Star and Path Perfect Graphs A Dissertation Submitted in Partial Fulfillment of the Requirements for the Award of the Degree of Master of Philosophy in Mathematics by Sanghita Ghosh (Reg. No. 1740040) Under the Supervision of Abraham V M Professor Department of Mathematics CHRIST (Deemed to be University) BENGALURU, INDIA February 2019 Approval of Dissertation The dissertation entitled Star and Path Perfect Graphs by Sanghita Ghosh, Reg. No. 1740040 is approved for the award of the degree of Master of Philosophy in Mathematics. Supervisor: Chairperson: General Research Coordinator: Place: Bengaluru Date: ii DECLARATION I, Sanghita Ghosh, hereby declare that the dissertation, titled Star and Path Perfect Graphs is a record of an original research work done by me under the supervision of Dr Abraham V M, Professor, Department of Mathematics. This work has been done for the award of the degree of Master of Philosophy in Mathematics. I also declare that the results embodied in this dissertation has not been submitted to any other University or Institute for the award of any degree, diploma, associateship, fellowship or other title. I hereby confirm the originality of the work and that there is no plagiarism in any part of the dissertation. Place: Bengaluru Date: Sanghita Ghosh Reg. No. 1740040 Department of Mathematics CHRIST (Deemed to be University), Bengaluru iii CERTIFICATE This is to certify that the dissertation submitted by Sanghita Ghosh, Reg. No. 1740040, titled Star and Path Perfect Graphs is a record of research work done by her during the academic year 2017-2018 under my supervision in partial fulfillment for the award of Master of Philosophy in Mathematics. This dissertation has not been submitted to any other University or Institute for the award of any degree, diploma, associateship, fellowship or other title. I hereby confirm the originality of the work and that there is no plagiarism in any part of the dissertation. Place: Bengaluru Date: Dr Abraham V M Professor Department of Mathematics CHRIST (Deemed to be University), Bengaluru Dr T V Joseph Head of the Department Department of Mathematics CHRIST (Deemed to be University), Bengaluru iv ACKNOWLEDGEMENT Foremost, I would like to express my deepest gratitude to Dr G. Ravindra without whom this dissertation would never have come to fruition. His support, keen interest and deep involvement helped me at every stage of my work. It is with a sense of deep appreciation that I place on record my earnest gratitude to him. My unfeigned thankfulness to Dr (Fr) Abraham V M for his profound guidance and support in the completion of the dissertation. My sincerest thanks to Dr T V Joseph, Head of the Department of Mathematics, Dr Mayamma Joseph, Coordinator of MPhil program, and Dr (Fr) Joseph Varghese, Department of Mathematics, CHRIST (Deemed to be University), for their support and encouragement. I am indeed indebted to all my teachers who have taught me over the years. My heartfelt regards to my family for their continuous and unparalleled love, help, support and encouragement to follow my passions. I am also truly grateful to my fellow classmates for their extended support. Sanghita Ghosh v Abstract Inspired by perfect graphs introduced by Berge in 1960, the concepts of f -perfect and F-perfect graphs were introduced by Ravindra in 2011. Let G be any graph and F( f ) be any subgraph (induced subgraph) of G. If for every induced subgraph of G, F( f )-partition number is equal to F( f )-independence number, we say G is F( f )-perfect. Here we characterize graphs that are F( f )-perfect when F( f ) is a path. Moreover, we characterize F( f )-perfect graphs when F( f ) is a star in the case of claw-free, P4-free, complete r-partite and some products of graphs. Graph complement satisfies perfectness preserving property, as per Lovasz´ Perfect Graph Theorem, but it need not be true in case of strongly perfectness, for example C2n;n ≥ 3. So, it is natural to study those strongly perfect graphs whose complements are also strongly perfect. With respect to this, Ravindra conjectured: A graph G is co-strongly perfect if G is C5 + e-free and Cn-free, n ≥ 5. Here we prove the conjecture for a few classes of graphs, study a few aspects of co-strongly perfect graphs and co-strongly perfect direct product graphs. vi Contents Approval of dissertation ii Declaration iii Certificate iv Acknowledgement v Abstract vi Contents vii List of Figures ix 1 Introduction 1 1.1 Origin of Graph Theory . 1 1.2 Outline of the Dissertation . 2 1.3 Basic Terminologies . 2 1.4 Motivation Behind Perfect Graphs and Applications . 