Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

Linear Time Recognition of P4-Indifferent Graphs

A simple graph is P4-indifferent if it admits a total order b > c > d. P4-indifferent graphs generalize indifferent graphs and are perfectly orderable. Recently, Hoang,Maray and Noy gave a characterization of P4-indifferent graphs interms of forbidden induced subgraphs. We clarify their proof and describe a linear time algorithm to recognize P4-indifferent graphs. Whenthe input is a P4-indifferent graph, then the algorithm computes an order < as above.Key words: P4-indifference, linear time, recognition, modular decomposition.

Results on perfect graphs

The chromatic number of a graph G is the least number of colours that can be assigned to the vertices of G such that two adjacent vertices are assigned different colours. The clique number of a graph G is the size of the largest clique that is an induced subgraph of G. The notion of perfect graphs was first introduced by Claude Berge in 1960. He defined a graph G to be perfect if the chromatic number of H is equal to the clique number of H for every induced subgraph H C G. He also conjectured that perfect graphs are exactly the class of graphs with no induced odd hole (a chordless odd cycle of greater than or equal to five vertices) or no induced complement of an odd hole, an odd anti-hole. This conjecture, that still remains an open problem, is better known as the Strong Perfect Graph Conjecture (or SPGC). An equivalent statement to SPGC is that minimal imperfect graphs are odd holes and odd anti-holes. Fonlupt conjectured that all minimal imperfect graphs with a minimal cutset that is the union of more than one disjoint clique, must be an odd hole. In this thesis we prove that any hole-free graph G with a minimal cutset C that is the union of vertexdisjoint cliques must have a clique in each component o f G — C that sees all of C. We further prove that minimal imperfect graphs with a minimal cutset that is the union of two disjoint cliques have a hole. Since the introduction of perfectly orderable graphs by Chvdtal in 1984, many classes of perfectly orderable graphs and their recognition algorithms have been identified. Perfectly ordered graphs are those graphs G such that for each induced ordered subgraph H of G, the greedy (or, sequential) colouring algorithm produces an optimal colouring of H. Hohng and Reed previously studied six natural subclasses of perfecdy orderable graphs that are defined by the orientations of the P4 ’s. Four of the six classes can be recognized in polynomial time. The recognition problem for the fifth class has been proven to be NP-complete. In this thesis, we discuss the problem o f recognition for the sixth class, known as one-in-one-out graphs. Also, we consider pyramid-free graphs with the same orientation as one-in-one-out graphs and prove that this class of graphs cannot contain a directed 3-cycle of more than one equivalence class

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image is isomorphic to a subgraph. We extend the definition of homomorphically full to oriented graphs in two different ways. For the first of these, we show that homomorphically full oriented graphs arise as quasi-transitive orientations of homomorphically full graphs. This in turn yields an efficient recognition and construction algorithms for these homomorphically full oriented graphs. For the second one, we show that the related recognition problem is GI-hard, and that the problem of deciding if a graph admits a homomorphically full orientation is NP-complete. In doing so we show the problem of deciding if two given oriented cliques are isomorphic is GI-complete

Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure

On the perfect orderability of unions of two graphs

A graph G is perfectly orderable if it admits an order < on its vertices such that the sequential coloring algorithm delivers an optimum coloring on each induced subgraph (H, <) of (G, <). A graph is a threshold graph if it contains no P4 , 2K2 or C4 as induced subgraph. A theorem of Chvatal, Hoang, Mahadev and de Werra states that a graph is perfectly orderable if it can be written as the union of two threshold graphs. In this thesis, we investigate possible generalizations of the above theorem. We conjecture that if G is the union of two graphs G1 and G2 then G is perfectly orderable whenever (i) G1 and G2 are both P4 -free and 2K2-free, or (ii) G1 is P4-free, 2K2-free and G2 is P4 -free, C4 -free. We show that the complement of the chordless cycle with at least five vertices cannot be a counter-example to our conjecture and we prove, jointly with Hoang, a special case of (i): if G1 and G2 are two edge disjoint graphs that are P4 -free and 2K2 -free then the union of G1 and G2 is perfectly orderable

Oriented coloring on recursively defined digraphs

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G=(V,A) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the chromatic number of an oriented graph is an NP-hard problem. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.Comment: 14 page