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 |V(G)| = n called the order of G and |E(G)| = m called the size of G. An edge uv ∈ E(G), if u is adjacent to v in G and uv ∈/ 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 {u ∈ V|uv ∈ E(G)} and the set of all non- neighbors of v in G, denoted as N(v,G) is the set {u ∈ V|uv ∈/ E(G)}. The closed neighborhood of a vertex v in G, denoted as N[v,G] is the set N(v,G) ∪ {v} and the closed non-neighborhood of a vertex v in G, denoted by N[v,G] is the set N(v,G) ∪ {v}. The open neighborhood of an edge e = uv in G, denoted by N(e,G) is the set consisting of the endpoints of e (that is u and v) and the edges adjacent to e in G. The closed neighborhood of an edge e = uv in G, denoted by N[e,G] is N(e,G) ∪ {e}.

In the graph shown in Figure 1.1, N(v1,G) = {v2,v3}; N(v1,G) = {v3,v4,v6,v7};

N[v1,G] = {v1,v2,v3}; N[v1,G] = {v1,v3,v4,v6,v7} 2 Figure 1.1: Example for neighbors

Deﬁnition 1.3.2. [1] A graph H is called subgraph of a graph G if H exists in the graph G. A subgraph H of a graph G is proper subgraph of G if either V(H) 6= V(G) or E(H) 6= E(G). For H ⊆ G, the induced subgraph denoted by G[H] is the maximal subgraph of G with respect to the vertex set H. Thus two points in H are adjacent in the induced subgraph G[H] if and only if they are adjacent in G.

G −U is the graph obtained by removing U ⊆ V from G. G − e is the graph obtained by removing e ∈ E from G. G + e is the graph obtained by addition of the edge e ∈ E.

In Figure 1.2, C5 is a subgraph of G, in fact a spanning subgraph of G. But C5 is not an induced subgraph of G. C3,C4 are induced subgraphs of G.

Figure 1.2: Example for subgraphs

Deﬁnition 1.3.3. [1] A graph invariant is a function f from the set of all graphs to any range of values (numerical, vectorial or any other) such that f takes the same value on isomorphic graphs. When the range of values is numerical (real, rational or intergral) the invariant is called a parameter.

Deﬁnition 1.3.4. [1] A chord of a cycle is an edge that joins two non-consecutive vertices of the cycle. Two chords viv j and vlvm of a cycle are called crossing chords if both paths of the 3 cycle joining vi and v j contain one of the endpoints of vlvm. A chordless cycle in G is a cycle of length at least 4 in G that has no chord (that is, the cycle is an induced subgraph). A graph is chordal if it is simple and has no chordless cycle.

In graph G shown in Figure 1.2, v2v3 is a chord in G.

Deﬁnition 1.3.5. [1] A complete graph on n vertices, denoted by Kn is a graph with edges between each pair of vertices. These graphs are also called as universal graphs. A clique is a maximal complete subgraph of G. The density of a graph G, denoted by ω(G), is the maximum cardinality of a clique in G. A unicliqual vertex (edge) is a vertex (edge) belonging to exactly one clique. A free clique is a clique having a unicliqual vertex.

Deﬁnition 1.3.6. [1] The minimum number of cliques covering all the vertices of G is called the partition number of G, denoted by θ(G).

Deﬁnition 1.3.7. [1] The degree of a vertex v in a simple graph G, written dG(v) or d(v), is the number of edges incident to v in G. The maximum and minimum of all the degree of vertices is ∆(G) and δ(G) respectively. A graph G is called k-regular if ∆(G) = δ(G) = k. An isolated vertex is a vertex of degree 0.

A vertex v ∈ V(G) is a pendent vertex (also called end point) if d(v) = 1 and e = uv ∈ E(G) is a pendent edge if d(u) = 1 or d(v) = 1.

Deﬁnition 1.3.8. [1] The complement G¯ of a graph G = (V,E) of order n is a graph with vertex set V(G) and edge set E(G) = E(Kn) − E(G).

Deﬁnition 1.3.9. [1] An isomorphism from a simple graph G to a simple graph H is a bijection f : V(G) −→ V(H) such that uv ∈ E(G) if and only if f (u) f (v) ∈ E(H). We say G is isomorphic to H, written G =∼ H, if there is an isomorphism from G to H.

Deﬁnition 1.3.10. [1] A graph G is H-free if G has no induced subgraph isomorphic to H.

For example, a claw-free graph is a graph having no K1,3 as an induced subgraph.

Deﬁnition 1.3.11. [1] A set of vertices which cover all the edges of G is called vertex cover of G and the set of edges which cover all the vertices of G is called edge cover of G.

Deﬁnition 1.3.12. [1] A set of vertices in a graph G is independent if no two of them are adjacent. The independence number, denoted by α(G) is the maximum number of vertices in an independent set of G.

4 Deﬁnition 1.3.13. [1] The chromatic number of a graph G, written as χ(G), is the minimum number of colors needed to label the vertices so that no two adjacent vertices are monochromatic.

Deﬁnition 1.3.14. [1] The circumference of the graph with a cycle is the length of its longest induced cycle.

Deﬁnition 1.3.15. [1] A block of a graph G is a maximal connected subgraph of G that has no cut-vertex. If G itself is connected and has no-cut vertex, then G is a block.

Deﬁnition 1.3.16. [1] A dominating set of a graph G is a set of vertices S ⊆ V(G) such that all vertices in the graph G are either in S or adjacent to some vertex of S. The domination number, denoted by γ(G) is the cardinality of the smallest dominating set. The domination number of a graph G can also be deﬁned as the minimum number of stars covering all the vertices of G.

Deﬁnition 1.3.17. [1] A graph G is perfect if χ(H) = ω(H) for every induced H ⊆ G. Equivalently, χ(G[A]) = ω(G[A]) for all A ⊆ V(G).