6 1.5 Perfect Graphs . 7 1.6 Review of Literature . 8 vii 2 Star and Path Perfect Graphs 12 2.1 Introduction . 12 2.2 Basic Definitions . 14 2.3 Star Perfect Graphs . 14 2.4 Path Perfect Graphs . 18 2.5 Conclusion . 19 3 Some aspects of Co-strongly Perfect Graphs 20 3.1 Introduction . 20 3.2 The Results . 21 3.2.1 Complete Multipartite Co-strongly Perfect Graphs . 21 3.2.2 Paw-free Co-strongly Perfect Graphs . 25 3.2.3 Co-strongly Perfect Tensor Product Graphs . 25 3.3 Conclusion . 26 Bibliography 26 viii List of Figures 1.1 Example for neighbors . 3 1.2 Example for subgraphs . 3 1.3 Forbidden graphs G1, G2, G3 for K1;3-free graphs . 10 1.4 Forbidden graphs . 11 2.1 A graph G ..................................... 13 2.2 P3 × P3 has an induced C4 ............................. 17 2.3 P3 K3 ...................................... 18 3.1 G is co-strongly perfect . 21 3.2 Every block of G is co-strongly perfect, but G is not co-strongly perfect. 23 ix Chapter 1 Introduction In the first part of this introductory chapter, we familiarize the reader with the origin of graph theory, the terminologies and notations that we shall use in this dissertation and in the later part we mention the definitions and motivation behind perfect graphs and perform a detailed review of literature on the area of our interest. 1.1 Origin of Graph Theory Graph theory is rapidly moving into the mainstream of mathematics and hence drawing attention of budding scientists. The theory of graphs is a well knit and has shown high prospects of further development which discerns and justifies its emergence at the mathematics foreground. Its scientific and engineering applications, especially to computer science, system theory and biology have already been accorded a place of pride in applied mathematics. The origin of graph theory stems back to a number of fundamental problems, the oldest being Konigsberg¨ seven bridges problem. It was in the year 1735 when Leonhard Euler figured the difficulty at that time which involved crossing all the bridges over river Pregel such that none of them is crossed twice. He solved the problem thereby proving the first theorem in graph theory to say that all the bridges can’t be crossed if the citizen wished to begin and end at the same place. Graph theory has advanced since then, solving other situations like knight’s tour problem, Kirchoff’s evolvement of the concept of trees in graph theory in 1847, Four color theorem in 1852 and so on. The area of graph theory has been an aid in solving many relevent problems and thereby making it a fascinating subject for further research. 1.2 Outline of the Dissertation The chapters in the dissertation are organized as follows: The first part of chapter one is purely introductory in nature. It familiarizes the reader with the origin of graph theory, the terminologies and notations used in this dissertation, the motivation behind perfect graphs and a few examples which aid in understanding the succeeding chapters. The later part of the chapter articulates detailed study of literature in this area. Chapter two introduces us to f perfect and F perfect graphs with illustrations. We present our work on star and path perfect graphs and a few characterization theorems for each of them are studied. Chapter three deals with the study of few aspects of co-strongly perfect graphs, proof of the conjecture given by Ravindra for a few classes graphs and characterize co-strongly direct product graphs. 1.3 Basic Terminologies In this section, we present the basic terminologies relevant for the material presented in the subsequent chapters. For further theoretic terminologies, we refer to West[1], Harary[2] and Reed[3]. Definition 1.3.1. [1] A simple graph G = (V;E) is a pair where V is a finite set of elements called vertices (or nodes, points) and E is a prescribed subset of the set of distinct unordered pairs of distinct elements of V. Elements of E are called edges (or lines). The definition of G, clearly precludes the occurrence of multiple edges and loops. For a graph G, we write jV(G)j = n called the order of G and jE(G)j = m called the size of G. An edge uv 2 E(G), if u is adjacent to v in G and uv 2= G, if u is not adjacent to v in G. The set of all neighbors of a vertex v in G, denoted as N(v;G) is the set fu 2 Vjuv 2 E(G)g and the set of all non- neighbors of v in G, denoted as N(v;G) is the set fu 2 Vjuv 2= E(G)g. The closed neighborhood of a vertex v in G, denoted as N[v;G] is the set N(v;G) [ fvg and the closed non-neighborhood of a vertex v in G, denoted by N[v;G] is the set N(v;G) [ fvg.