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Camby, E. (2015). Connecting hitting sets and hitting paths in graphs (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Mathématiques, Bruxelles.

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D 04084

CONSULTATION REFUSEE

ULB

LIBRE DE BRUXELLES

Faculté des Sciences Département de Mathématique

Connecting hitting sets and hitting paths

in graphs

Eglantine CAMBY

Université Libre de Bruxe

0035BE1

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UNIVERSITÉ LIBRE DE BRUXELLES Faculté des Sciences

Département de Mathématique

Connecting hitting sets and hitting paths

in graphs

Eglantine CAMBY

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences Juin 2015

Jury de Thèse

G. Joret (président), Université libre de Bruxelles, Belgium S. Fiorini (promoteur). Université libre de Bruxelles, Belgium

J. Cardinal (co-promoteur & secrétaire), Univ. libre de Bruxelles, Belgium A. Hertz, Polytechnique Montréal, Canada

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“Hard try and never give up” Author Unknown. ‘Ail things are difficult before they are easy”

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Remerciements

Après cinq années de thèse, je tiens à remercier les personnes qui m’ont accompagnée et soutenue durant cette longue période.

Cette thèse a vu le jour sous la direction de Samuel Fiorini, mon promo teur. Je tiens à le remercier de m’avoir suivi dans ce projet, d’avoir cru en mes idées et de m’avoir laissé suffisamment de liberté pour découvrir l’univers scientifique à travers le monde. Ce premier a essentiellement mis l’accent sur le côté structurel de la recherche.

Par la suite, j’ai eu l’honneur d’avoir Jean Cardinal comme co-promoteur. Etant informaticien, Jean Cardinal a attiré mon attention sur le côté algo rithmique de la recherche. Je tiens à le remercier de m’avoir initié, avec Hadrien Mélot, à la recherche via le mémoire de Master, et aussi de m’avoir guidé à travers ces années.

Ce tandem est d’abord une richesse, ensuite une occasion d’apprendre deux fois plus. Il m’a donné le moyen de me forger ma propre opinion sur base de leurs opinions, souvent complémentaires. Ce duo m’a enseigné une précision d’écriture tant au point de vue du contenu que de la forme.

Je tiens à remercier Gwenaël Joret, Alain Hertz, Oliver Schaudt et loan Todinca pour avoir accepté de faire partie de mon jury de thèse et pour leur lecture attentive de ma thèse ainsi que leurs commentaires. I would like to address spécial thanks to Oliver Schaudt for his kind and rewarding collaboration. De même, j’aimerais remercier tout particulièrement Alain Hertz et Hadrien Mélot pour m’avoir appris à accorder du crédit à lires idées.

Je remercie l’ensemble des professeurs de l’UMons pour m’avoir initié aux mathématiques et pour m’avoir donné le goût de la recherche. J’adresse mes remerciements à Olivier et Stéphanie pour m’avoir épaulé dans des moments de tristesse.

J’aimerais aussi remercier les personnes qui m’ont aidée avec des relec tures tant au niveau scientifique qu’au niveau linguistique.

Je remercie également le Fonds de la Recherche Scientifique, l’Université Libre de Bruxelles, la Fédération Wallonie-Bruxelles ainsi que mes deux pro moteurs pour le soutien financier apporté durant mes nombreuses missions scientifiques à travers le monde.

Partager le Savoir est primordial. Avec ce but, j’ai pu m’épanouir dans

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le métier d’assistante. Enseigner est une expérience enrichissante sur le plan intellectuel mais aussi humain. Je remercie tous mes étudiants, en partic ulier, Bob, Frank, Mélanie, Alex, Cédric, Biaise, Vincent, Sofiane, Lancelot, Matvei, Philippe, Corentin, Bâlint, ... Je tiens spécialement à remercier Yan et Georges pour leur soutien dans les moments difficiles mais aussi Eileen dont le chemin est intimement lié au mien.

Ces cinq dernières années n’ont pas toujours été dans la joie. Cependant, j’ai rencontré une personne formidable durant mes premiers cours du soir d’anglais : Seher, ma meilleure amie. Même si les maths sont pour elle comme une langue étrangère, Seher m’a toujours soutenue dans mes décisions et m’a toujours entourée de son amour inconditionnel. Peu importe l’endroit, peu importe le moment, je peux compter sur elle. Merci with love, Seher, de m’avoir accompagné dans ces moments agréables et/ou difficiles.

Parmi mes collègues, je remercie Thomas, Selim, Rémi, Robson, Nathann et Audrey. J’ai pu particulièrement compter sur Sarah et Carine, aliris my

Cherry et Mademoiselle, avec qui j’ai grandi, rigolé mais aussi pleuré. Je les

remercie infininement pour leur gentillesse et leur amour.

De plus, je tiens spécialement à remercier Jean-Paul. Je n’aurais pas pu espérer un meilleur supérieur que Jean-Paul. Merci pour ta bienveillance et ton amabilité.

Parmi mes artistes favoris, je cite Nadège. Merci Nadj de m’avoir ouvert ta porte lorsque j’en avais le plus besoin. Merci pour ta bonne humeur et tes délires. Merci de m’avoir parlé au bord de la piscine de Soignies il y a maintenant huit ans.

Je remercie également Alain, un autre de mes artistes favoris. Il m’a ouvert au monde du spectacle et restera probablement un de mes meilleurs fans. Merci de garder un telle admiration à mon égard.

Merci à Marie-Eve, Italia et Nadir pour tous les rayons de soleil envoyés depuis tant d’années.

Depuis l’autre bout du monde, je remercie Corine de m’avoir accueilli si chaleureusement à Montréal et de m’avoir encouragé lorsque le doute m’envahissait.

J’adresse un tout grand merci à chacun des membres de ma famille : Papa, Maman, Gabriel, Anémone, Amélie, Pierre. Chacun de vous m’a aidée à sa façon à garder le cap dans ma vie, et en particulier dans ce projet qu’est la thèse. Merci à Parrain, Charlotte et leurs enfants pour m’avoir accueilli dans leur maison comme si j’étais leur propre enfant.

Je tiens aussi à remercier Dora. Merci pour ces sept années de vie com mune, merci de me câliner lorsque mon moral est au plus bas, merci de veiller avec moi durant ces longues nuits de travail, merci d’avoir partagé ce stress qui m’habitait.

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Chapter 1 Introduction

In 1735, Leonhard Euler (1707-1783) solved a historically notable problem in discrète mathematics; the seven bridges of Kônigsberg, establishing the first resuit in graph theory. The problem was to find a closed walk Crossing exactly once each bridge in the city of Kônigsberg, in Prussia (now Kaliningrad, in Russia). We can model this with a graph; each area is represented by a

vertex and each bridge by an edge (see Figure 1.1). The desired closed walk

is called a Eulerian cycle. Euler established a necessary condition for the existence of a Eulerian cycle: the number of edges incident to every vertex must be even. Actually this condition is also sufficient, provided the graph is connected.

Figure 1.1: The seven bridges of Kô'nigsberg and the corresponding graph. Dénes Kônig (1884-1944) was another forerunner of graph theory. In 1936, he wrote the first textbook on the field and proved a well-known theorem named after him: in every bipartite graph, the maximum number of vertex-disjoint edges equals the minimum number of vertices meeting ail the edges, or more formally, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.

Another founding father of graph theory is Claude Berge (1926-2002). He is well-known in particular for his contributions to perfect graphs. A

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2 Chapter 1. Introduction

graph is perfect if the chromatic number of every induced subgraph equals the size of the largest clique of this subgraph (the chromatic number is the minimum number of colors to assign to vertices such that any two adja cent vertices hâve distinct colors, and a clique is a set of pairwise adjacent vertices). Berge proposed two conjectures on perfect graphs. Berge’s first conjecture stated that a graph is perfect if and only if its complément is perfect. The complément G of a graph G is the graph on the same vertices such that two vertices are adjacent in G if and only if they are not in G. The second conjecture is a characterization of perfect graphs in ternis of forbid- den induced subgraphs: a graph G is perfect if and only if neither G nor G contains an induced cycle Ck on k vertices, with A: ^ 5 an odd number.

After Berge, (structural) graph theory emerged as an important discipline of discrète mathematics. At the same time, major progress was achieved in computer science, especially in computational theory, whose main focus is to détermine whether a computational problem can be solved efficiently in standard models of computation.

The introduction of the class P goes back to Jack Edmonds (1934-) and Alan Cobham (1927-). This class contains ail decision problems that can be solved by a polynomial-time algorithm. Edmonds and Cobham first proposed polynomial-time solvability as a synonym for tractable.