A graph G is said to be minimal imperfect if G is not perfect but for any v ∈ G, G − v is perfect.

Deﬁnition 1.3.18. [3] An odd hole or odd antihole in G is an induced subgraph of G that is

C2k+1 or C2k+1 (for some k ≥ 2), respectively. A graph having no odd hole or antihole is a Berge graph.

Deﬁnition 1.3.19. [3] A set of minimum number of vertex disjoint complete subgraphs covering the vertices of G is called a θ-cover of G. A set of minimum number of colors used to label all the vertices of G such that no two adjacent vertices are monochromatic is called a χ-cover of G.

Deﬁnition 1.3.20. [3] A family of graphs G is hereditary if every induced subgraph of the graph G ∈ G is also a graph in G .

In order to prove that every graph in a hereditary class G is perfect, it sufﬁces to verify that χ(G) = ω(G) for every G ∈ G . Doing so induces the proof of equality for the induced subgraphs of G.

Deﬁnition 1.3.21. [3] A graph G is said to be critical or minimal imperfect if G is not perfect but G − v is perfect for every v ∈ V.

5 Following are the definitions of operations on two graphs (the definitions can easily be extended to any finite number of graphs).

Deﬁnition 1.3.22. [1] The cartesian product of G and H, written GH, is the graph with vertex set V(G) ×V(H) speciﬁed by putting (u,v) adjacent to (u0,v0) if and only if

1. u = u0 and vv0 ∈ E(H), or

2. v = v0 and uu0 ∈ E(G).

∼ The cartesian product operation is symmetric; GH = HG.

Deﬁnition 1.3.23. [1] The direct product of G and H, written G × H, is the graph with vertex set V(G) ×V(H) speciﬁed by putting (u,v) adjacent to (u0,v0) if and only if uu0 ∈ E(G) and vv0 ∈ E(H)

The direct product operation is symmetric; G × H =∼ H × G.

Deﬁnition 1.3.24. [1] The strong product of G and H, written G H, is the graph with vertex set V(G) ×V(H) and edge set E(GH) ∪ E(G × H). ∼ The strong product operation is symmetric; G H = H G.

Deﬁnition 1.3.25. [1] The composition or lexicographic product of two simple graphs G and H, written as G ◦ H, is the simple graph whose vertex set is V(G) ×V(H), with edges given by (u,v) ↔ (u0,v0) if and only if

1. uu0 is an edge of G, or

2. u = u0 and vv0 is an edge of H.

The lexicographic product operation of two graphs is symmetric, that is G ◦ H =∼ H ◦ G if and only if G = H.

1.4 Motivation Behind Perfect Graphs and Applications

Perfect graphs were a very important part of Claude Berge’s life work. When Claude Berge deﬁned perfect graphs in 1960s, he was motivated by a very practical problem worked by Shannon[4]: How do we maximize the rate at which information is sent through a transmission channel while avoiding the introduction of errors because of the physical imperfections of the system? 6 According to Shannon’s work, capacity of the channel is concerned with rates of transmission i.e. to check if what was sent during transmission is same as to what is recieved at the end of transmission. One example for that is the internet connection which we use at home, the spped of which is an example of the rate of transmission from the source of the information to the computer and also that the information transferred is 100 percent error-free and not confounded. One of the main reasons behind the investigation of perfect graphs is that the various classes of perfect graphs have a whole lot of applications in diverse areas of research and in different aspects of real life problems. The chief reason behind the use of perfect graphs in the wide application territory is because of its distinctive nature of determinablilty of different graph invariants. The importance of perfect graphs is both conceptual (as it has applications in coloring problems) and realistic (applications in communication theory and operations research, municipal services optimization problems, maintainance of temperatures of chemicals, etc). Tucker[5] found the application of perfect graphs in optimizing municipal services and domination problems.

1.5 Perfect Graphs

According to the definition, Berge’s class of perfect graph is a class of graphs for which independence number equals partition number (or chromatic number equals density number). One more prime characterization of perfect graphs is that it is hereditary in nature and the equality holds true for every induced subgraph of the graph. Also, perfect graphs are closed with respect to complementation. This characterization of perfect graphs is called Weak Perfect Graph conjecture and was proved by Lovasz.´ The only graphs for which the aforesaid invariants differ are odd cycle or odd anticycle of length greater than or equal to five. This characterization of perfect graphs is also known as Strong Perfect Graph Conjecture. Doing research in perfect graphs is interesting since the domain is growing very fast and it also allows us to ponder on different existing classes of perfect graphs. Some of the classes of perfect graphs are as follows: triangulated graphs, bipartite graphs, P4-free graphs, nearly bipartite graphs, comparibility graphs, complete graphs, etc. Hougardy[6] enlisted nearly 120 classes of perfect graphs. and the strong perfect graph conjecture given by Berge has also led to defining and studying many new classes of graphs for which the validity of the conjecture is asserted.

