The Strong Perfect Graph Conjecture for Pan-Free Graphs

A graph G is perfect if for every induced subgraph F of G, the chromatic number χ(F) equals the largest number ω(F) of pairwise adjacent vertices in F. Berge\u27s famous Strong Perfect Graph Conjecture asserts that a graph G is perfect if and only if neither G nor its complement G contains an odd chordless cycle of length at least five. Its resolution has eluded researchers for more than twenty years. We prove that the conjecture is true for a class of graphs which strictly contains the claw-free graphs

Structural Characterisations of Hereditary Graph Classes and Algorithmic Consequences

A hole is a chordless cycle of length at least four, and is even or odd depending onthe parity of its length. Many interesting classes of graphs are defined by excluding (possibly among other graphs) holes of certain lengths. Most famously perhaps is the class of Berge graphs, which are the graphs that contain no odd hole and no complement of an odd hole. A graph is perfect if the chromatic number of each of its induced subgraphs is equal to the size of a maximum clique in that subgraph. It was conjectured in the 1960’s by Claude Berge that Berge graphs and perfect graphs are equivalent, that is, a graph is perfect if and only if it is Berge. This conjecture was finally resolved by Chudnovsky, Robertson, Seymour and Thomas in 2002, and it is now called the strong perfect graph theorem. Graphs that do not contain even holes are structurally similar to Berge graphs, and for this reason Conforti, Cornuéjols, Kapoor and Vušković initiated the study of even-hole-free graphs. One of their main results was a decomposition theorem and a recognition algorithm for even-hole-free graphs, and many techniques developed in the pursuit of a decomposition theorem for even-hole-free graphs proved useful in the study of perfect graphs. Indeed, the proof of the strong perfect graph theorem relied on decomposition, and many interesting graph classes have since then been understood from the viewpoint of decomposition. In this thesis we study several classes of graphs that relate to even-hole-free graphs. First, we focus on β-perfect graphs, which form a subclass of even-hole-free graphs. While it is unknown whether even-hole-free graphs can be coloured in polynomial time, β-perfect graphs can be coloured optimally in polynomial time using the greedy colouring algorithm. The class of β-perfect graphs was introduced in 1996 by Markossian, Gasparian and Reed, and since then several classes of β-perfect graphs have been identified but no forbidden induced subgraph characterisation is known. In this thesis we identify a new class of β-perfect graphs, and we present forbidden induced subgraph characterisations for the class of β-perfect hyperholes and for the class of claw-free β-perfect graphs. We use these characterisations to decide in polynomial time whether a given hyperhole, or more generally a claw-free graph, is β-perfect. A graph is l-holed (for an integer l ≥ 4) if every one of its holes is of length l. Another focus of the thesis is the class of l-holed graphs. When l is odd, the l-holed graphs form a subclass of even-hole-free graphs. Together with Preissmann, Robin, Sintiari, Trotignon and Vušković we obtained a structure theorem for l-holed graphs where l ≥ 7. Working independently, Cook and Seymour obtained a structure theorem for the same class of graphs. In this thesis we establish that these two structure theorems are equivalent. Furthermore, we present two recognition algorithms for l-holed graphs for odd l ≥ 7. The firs uses the structure theorem of Preissmann, Robin, Sintiari, Trotignon, Vušković and the present author, and relies on decomposition by a new variant of a 2-join called a special 2-join, and the second uses the structure theorem of Cook and Seymour, and relies only on a process of clique cutset decomposition. We also give algorithms that solve in polynomial time the maximum clique and maximum stable set problems for l-holed graphs for odd l ≥ 7. Finally, we focus on circular-arc graphs. It is a long standing open problem to characterise in terms of forbidden induced subgraphs the class of circular-arc graphs, and even the class of chordal circular-arc graphs. Motivated by a result of Cameron, Chap-lick and Hoàng stating that even-hole-free graphs that are pan-free can be decomposed by clique cutsets into circular-arc graphs, we investigate the class of even-hole-free circular-arc graphs. We present a partial characterisation for the class of even-hole-free circular-arc graphs that are not chordal

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

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.Comment: Accepted to appear in the Journal of Graph Theor

Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

Circuits, Perfect Matchings and Paths in Graphs

We primarily consider the problem of finding a family of circuits to cover a bidgeless graph (mainly on cubic graph) with respect to a given weight function defined on the edge set. The first chapter of this thesis is going to cover all basic concepts and notations will be used and a survey of this topic.;In Chapter two, we shall pay our attention to the Strong Circuit Double Cover Conjecture (SCDC Conjecture). This conjecture was verified for some graphs with special structure. As the complement of two factor in cubic graph, the Berge-Fulkersen Conjecture was introduced right after SCDC Conjecture. In Chapter three, we shall present a series of conjectures related to perfect matching covering and point out their relationship.;In last chapter, we shall introduce the saturation number, in contrast to extremal number (or known as Turan Number), and describe the edge spectrum of saturation number for small paths, where the spectrum was consisted of all possible integers between saturation number and Turan number

Digital Discovery of 100 diverse Quantum Experiments with PyTheus

Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantum-enhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computer-designed experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closed-source, inefficient, or targeted to very specific use-cases that are difficult to generalize. Here, we overcome these problems with a highly-efficient, open-source digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multi-particle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries -- one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology