The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

On vertex neighborhood in minimal imperfect graphs

AbstractLubiw (J. Combin. Theory Ser. B 51 (1991) 24) conjectures that in a minimal imperfect Berge graph, the neighborhood graph N(v) of any vertex v must be connected; this conjecture implies a well known Chvátal's conjecture (Chvátal, First Workshop on Perfect Graphs, Princeton, 1993) which states that N(v) must contain a P4. In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P4-free if, and only if, every induced subgraph with at least two vertices either is disconnected or its complement is disconnected; this characterization implies that P4-free graphs can be constructed by complete join and disjoint union from isolated vertices. We propose to replace P4-free graphs by a similar construction using bipartite graphs instead of isolated vertices

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

Structure and coloring of (P7P_7, C5C_5, diamond)-free graphs

We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices, respectively. A diamond consists of two triangles that share exactly one edge, a kite is a graph obtained from a diamond by adding a new vertex adjacent to a vertex of degree 2 of the diamond, a paraglider is the graph that consists of a $C_4$ plus a vertex adjacent to three vertices of the $C_4$, a paw is a graph obtained from a triangle by adding a pendant edge. A comparable pair $(u, v)$ consists of two nonadjacent vertices $u$ and $v$ such that $N(u)\subseteq N(v)$ or $N(v)\subseteq N(u)$. A universal clique is a clique $K$ such that $xy \in E(G)$ for any two vertices $x \in K$ and $y\in V (G)\setminus K$. A blowup of a graph H is a graph obtained by substituting a stable set for each vertex, and correspondingly replacing each edge by a complete bipartite graph. We prove that 1) there is a unique connected imperfect $(P_7, C_5$, kite, paraglider)-free graph G with \delta(G) \geq \omega(G)+ 1 which has no clique cutsets, no comparable pairs, and no universal cliques; 2) if G is a connected imperfect $(P_7, C_5$, diamond)-free graph with \delta(G) \geq max{3, \omega(G)} and without comparable pairs, then G is isomorphic to a graph of a well defined 12 graph families; and 3) each connected imperfect $(P_7, C_5$, paw)-free graph is a blowup of $C_7$. As consequences, we show that \chi(G) \leq \omega(G)+1 if G is (P7, C5, kite, paraglider)-free, and \chi(G) \leq max{3, \omega(G)} if G is $(P_7, C_5$, H)-free with H being a diamond or a paw. We also show that \chi(G) \le

Wings and Perfect Graphs

An edge uv of a graph G is called a wing if there exists a chordless path with vertices u, v, x, y and edges uv, vx, xy. The wing-graph W(G) of a graph G is a graph having the same vertex set as G; uv is an edge in W(G) if and only if uv is a wing in G. A graph G is saturated if G is isomorphic to W(G). A star-cutset in a graph G is a non-empty set of vertices such that G — C is disconnected and some vertex in C is adjacent to all the remaining vertices in C. V. Chvátal proposed to call a graph unbreakable if neither G nor its complement contain a star-cutset. We establish several properties of unbreakable graphs using the notions of wings and saturation. In particular, we obtain seven equivalent versions of the Strong Perfect Graph Conjecture

Colouring on hereditary graph classes

The graph colouring problems ask if one can assign a colour from a palette of colour to every vertex of a graph so that any two adjacent vertices receive different colours. We call the resulting problem k-Colourability if the palette is of fixed size k, and Chromatic Number if the goal is to minimize the size of the palette. One of the earliest NP-completeness results states that 3-Colourability is NP-complete. Thereafter, numerous studies have been devoted to the graph colouring problems on special graph classes. For a fixed set of graphs H we denote F orb(H) by the set of graphs that exclude any graph H ∈ H as an induced subgraph. In this thesis, we explore the computational complexity of graph colouring problems on F orb(H) for different sets of H.In the first part of this thesis, we study k-Colourability on classes F orb(H) when H contains at most two graphs. We show that 4-Colourability and 5-Colourability are NPcomplete on F orb({P7}) and F orb({P6}), respectively, where Pt denotes a path of order t. These results leave open, for k ≥ 4, only the complexity of k-Colourability on F orb({Pt}) for k = 4 and t = 6. Secondly, we refine our NP-completeness results on k-Colourability to classes F orb({Cs, Pt}), where Cs denotes a cycle of length s. We prove new NP-completeness results for different combinations of values of k, s and t. Furthermore, we consider two common variants of the k-colouring problem, namely the list k-colouring problem and the pre-colouring extension of k-colouring problem. We show that in most cases these problems are also NP-complete on the class F orb({Cs, Pt}). Thirdly, we prove that the set of forbidden induced subgraph that characterizes the class of k-colourable (C4, P6)-free graphs is of finite size. For k ∈ {3, 4}, we obtain an explicit list of forbidden induced subgraphs and the first polynomial certifying algorithms for k-Colourability on F orb({C4, P6}).We also discuss one particular class F orb(H) when the size of H is infinite. We consider the intersection class of F orb({C4, C6, . . .}) and F orb(caps), where a cap is a graph obtained from an induced cycle by adding an additional vertex and making it adjacent to two adjacent vertices on the cycle. Our main result is a polynomial time 3/2-approximation algorithm for Chromatic Number on this class

On hereditary graph classes defined by forbidding Truemper configurations: recognition and combinatorial optimization algorithms, and χ-boundedness results

Truemper configurations are four types of graphs that helped us understand the structure of several well-known hereditary graph classes. The most famous examples are perhaps the class of perfect graphs and the class of even-hole-free graphs: for both of them, some Truemper configurations are excluded (as induced subgraphs), and this fact appeared to be useful, and played some role in the proof of the known decomposition theorems for these classes. The main goal of this thesis is to contribute to the systematic exploration of hereditary graph classes defined by forbidding Truemper configurations. We study many of these classes, and we investigate their structure by applying the decomposition method. We then use our structural results to analyze the complexity of the maximum clique, maximum stable set and optimal coloring problems restricted to these classes. Finally, we provide polynomial-time recognition algorithms for all of these classes, and we obtain χ-boundedness results