7 1.6 Review of Literature

The domain of perfect graphs is vast owing to its numerous applications. The theory of perfect graphs started in early 1960s, when the well recognized mathematician Claude Berge, known as the modern founders of combinatorics and graphs, got motivated by Shannon’s work on communication theory and started to work on certain invariants of graphs. This part provides an general overview listing the major developments in the the area of Perfect graphs over the last six decades. The coverage of the content has been limited as the summarization has condensed the content into a shorter form, communicating only the essential facts of the subject. 1956- Shannon’s work on communication theory, particularly by the notion of zero-error capacity of a noisy channel where he noticed that only graphs with θ = α can be used for perfect channels in communication theory[4]. 1958- Hajnal and Suryani[7] proved that triangulated graphs belong to the aforesaid class of graphs mentioned by Shannon. 1960- Berge’s[8] introductory paper on Perfect graphs where he connected the well known information theoretical problems by Shannon to the invariants of graphs. 1962- Berge[9] proposed the two famous conjectures on Perfect graphs and called them the Strong and Weak Perfect Graph Conjectures. Gallai[10] proved that a graph with each of its odd cycles of length at least ﬁve having two non crossing chords is perfect. 1972- Lovasz[11]´ made the ﬁrst breakthrough by settling Weak Perfect Graph Conjecture and strengthened it by calling it Perfect Graph Theorem. 1971-1972- Fulkerson[12, 13] attacked and solved the Weak Perfect Graph Conjecture independently using the concepts of blocking and antiblocking pairs of polyhedra. 1978- Berge posed the following problem in Monday Seminars of Graph Theory Research Group at MSH, Paris[14]: Does every perfect graph contain an independent set which meets all the cliques in it?

C¯6 was the ﬁrst counter example given by Ravindra to the above question. In this context, Berge called a graph ‘Strongly Perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. There is no complete characterization of strongly perfect graphs, although Berge and Duchet, Ravindra, Chvatal[15]´ have obtained several interesting results in the area of strongly perfect graphs. Perfect graph theorem asserts that a graph is perfect if and only if its complement is perfect. But the complement of a strongly perfect graph is not necessarily strongly perfect. For example,

C2n(n ≥ 3) is strongly perfect but not C¯2n. It is natural to investigate those strongly perfect

8 graphs whose complements are also strongly perfect. The graphs for which G and its complement G are strongly perfect were termed as co-strongly perfect. The characterization of co-strongly perfect graphs like triangulated graphs, nearly-bipartite, line, total, cartesian product and tensor product graphs has been done and analogous to these very strongly and very co-strongly perfect graphs were also deﬁned and characterized by Bassavaya and Ravindra[14]. 1987- Olaru[16] proved that a graph with each of its odd cycles of length at least ﬁve having two crossing chords is perfect. 1988- Meyniel[17] generalized the result of Gallai and Olaru by characterizing perfect graphs. 1976-1987- Strong Perfect Graph Conjecture(SPGC) was proved for many classes of graphs:

K1,3-free graphs by Ravindra[18], K4 − e-free graphs by Parthasarthy and Ravindra[19] and

Tuckar[19] independently, K4-free graphs by Tucker, cartesian products by Ravindra and Parthasarthy, etc[19]. 2002- SPGC was settled by Chudnovasky, Robertson, Seymour and Thomas[20]. Berge’s famous strong perfect graph theorem is an excellent characterization of perfect graphs in terms of forbidden subgraphs. However, there is no elegant characterization of strongly perfect graphs in terms of forbidden subgraphs. Thus the problem of attempting to characterize strongly perfect graphs in terms of forbidden subgraphs, originally posed by Berge is worth considering. 1982, 1999- Ravindra deﬁned sp-critical graphs as those graphs which are not strongly perfect but every proper induced subgraph are strongly perfect and conjectured[21, 22]:

Conjecture 1.6.1. If G is C5 + e-free and Cn-free, n ≥ 5, then G is co-strongly perfect.

Conjecture 1.6.2. The only K1,3-free sp-critical graphs are C2n+1,n ≥ 2, C¯n,n ≥ 5, the graphs of Figure 1.3.

Conjecture 1.6.2 has been settled in afﬁrmative by Wang[23].

Conjecture 1.6.3. The only sp-critical graphs are C2n+1,n ≥ 2, C¯n,n ≥ 5 and in Figure 1.3 and Figure 1.4.

1977-1978- Ravindra and Parthasarthy[28] studied the perfectness of cartesian products, tensor products and lexicographic productsand obtained necessary and sufficient conditions for these products to be perfect. Ravindra[29] characterized perfect normal product of bipartite graphs and obtained a few sufficient conditions for perfectness of normal products. However, the problem of finding a necessary and sufficient condition for perfectness of an arbitrary normal product graph remains open till today. 9 Figure 1.3: Forbidden graphs G1, G2, G3 for K1,3-free graphs

1986- Tuza and Lehel[30] defined and characterized Neighborhood Perfect Graphs. 2000- Rangan and Guruswami[31] defined and characterized clique transversal and clique perfect graphs. 2011- Inspired by perfect graphs, Ravindra introduced F-perfect graphs in 2011[32, 33]. If F is complete or the complement of complete graph, then F-perfect graph means usual perfect graphs. Much work has been done over four decades on perfect graphs, but all the characterizations done for perfect graphs are considering F = Kn i.e. complete graphs. So the new definition of F- perfect graphs generalize perfect graphs and this new concept of F-perfect graphs extend the boundary of perfect graphs to F-perfect graphs where any class of graphs or union of various graphs can be considered as F and new characterizations can be made. Based on this review of literature, the present study focuses on the following problems.

10 Figure 1.4: Forbidden graphs

1. Characterization of F( f )-perfect claw-free graphs, complete r-partite graphs and P4- free graphs and product graphs, considering F( f ) as star.

2. Characterization of F( f )-perfect graphs considering F( f ) as path.

3. Proof of Conjecture 1.6.1 for a few classes of graphs.

4. Characterization of co-strongly perfect tensor product graphs.

11 Chapter 2

Star and Path Perfect Graphs

2.1 Introduction

Berge defined perfection as “freedom from fault or defect, an exemplification of supreme excellence, an unsurpassable degree of accuracy” [8] and posed the question, “Can perfection be achieved?” A lot of work has been done on perfect graphs over the decades. Inspired by perfect graphs, the concepts of f -perfect and F-perfect graphs were introduced by Ravindra in 2011[32, 33]. The definition of f -perfect and F-perfect graphs generalizes perfect graphs and hence extends the horizon of further research work in this area. The following part presents discussion of the basic definitions of f -perfect and F-perfect graphs.

