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

Recognizing graphs close to bipartite graphs.

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

Cluster Deletion on Interval Graphs and Split Related Graphs

In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (P_5-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of the algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of P_5-free chordal graphs. Although the later result arises from the already-known reduction for P_5-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs

The Orthology Road: Theory and Methods in Orthology Analysis

The evolution of biological species depends on changes in genes. Among these changes are the gradual accumulation of DNA mutations, insertions and deletions, duplication of genes, movements of genes within and between chromosomes, gene losses and gene transfer. As two populations of the same species evolve independently, they will eventually become reproductively isolated and become two distinct species. The evolutionary history of a set of related species through the repeated occurrence of this speciation process can be represented as a tree-like structure, called a phylogenetic tree or a species tree. Since duplicated genes in a single species also independently accumulate point mutations, insertions and deletions, they drift apart in composition in the same way as genes in two related species. The divergence of all the genes descended from a single gene in an ancestral species can also be represented as a tree, a gene tree that takes into account both speciation and duplication events. In order to reconstruct the evolutionary history from the study of extant species, we use sets of similar genes, with relatively high degree of DNA similarity and usually with some functional resemblance, that appear to have been derived from a common ancestor. The degree of similarity among different instances of the “same gene” in different species can be used to explore their evolutionary history via the reconstruction of gene family histories, namely gene trees. Orthology refers specifically to the relationship between two genes that arose by a speciation event, recent or remote, rather than duplication. Comparing orthologous genes is essential to the correct reconstruction of species trees, so that detecting and identifying orthologous genes is an important problem, and a longstanding challenge, in comparative and evolutionary genomics as well as phylogenetics. A variety of orthology detection methods have been devised in recent years. Although many of these methods are dependent on generating gene and/or species trees, it has been shown that orthology can be estimated at acceptable levels of accuracy without having to infer gene trees and/or reconciling gene trees with species trees. Therefore, there is good reason to look at the connection of trees and orthology from a different angle: How much information about the gene tree, the species tree, and their reconciliation is already contained in the orthology relation among genes? Intriguingly, a solution to the first part of this question has already been given by Boecker and Dress [Boecker and Dress, 1998] in a different context. In particular, they completely characterized certain maps which they called symbolic ultrametrics. Semple and Steel [Semple and Steel, 2003] then presented an algorithm that can be used to reconstruct a phylogenetic tree from any given symbolic ultrametric. In this thesis we investigate a new characterization of orthology relations, based on symbolic ultramterics for recovering the gene tree. According to Fitch’s definition [Fitch, 2000], two genes are (co-)orthologous if their last common ancestor in the gene tree represents a speciation event. On the other hand, when their last common ancestor is a duplication event, the genes are paralogs. The orthology relation on a set of genes is therefore determined by the gene tree and an “event labeling” that identifies each interior vertex of that tree as either a duplication or a speciation event. In the context of analyzing orthology data, the problem of reconciling event-labeled gene trees with a species tree appears as a variant of the reconciliation problem where genes trees have no labels in their internal vertices. When reconciling a gene tree with a species tree, it can be assumed that the species tree is correct or, in the case of a unknown species tree, it can be inferred. Therefore it is crucial to know for a given gene tree whether there even exists a species tree. In this thesis we characterize event-labelled gene trees for which a species tree exists and species trees to which event-labelled gene trees can be mapped. Reconciliation methods are not always the best options for detecting orthology. A fundamental problem is that, aside from multicellular eukaryotes, evolution does not seem to have conformed to the descent-with-modification model that gives rise to tree-like phylogenies. Examples include many cases of prokaryotes and viruses whose evolution involved horizontal gene transfer. To treat the problem of distinguishing orthology and paralogy within a more general framework, graph-based methods have been proposed to detect and differentiate among evolutionary relationships of genes in those organisms. In this work we introduce a measure of orthology that can be used to test graph-based methods and reconciliation methods that detect orthology. Using these results a new algorithm BOTTOM-UP to determine whether a map from the set of vertices of a tree to a set of events is a symbolic ultrametric or not is devised. Additioanlly, a simulation environment designed to generate large gene families with complex duplication histories on which reconstruction algorithms can be tested and software tools can be benchmarked is presented

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Graph Algorithms and Complexity Aspects on Special Graph Classes

Graphs are a very flexible tool within mathematics, as such, numerous problems can be solved by formulating them as an instance of a graph. As a result, however, some of the structures found in real world problems may be lost in a more general graph. An example of this is the 4-Colouring problem which, as a graph problem, is NP-complete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})-minor-free which can be exploited and as such there exist algorithms which can find 4-colourings of maps in polynomial time. This thesis looks at problems which are NP-complete in general and determines the complexity of the problem when various restrictions are placed on the input, both for the purpose of finding tractable solutions for inputs which have certain structures, and to increase our understanding of the point at which a problem becomes NP-complete. This thesis looks at four problems over four chapters, the first being Parallel Knock-Out. This chapter will show that Parallel Knock-Out can be solved in O(n+m) time on P_4-free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5-free graphs. From this a dichotomy is shown on $P_k$-free graphs for any fixed integer $k$. The second chapter looks at Minimal Disconnected Cut. Along with some smaller results, the main result in this chapter is another dichotomy theorem which states that Minimal Disconnected Cut is polynomial time solvable for 3-connected planar graphs but NP-hard for 2-connected planar graphs. The third chapter looks at Square Root. Whilst a number of results were found, the work in this thesis focuses on the Square Root problem when restricted to some classes of graphs with low clique number. The final chapter looks at Surjective H-Colouring. This chapter shows that Surjective H-Colouring is NP-complete, for any fixed, non-loop connected graph H with two reflexive vertices and for any fixed graph H’ which can be obtained from H by replacing vertices with true twins. This result enabled us to determine the complexity of Surjective H-Colouring on all fixed graphs H of size at most 4