$ restricted by forbidden induced subgraphs. We show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable $H$-free graph $G$ if and only if $H$ is an induced subgraph of $P_4$ or $P_3+P_1$. We also start an investigation into this problem for classes of graphs defined by two forbidden induced subgraphs. We show that if $G$ is a $k$-colourable ($2K_2$, $C_4$)-free graph, then $\mathcal{R}_{k+1}(G)$ is connected with diameter at most $4n$. Furthermore, we show that $\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable ($P_5$, $C_4$)-free graph $G$.

math.CO↗
arXiv2020

$k$-Critical Graphs in $P_5$-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices. A graph $G$ is $k$-vertex-critical if $G$ has chromatic number $k$ but every proper induced subgraph of $G$ has chromatic number less than $k$. The study of $k$-vertex-critical graphs for graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there is a polynomial-time algorithm to decide if a graph in the class is $(k-1)$-colorable. In this paper, we initiate a systematic study of the finiteness of $k$-vertex-critical graphs in subclasses of $P_5$-free graphs. Our main result is a complete classification of the finiteness of $k$-vertex-critical graphs in the class of $(P_5,H)$-free graphs for all graphs $H$ on 4 vertices. To obtain the complete dichotomy, we prove the finiteness for four new graphs $H$ using various techniques -- such as Ramsey-type arguments and the dual of Dilworth's Theorem -- that may be of independent interest.

math.CO↗
arXiv2018

An Optimal $χ$-Bound for ($P_6$, diamond)-Free Graphs

Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, diamond)-free graph $G$ satisfies $χ(G)\le ω(G)+3$, where $χ(G)$ and $ω(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schläfli graph. Our result unifies previously known results on the existence of linear $χ$-binding functions for several graph classes. Our proof is based on a reduction via the Strong Perfect Graph Theorem to imperfect ($P_6$, diamond)-free graphs, a careful analysis of the structure of those graphs, and a computer search that relies on a well-known characterization of 3-colourable $(P_6,K_3)$-free graphs.

math.CO↗
arXiv2018

The class of $(P_7,C_4,C_5)$-free graphs: decomposition, algorithms, and $χ$-boundedness

As usual, $P_n$ ($n \geq 1$) denotes the path on $n$ vertices, and $C_n$ ($n \geq 3$) denotes the cycle on $n$ vertices. For a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to any graph in $\mathcal{H}$. We present a decomposition theorem for the class of $(P_7,C_4,C_5)$-free graphs; in fact, we give a complete structural characterization of $(P_7,C_4,C_5)$-free graphs that do not admit a clique-cutset. We use this decomposition theorem to show that the class of $(P_7,C_4,C_5)$-free graphs is $χ$-bounded by a linear function (more precisely, every $(P_7,C_4,C_5)$-free graph $G$ satisfies $χ(G) \leq \frac{3}{2} ω(G)$). We also use the decomposition theorem to construct an $O(n^3)$ algorithm for the minimum coloring problem, an $O(n^2m)$ algorithm for the maximum weight stable set problem, and an $O(n^3)$ algorithm for the maximum weight clique problem for this class, where $n$ denotes the number of vertices and $m$ the number of edges of the input graph.

math.CO↗
arXiv2017

On the structure of (pan, even hole)-free graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our $O(nm)$-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our $O(n^{2.5}+nm)$-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.

cs.DM↗
arXiv2016

Structure and algorithms for (cap, even hole)-free graphs

A graph is even-hole-free if it has no induced even cycles of length 4 or more. A cap is a cycle of length at least 5 with exactly one chord and that chord creates a triangle with the cycle. In this paper, we consider (cap, even hole)-free graphs, and more generally, (cap, 4-hole)-free odd-signable graphs. We give an explicit construction of these graphs. We prove that every such graph $G$ has a vertex of degree at most $\frac{3}{2}ω(G) -1$, and hence $χ(G)\leq \frac{3}{2}ω(G)$, where $ω(G)$ denotes the size of a largest clique in $G$ and $χ(G)$ denotes the chromatic number of $G$. We give an $O(nm)$ algorithm for $q$-coloring these graphs for fixed $q$ and an $O(nm)$ algorithm for maximum weight stable set. We also give a polynomial-time algorithm for minimum coloring. Our algorithms are based on our results that triangle-free odd-signable graphs have treewidth at most 5 and thus have clique-width at most 48, and that (cap, 4-hole)-free odd-signable graphs $G$ without clique cutsets have treewidth at most $6ω(G)-1$ and clique-width at most 48.

cs.DM↗
arXiv2014

Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: $\llcorner$,$\ulcorner$, $\urcorner$, $\lrcorner$, and we consider zero bend paths (i.e., | and $-$) to be degenerate $\llcorner$s. These graphs, called $B_1$-EPG graphs, were first introduced by Golumbic et al (2009). We consider the natural subclasses of $B_1$-EPG formed by the subsets of the four single bend shapes (i.e., {$\llcorner$}, {$\llcorner$,$\ulcorner$}, {$\llcorner$,$\urcorner$}, and {$\llcorner$,$\ulcorner$,$\urcorner$}) and we denote the classes by [$\llcorner$], [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$], and [$\llcorner$,$\ulcorner$,$\urcorner$] respectively. Note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [$\llcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$,$\urcorner$] $\subsetneq$ $B_1$-EPG; also, [$\llcorner$,$\ulcorner$] is incomparable with [$\llcorner$,$\urcorner$]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split $\cap$ [$\llcorner$].

cs.DM↗
arXiv2008

Asteroids in rooted and directed path graphs

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a strong path. Two non-adjacent vertices are linked by a strong path if either they have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A strong asteroidal triple is an asteroidal triple such that each pair is linked by a strong path. We prove that a chordal graph is a directed path graph if and only if it contains no strong asteroidal triple. We also introduce a related notion of asteroidal quadruple, and conjecture a characterization of rooted path graphs which are the intersection graphs of directed paths in a rooted tree.

cs.DM↗
arXiv2007

Coloring vertices of a graph or finding a Meyniel obstruction

A Meyniel obstruction is an odd cycle with at least five vertices and at most one chord. A graph is Meyniel if and only if it has no Meyniel obstruction as an induced subgraph. Here we give a O(n^2) algorithm that, for any graph, finds either a clique and coloring of the same size or a Meyniel obstruction. We also give a O(n^3) algorithm that, for any graph, finds either aneasily recognizable strong stable set or a Meyniel obstruction.

cs.DM↗