Deﬁnition 2.1.1. Let G be any graph and F( f ) be subgraph (induced subgraph) of G The invariants involved in F( f )-perfect graphs are as follows:

• The F( f )-independence number of G, denoted by αF( f )(G), is the maximum number of vertices in G such that no two of the vertices are in the same F( f )-subgraphs.

• The F( f )-density of G, denoted by ωF( f )(G), is the maximum size of a F( f )-subgraphs in G.

Then G is called F( f )-perfect if αF ( f )(H) = θF ( f )(H) (or χF( f )(H) = ωF( f )(H)) for every induced subgraph H of G.

Deﬁnition 2.1.2. A set of vertices in G is F( f )-independent if no two vertices of the set are in the same F( f )-subgraphs.

If F is Kn or Kn, then F( f )-perfect graph means Berge’s perfect graphs. 12 We use the graph G in Figure 2.1 to illustrate αF ,α f ,θF ,θ f of G, when F( f ) is star or path.

Figure 2.1: A graph G

1. αF (G) = 1, θF (G) = 2, α f (G) = 1, θ f (G) = 2 when F is a star.

Here, αF (G) = 1 and α f (G) = 1 since every pair of vertices are in some star subgraph or induced star subgraph. So the maximum size of F-independence and f -independence set is 1.

For the given graph, θF (G) = 2 and θ f (G) = 2. Since the graph does not have a spanning star, the minimum number of star subgraphs and induced star subgraph required to cover the vertices of G is obviously ≥ 2. We see that for the graph G, two star subgraphs and star induced subgraphs are required to cover all the vertices of G.

2. αF (G) = 1, θF (G) = 1, α f (G) = 1, θ f (G) = 2 when F is a path.

Since the graph G is connected, αF (G) = 1 and α f (G) = 1 since there exists a path between every pair of vertices. So the maximum size of F-independence and f -independence set is 1.

For the given graph, θF (G) = 1 and θ f (G) = 2. Since the graph does not have an induced

path subgraph covering the vertices of G, therefore obviously θ f (G) ≥ 2. We see that for the graph G, two induced path subgraphs are enough to cover all the vertices of G;

hence θ f (G) = 2. The graph has a path subgraph covering all the vertices of G. Hence,

θF (G) = 1.

Similarly we need to check for every induced subgraph H of G.

13 2.2 Basic Deﬁnitions

Deﬁnition 2.2.1. A graph G is minimal F-imperfect ( f -imperfect) if for any v, G−v is F-perfect

( f -perfect) for every induced subgraph H of G − v and αF (G) 6= θF (G) (α f (G) 6= θ f (G)).

Deﬁnition 2.2.2. [1] Let S be a set of vertices in a graph G. An S-lobe of G is an induced subgraph of G whose vertex set consists of S and the vertices of a component of G − S.

Deﬁnition 2.2.3. A set of minimum number of vertex disjoint F( f )-subgraphs covering the vertices of G is called a θF( f )-cover of G. A set of minimum number of colors used to label all the vertices of G such that no two vertices in same F( f )-subgraphs are monochromatic is called a χ f ( f )-cover of G.

Since the intersection of an F( f )-independent set (F( f )-subgraph) and a member of θF( f )-cover

(χF( f ) -cover) of G is at most one, it follows immediately that

αF( f )(G) ≤ θF( f )(G)

ωF( f )(G) ≤ χF( f )(G).

F-perfect and f -perfect graphs will have different properties with respect to F- subgraph or f -subgraph. For example if F is an induced cycle, tree or path of G, then F-perfectness will enjoy more non-triviality. One could also color G differently using F-independent sets of G.

2.3 Star Perfect Graphs

We consider the graph F( f ) as a star K1,n. If F( f ) is a star and αF( f )(G) = θF( f )(G) for every induced subgraph H of G, then we have star perfect graphs. We know that every graph G has

K1,n as a subgraph. Also one more interesting observation of star perfect graphs is that θF (G) is equal to the domination number of G (i.e. γ(G)) as domination number of a graph G can also be deﬁned as the minimum number of stars covering all the vertices of G.

Observations

In this subsection we list a few observations on star perfect graphs.

• Star perfect graphs are not perfect and the converse is not true.

For example, C3k is star perfect, but odd C3k are not perfect; even C3k+1 or C3k+2 are

14 perfect but not star perfect. Let C3k be an odd cycle where k ≥ 3 and let its vertices be

v1,v2,...,v3k. Then v1,v4,v7,v10,...,v3k−2 are f-independent(since any two of them are at a distance ≥ 3). Also the stars at these vertices are mutually disjoint and cover all

the vertices. Therefore θ f ≤ α f = k and so θ f = α f = k (since α f ≤ θ f always). Every

proper induced subgraph of C3k will be a forest, each of whose component is a tree. But

for a tree, θ f = α f .

• If G is has C3, C3k+1 or C3k+2 as an induced subgraph, then G is not f -perfect.

For C3, θ f (C3) = 2 and α f (C3) = 1. For C3k+1, there are k K1,2’s which are independent

and one extra vertex. Hence, θ f = k + 1. No two of these starts intersect. This implies

that there are k f -independent vertices, that is, α f (C3k+1) = k. The star at the extra vertex

intersects with one of the stars K1,2. Hence, α f ≤ k. Hence, θ f 6= α f . Similarly, we can

ﬁnd the proof for C3k+2 also.

• For C3-free graphs, f perfect and F-perfect are the same.

Theorem 2.3.1. If each lobe of G with respect to a vertex v is f -perfect then G is f -perfect.

Proof. Let α f and θ f respectively denote maximum size of f -independent set and minimum

star cover of G. Let α fi and θ fi refer to maximum f-independent set and minimum star cover of

each lobes say L1,L2,...Lk with respect to v. Then since each lobe is f -perfect, α fi = θ fi , for

all 1 ≤ i ≤ k. To prove α f = θ f . We have the following two cases. For any vertex v ∈ G,

Case I If each maximum α f set contains v, that is α f (G−v) = α f −1. then v together with α f −1

stars covering vertices of G − v will form independent set of size α f . Since α fi = θ fi ,∀i,

therefore, in this case α f = θ f .

Case II If there exists an α f set that does not contain v, then each component of G − v contains

minimum θ f -cover of size θ fi . θ f1 cover of L1 with θ fi of other components Lm − v,

(m 6= 1) is θ f -cover of size α f .

Combining both the cases we see that if each lobe of G with respect to a vertex v is f -perfect then G is f -perfect.

Theorem 2.3.1 is true for F- perfectness also.

Corollary 2.3.1. A minimal f -imperfect graph is a block, that is it has no cut vertices.

15 Proof. Let G be a minimal f -imperfect graph with v as a cut vertex. Then every lobe with respect to v is f -perfect (since G is minimal f -imperfect graph) and by Theorem 2.3.1, G is f -perfect, contradiction to minimal f -imperfection of G.

As a consequence we have,

Theorem 2.3.2. All trees are f -perfect and F-perfect.

Proof. Let T be the minimal star imperfect graph. Then by Corollary 2.3.1, T is a block. But we know that every tree has at least one cut vertex. Hence T is star perfect.

Theorem 2.3.3. AP4−free graph is F-perfect if and only if G is C4−free.

Proof. Let G be a P4−free graph and H be its complement. If G is disconnected, by induction hypothesis each of its components is F-perfect, and so G. If not, i.e. G is connected then H is disconnected. If two of its component have an edge, then the two edges will induce a C4 in G, which is forbidden. Therefore at most one component has an edge and the rest K1s. Then G is Kn or Kn − e. Each of them has spanning star, so θF = 1 for each of them. Therefore G is F-perfect.

Corollary 2.3.2. AP4−free graph is f -perfect if and only if G is a tree.

Theorem 2.3.4. A claw-free graph is f -perfect and F-perfect if and only if G is C3-free,

C3k+1-free and C3k+2-free.

Proof. Let G be a claw-free graph which is C3-free, C3k+1-free and C3k+2-free. Since G is claw-free with no C3, the degree of every vertex is ≤ 2. Hence G can be either tree or cycle. We know trees are star perfect and the only possibility of cycles being star perfect is C3k, k ≥ 1.

Theorem 2.3.5. If G is a complete r-partite graph, then G is F- perfect if and only if at most one of the partitions has two vertices.

Proof. Let G be a complete r-partite graph which is F- perfect. Then G will not have C4 as an induced subgraph. Let G have two partitions with more than one vertex each, then G will have C4 as an induced subgraph, a contradiction. Hence G has at most one of partition with two vertices. Conversely, let G is a complete r-partite graph and at most one of the partitions have two vertices. Then the graph is K1,1,1,...,2. and αF (H) = θF (H) = 1, for every induced subgraph

H of K1,1,1,...,2. Hence the graph is F-perfect.

16 Deﬁnition 2.3.6. [34] θ1(G) is the minimum number of cliques containing all the edges of G. A graph G is θ1-perfect if θ1(H) = α(H) for every induced subgraph H of G. θ1-perfect graphs were characterized as those graphs which do not have P4 or C4 as an induced subgraph.

Theorem 2.3.7. A connected graph G = G1G2 is F-perfect if and only if one of them is F-perfect and the other is K1.

Proof. Since K2K2 is C4, therefore two graphs with K2 as as induced subgraph will form a cycle of length 4 in the cartesian product. Since C4 = (C3.1+1) is a forbidden subgraph for

F-perfectness, so none of G1 or G2 will have K2 as an induced subgraph. So one of G1 or G2 is

K1 (isolated vertex).

By the deﬁnition of cartesian product of two graphs, the product of any graph H and K1 is the graph H itself. Therefore, G to be F-perfect if and only if one of G1 or G2 is F- perfect and the other is K1.

Theorem 2.3.8. A connected graph G = G1 × G2 is F-perfect if and only if one of them is

F-perfect and the other is K1 or K2.

If G1 and G2 are not K2, then it will have P3 or K3 as an induced subgraph. K3 × K3 or

P3 × P3 or K3 × P3 will have C4 as an induced subgraph.

Figure 2.2: P3 × P3 has an induced C4

So, G1 or G2 has to be K2. Let G1 be star perfect and G2 = K2, the proof of the theorem utilizes the fact that G1 ×G2 is bipartite. Now, C2n ×K2 gives two copies of C2n and C2n+1 ×K2 gives C4n+2.

Theorem 2.3.9. If G1 and G2 are θ1-perfect, then G1 G2 is f -perfect and F-perfect if and only if one of them is complete.

Proof. G1 and G2 are θ1-perfect, so neither G1 or G2 will have any induced cycles of length ≥

4 (since, a graph G is θ1-perfect if and only if G is P4 and C4 free). 17 If not, since one of G1 and G2 is a complete graph, there exists a chord in every induced cycle of length ≥ 4. Hence the graph is triangulated and every induced subgraph has a vertex adjacent to all other vertices. Therefore θF = αF = 1, for every induced subgraph of G.

Figure 2.3: P3 K3

If u is a vertex adjacent to all other vertices in G1, (u,v) where v ∈ V(G2) is adjacent to all other vertices in G G2 (by deﬁnition of strong product). Hence, θF = αF = 1 for every induced subgraph of G following that the graph G is f -perfect and F-perfect.

Theorem 2.3.10. G1 ◦G2 is f -perfect and F-perfect if and only if both G1 and G2 are f -perfect or F-perfect.

The result is obvious as according to the deﬁnition of lexicographic product, if there exist a non f -perfect or F perfect subgraph in G1 and G2, then those structures will appear in the product G1 ◦ G2. Hence G1 and G2 are necessarily f -perfect or F-perfect.

2.4 Path Perfect Graphs

We consider the graph F( f ) as any path Pn. If F( f ) is path and αF( f )(H) = θF( f )(H) for every induced subgraph H of G, then we have path perfect graphs. Let G be a connected graph. Then there exists a path between every pair of vertices. If G is not connected, then we can consider each component of G separately. The shortest path between every pair of vertices in a connected graph G is an induced path. Hence, every pair of vertices is in an induced path. This implies that for a connected graph G, α f (G) and αF (G) is always equal to 1.

However, if the given graph G is a path Pn, then θ f (G) = 1. These observations lead us the following theorem.

Theorem 2.4.1. A connected graph is f -perfect if and only if it is a path.

18 Proof. Let G be a connected graph other than path Pn. Then as G is a connected graph, α f (G) = 1 and since the graph does not have an induced path subgraph covering all vertices, therefore

θ f (G) ≥ 2. Hence, G is not f -perfect.

Conversely, let G be any path Pn. Then α f (Pn) = 1 = θ f (Pn). Therefore, G is f -perfect.

Theorem 2.4.2. A claw free, net-free graph G has a Hamiltonian path[35].

Theorem 2.4.3. A connected graph G is F-perfect if and only it is claw-free and net-free.

Proof. Let G be a connected graph which is not claw-free or net-free. Then G has a claw or net as an induced subgraph. For a claw, αF = 1 and θF = 2 and for a net, we have αF = 1.

However, θF = 2. Hence αF (G) 6= θF (G) for claw and net. Since the graph G has claw or net as an induced subgraph, αF 6= θF for every induced subgraph H of G. Hence, the a connected graph G containing claw or net is not F-perfect. Conversely, let us assume that the graph a connected graph G is claw-free and net-free.

Duffus et al.[35] proved that G has a spanning path. Therefore, αF (H) = 1 = θF (H), for every induced subgraph H of G and hence the graph G is F-perfect.

2.5 Conclusion

In this chapter, we found a few forbidden subgraphs for star perfect graphs and obtained characterizations for star perfectness for a few classes of graphs and product graphs. We also obtained characterizations for path perfect graphs.

19 Chapter 3

Some aspects of Co-strongly Perfect Graphs

3.1 Introduction

The concept of perfect graphs was introduced by Berge[8] having some connections with a well-known information-theoretical problem by Shannon[4]. Perfect graphs have varied applications in coloring problems, channel assignment problems, etc. A graph is strongly perfect if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. Since complement of every strongly perfect graphs need not be perfect, for example C2n is strongly perfect but C2n,n ≥ 3, so it is intriguing to study those strongly perfect graphs G whose complement are also co-strongly perfect. The following deﬁnitions are used throughout in this chapter.

Deﬁnition 3.1.1. [36] A graph is strongly perfect if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it.

Strongly perfect graphs form an interesting class of perfect graphs as the complement of strongly perfect graphs need not be perfect, unlike perfect graphs.

Deﬁnition 3.1.2. [36] A graph G co-strongly perfect if G and its complement are strongly perfect.

Deﬁnition 3.1.3. [1] A graph G is H−free if G does not contain any induced subgraph isomorphic to H.

Deﬁnition 3.1.4. [36] A graph is G is Meyniel if every odd cycle of length atleast 5 in G has at most two chords. 20 Deﬁnition 3.1.5. [1] Circumference of a graph G, denoted by c(G), is the length of largest induced cycle in G.

Deﬁnition 3.1.6. [1] The total graph T(G) of G has vertex set V(G) ∪ E(G), and two points of T(G) are adjacent whenever they are neighbors in G.

Deﬁnition 3.1.7. [36] A complete subgraph (or independent set) in G is good if it meets all the maximal stable sets (or complete subgraphs) in G.

3.2 The Results

In major works of graph theory, we often ﬁnd the description or characterization of graph families with respect to the set of graphs that do not belong to the family or are forbidden to appear. Different graph families vary in the forbidden structures. Berge’s famous strong perfect graph theorem is an excellent characterization of perfect graphs in terms of forbidden subgraphs. However, the problem of attempting to characterize strongly perfect graphs in terms of forbidden subgraphs, originally posed by Berge has remained an open question for a long time and is worth considering. Musing on this, one of the conjectures given by Ravindra[21] is as follows:

Conjecture 3.2.1. If G is C5 + e-free and Cn-free, n ≥ 5, then G is co-strongly perfect. The graph G in Figure 3.1 shows that the converse of the conjecture need not be true.

Figure 3.1: G is co-strongly perfect

3.2.1 Complete Multipartite Co-strongly Perfect Graphs

Let every block of a graph G is complete multipartite, we show that it satisﬁes the statement of Conjecture 3.2.1. Let C be a cycle in G. Then C will not have a chord (or else the maximal 21 subgraph of C will be C + e which will form a block, a contradiction). Hence G is C5 + e-free and Cn−free, n ≥ 5. The following theorem shows that if every block of a graph is complete multipartite, then it is co-strongly perfect.

Theorem 3.2.1. If every block of a graph G is complete multipartite (Kr1,r2,...,rk , k ≥ 2), then G is co-strongly perfect.

Proof. It is enough to prove that for every connected induced subgraph H of G, H has a good clique and a good independent set. If H has no cut vertex, then H is a block and is complete multipartite graph. Let S be a partition of H. Since H is complete multipartite, every maximal clique will have a vertex in S. So S is good independent set in H. Let Q be a maximal complete subgraph of H. Then Q meets each of the partitions of H which are the only maximal independent sets of H, since it is complete multipartite.

So let G have a cut vertex, say v. Then we ﬁrst show that H has a good clique. Let B1,B2,...,Bl be the blocks of H. Without loss of generality B1 be an end block and v be the cut vertex in B1. This implies that

1. Every vertex in the component B1 −v is non-adjacent to every vertex in the other components of G − v.

2. The partitions Vik of a block Bi,∀i are the only maximal independent sets of H and each

of the maximal cliques of Bi meets all the Vik s.

Let Q be a maximal clique in B1 containing v. By (1), if S is an independent set of H, then either V1l ⊆ S, v ∈/ V1l or V1v ⊆ S, V1v is the partition in Bi containing v. So by (2), Q meets S and Q is a good clique in H. Next we show H has a good independent set. Let S be a maximal independent set of H S S 0 containing v. Then S = Vi ∪ Vi 0 , k is ﬁxed. Every maximal clique Q meets either v k v∈/Vi 0 k

Viv or Vik , i.e. S meets all the maximal cliques Q. So S is a good independent set in H.

Note 3.2.2. If every block of G is co-strongly perfect, then G need not be co-strongly perfect.

The graph in Figure 3.2 is a forbidden subgraph for a line graph G to be co-strongly perfect[37]. Hence even if every block of G is co-strongly perfect, G is not co-strongly perfect.

Corollary 3.2.1. If m ≤ n, G is co-strongly perfect if and only if at most one block is C3 or C4 and others are K2.

22 Figure 3.2: Every block of G is co-strongly perfect, but G is not co-strongly perfect.

Corollary 3.2.2. G is co-strongly perfect if every block of G is K2 or K3.

These graphs (every blocks are K2 or K3) are of special interest because of the following fact:

Fact 3.2.3. For a total graph T(G) of G the following properties are equivalent.[14, 37]

(i) T(G) is perfect.

(ii) T(G) is strongly perfect.

(iv) Every block of G is either K2 or K3.

Theorem 3.2.4. If every block of G is either K2 or K3, then every induced subgraph of T(G) has a clique which meets all its maximal independent sets.

Proof. Let H be an induced subgraph of T(G). Let H0 be a subgraph of G such that T(H0) = H. 0 Since every block of G is either K2 or K3, the same holds good for H . Since every block of 0 0 G is either K2 or K3, the same holds good for H also. If H has K2 as an end block, this gives rise to unicliqual point in H and since unicliqual vertex meets every maximal independent set, hence the clique containing the point meets all the maximal stable sets in H. So consider an end 0 block T of H which is K3. Let V(T) = {v1,v2,v3}, e1 = (v1,v2), e2 = (v2,v3), e3 = (v3,v1) and also let v1 and v2 be unicliqual. We now show that the clique Q induced by the points v1,e1 and v2 in H meets all the maximal stable sets in H. Now N(e1,H) = {v1,v2,e2,e3}. Let S be a maximal independent set in H. If e1 ∈/ S, N(e1,H) ∩ S 6= φ. By the choice of T obviously

N(v1,H) ⊆ N(e2,H) and N(v2,H) ⊆ N(e3,H). This implies that

(i) if e2 ∈ S, then v1 ∈ S and

(ii) if e3 ∈ S, then v2 ∈ S. 23 Thus we conclude that v1 or v2 ∈ S, whenever e1 ∈/ S and hence the clique Q meets all the stable sets of H.

The previous theorem asserts that T(G) is co-strongly perfect is also an equivalent statement to the ones mentioned in Fact 3.2.3.

Meyniel in 1987 proved that if every odd cycle of length at least 5 in a graph G has at most two chords, then the graph is perfect. Ravindra[38] called these graphs Meyniel graphs. Meyniel graphs are one of the most interesting classes of graphs in the theory of perfect graphs as they are perfect[39], strongly perfect[38] and very strongly perfect[40]. In the following theorem, we see that the graph which satisﬁes the statement of Conjecture 3.2.1 are Meyniel.

Theorem 3.2.5. If G is C5 + e−free and Cn-free, n ≥ 5, then G is Meyniel.

Proof. Suppose G is not Meyniel. Then every odd cycle of length at least ﬁve has at most one chord. If G has no odd cycles, then G is bipartite and hence Meyniel. So, let G has an odd cycle of length at least 5; call it C. If C has no chords, then C is induced odd cycle of length at least 5 in G. If C has a chord, then G has either C5 +e or a cycle of length at least 5. Thus G is Meyniel if G satisﬁes C5 + e−free and Cn-free, n ≥ 5.

If A is the set of all graphs which are Meyniel and B is the set of all graphs which satisfy the condition of the conjecture, i.e C5 + e−free and Cn-free, n ≥ 5, then by Theorem 3.2.5, B ⊆ A whereas A * B since C6 is Meyniel but C6 is not C5 +e−free and Cn-free, n ≥ 5. Meyniel graphs are not generally co-strongly perfect. Also according to the deﬁnition of Meyniel graphs it is evident that it will not contain C5 + e or C2n+1,n ≥ 2 as an induced subgraph. So the following Conjecture 3.2.2, if proved, can be the structural characterization of co-strongly perfect Meyniel graphs.

Conjecture 3.2.2. A Meyniel graph is co-strongly perfect if and only if the graph is C2n-free, n ≥ 3.

The statement of the next theorem too satisﬁes the conjecture. The next theorem encounters bipartite graphs with circumference four. Such graphs are also complete multipartite graphs.

Theorem 3.2.6. If every block is either multipartite or a bipartite graph having circumference 4, then G is co-strongly perfect.

Proof. G is obviously Meyniel, so G is strongly perfect and has a good independent set[38]. It is enough to prove that every induced subgraph of G has a good clique. Let v be a cut vertex 24 and let G1,G2,...,Gk, k ≥ 2 be the components of G − v and H1,H2,...,Hk, k ≥ 2 be the lobes of G with respect to v. If one of the blocks of G say H1 is complete multipartite with a triangle and if S is a maximal independent et of G1 and Q is a maximal clique of H1, then Q or Q − v will meet S depending on if Q or Q − v will meet S depending upon v ∈ S or v ∈/ S. Otherwise, if each of the blocks Hi is bipartite with c(G) = 4. Then by the theorem on co-strongly perfect bipartite graphs by Ravindra and Bassavaya[36], G is co-strongly perfect.

3.2.2 Paw-free Co-strongly Perfect Graphs

Lemma 3.2.7. If G has a triangle and G is paw-free, then G is complete multipartite.

Proof. Let K be a complete r−partite graph in G with 3 ≤ s ≤ r and having maximum number of vertices. Let V1 ∪V2 ∪ ··· ∪Vs be the s−partitions of G. To prove: G = K, i.e V(G) = V(K). If not, say V(G) −V(K) 6= φ and let v ∈ V(G) −V(K) such that v is adjacent to some points in K. If N(V) = N(K), then V(K) ∪ v is a complete (s + 1)−partition graph, contradicting construction of K. If otherwise let v ∈ V(G) −V(K) and v ∈ V1, such that vv1 ∈/ E(G). Since

G is paw-free and G has a triangle, therefore N(v1,K) ⊆ N(v,G), i.e. v is adjacent to all other 0 partitions of K except V1. Furthermore, v is not adjacent to any vertex of V1, otherwise if v1 ∈ V1 0 0 such that vv1 ∈ E(G), then the graph induced by {v1,v1,v2,v} forms a paw, where v2 ∈ V2. But then v can be added to V1 in K to get a complete s−partite graph with more number of vertices, contradicting the choice of K. Hence V(G) = V(K), i.e. G is complete multipartite.

Corollary 3.2.3. If G is paw-free and G has circumference 4, then G is co-strongly perfect.

Proof. If G has no triangle, then G is a bipartite graph of circumference 4 and hence G is co-strongly perfect. If G has a triangle and G is paw-free, then by Lemma 3.2.7, G is complete multipartite and So by Theorem 3.2.1, G is co-strongly perfect.

3.2.3 Co-strongly Perfect Tensor Product Graphs

Theorem 3.2.8. G = G1 × G2 is co-strongly perfect if and only if either

(i) G1 or G2 is compete r−partite, r ≥ 3 and other is K1.

(ii) G1 or G2 is bipartite of circumference 4

Proof. Let G = G1 ×G2 be co-strongly perfect. To prove that either (i) or (ii) is true. If (i) is not true, then G1 and G2 will contain K2. If say G1 is not r−partite and r ≥ 3, then G1 has a paw.

This implies G1 has an induced paw and G2 has K2. But then G = G1 ×G2 has C6 as an induced 25 C6 as an induced subgraph, contradicting that G is co-strongly perfect. If G1 is not r−partite and r ≤ 2, then G1 is bipartite. Moreover G1 and G2 cannot have induced cycle of length at least 5. If not, let G1 have Cn,n ≥ 5 as an induced subgraph. Obviously G2 will have K2 as an induced subgraph. But then, Cn × K2 will have induced cycle of length at least 6, contradicting that G is co-strongly perfect. This implies c(G1) = c(G2) = 4, and by theorem on co-strongly perfect bipartite graphs by Ravindra and Bassavaya[36], G is co-strongly perfect. Conversely, suppose either (i) or (ii) is true. If (i) is true, then G is a set of isolated vertices and hence G is co-strongly perfect. If (ii) is true, i.e. G1 and G2 are bipartite graphs and c(G1) = c(G2) = 4, then c(G1 × G2) = 4. Then by theorem on co-strongly perfect bipartite graphs by Ravindra and Bassavaya[36], G is co-strongly perfect.

Lemma 3.2.9. If G is a triangle free graph and if it does not contain an odd induced cycle of length at least ﬁve, then G is bipartite.

Proof. If G has an odd cycle, let C be an odd cycle of G with minimum (least) length (≥ 5). If C has a chord e, then e will divide the vertices of C into an odd and even cycles of lengths at least 5 and 4 respectively, contradicting minimality of C. Therefore, C is an induced odd cycle of length at least 5, a contradiction. Thus G has no odd cycle and hence it is bipartite.

Lemma 3.2.10. Cn,n ≥ 5 is not strongly perfect.

Proof. Let v1v2 ...vn be the cycle Cn. Let ei = vivi+1 (subscripts taken addition mod n) be an edge of C. e cannot meet the maximal independent set containing vi−1vi+2. This implies that no edge of Cn meets all the maximal independent sets in it. That is, Cn does not contain an independent set which meets all the cliques in it. Thus G is not strongly perfect.

3.3 Conclusion

In this chapter, we proved Conjecture 3.2.1 given by Ravindra for a few classes of graphs and studied few aspects of co-strongly perfect graphs. We also conjectured a structural characterization for co-strongly perfect Meyniel graphs and further obtained a few new results on co-strongly perfect product graphs.

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30 Paper Presentations

1. A paper entitled “Co-strongly Perfect Graphs” presented in the Second International Conference on Science, Engineering and Social Science on 18th and 19th January 2018, organized by Department of Mathematics, Kuriakose Elias College, Kottayam, Kerala.

2. A paper entitled “Path and Star Perfect Graphs” presented in the National Conference on “Advances in Applied Mathematics” on 5th February 2018, organized by Department of Mathematics, University of Science, Tumkur University, Tumakuru.