Beyond P, the class N P contains ail decision problems whose certificate can be checked in polynomial-time. By définition, P is a subset of NP. An important open question is whether this inclusion is strict, that is, whether P 7^ N P. Stephen Cook (1939-) showed that inside NP, there are problems which are at least as difhcult as ail the problems in N P because every such problem can be reduced to it. Richard Karp (1935-) brought the concept of AP-completeness to the attention of a larger public with his famous list of 21 A^P-complete problems coming from different fields [83]. Among these was the vertex cover problem: given a graph G and a positive integer fc, détermine whether G has a vertex cover of size at most k.

The class of AP-complete problems includes the vertex cover problem and the dominating set problem, which will play a major rôle in our inves tigations. A dominating set is a set S of vertices such that every vertex not in S is adjacent to a vertex in S. The dominating set problem can be formulated as follows: given a graph G and a positive integer k, détermine whether G has a dominating set of size at most k. Both problems are spécial cases of the Jf’-hitting set problem.

Given a graph G and a collection Jif of subgraphs of G, we define an

Jif-hitting set as a set of vertices of G meeting ail subgraphs of Jif. For the

vertex cover problem, the collection Jif contains ail the edges while, for the dominating set problem, a subgraph in Jif is induced by a vertex and ail its neighbors. The J^?^-hitting set problem consists in finding an ^?^-hitting set of minimum size.

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3

A^P-complete problems, the computer science community gave up on trying to find polynomial-time algorithms for solving these problems exactly and, instead, investigated heuristics. These are efficient algorithms designed to find a “good” feasible solution to a given optimization problem. Of particular interest are approximation algorithms, which are heuristics with a guarantee on the quality of the solution.

A canonical example of an approximation algorithm is the following sim ple greedy procedure for the vertex cover problem: iteratively find an uncovered edge and add both endpoints to the vertex cover, until no uncovered edge remains. Since the number of edges in any matching is a lower bound on the size of every vertex cover, the resulting vertex cover is at most twice as large as the optimal one. This is an approximation algorithm with a

performance ratio of 2.

This thesis involves two points of view on graph theory. The first one is structural as illustrated by the contributions of Euler, Kônig and Berge described above. The second one is algorithmic. Both sides hâve their im portance and are closely related: their interplay fosters the development of graph theory. Indeed, structural results give tools to design algorithms, while algorithmic problems motivate the study of structural questions.

A typical instance of such an interaction between structural and algo rithmic results cornes from the theory of graph minors. The concept of mi- nor generalizes the concept of subgraph by allowing, besides the operations of edge-deletion and vertex-deletion, that of edge-contraction. Kuratowski proved the following theorem about planar graphs (defîned as graphs that can be drawn in the plane without Crossing internally edges); a graph is planar if and only if none of its minors includes or A3 3.

Figure 1.2: on the left and K3 3 on the right.

Kuratowski’s theorem can be generalized to graphs embedded in any given surface of More generally, from 1983 to 2004, Neil Robertson and Paul Seymour proved inter alia that every family of graphs that is closed

under minors can be characterized by a finite set of forbidden minors.

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treewidth of a graph is large, then it contains a somewhat large grid minor. In this instance, structural graph theory gives tools to develop algorithms on graphs. Typically, when a problem is easy to solve on trees, it is expected to be easy to solve on bounded-treewidth graphs.

Before explaining how this interaction materializes in this thesis, we introduce the connectivity constraint, which will be présent throughout our work. We consider the connected version of the ^-hitting set problem, in which we require the J^-hitting set to induce a connected subgraph. Clearly, the minimum size of a connected Jf’-hitting set is always at least that of an ^-hitting set. We defîne the price of connectivity as the following ratio:

„ minimum size of a connected Jf'-hitting set price of connectivity =--- ---

——;---minimum size of an ^-hittmg set

The main goal of this thesis is to study the price of connectivity for the vertex cover problem and the dominating set problem.

This is a natural structural question motivated, for instance, by the fol lowing two-phase algorithmic approach to the connected ^;?f'-hitting set prob lem: first lind an optimal solution to the Jf’-hitting set problem and then transform it into a connected o?^^-hitting set without increasing its size too much. Thus, in some sense, the structural results obtained in this thesis are motivated by an algorithmic problem. The algorithmic origin of the questions survives in our proof techniques.

As we will prove, computing the price of connectivity is complété for a class which is above NP. thus not much can be said in general about graphs with price of connectivity bounded by a given rational number r. However, the situation is completely different once we consider restricted classes of graphs, such as those defined by forbidden induced subgraphs. This explains why this thesis focuses mainly on such hereditary classes of graphs.

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1.1 OUTLINE 5

1.1 Outline

This thesis tackles three distinct topics:

1. the price of connectivity for the =;?^-hitting set problem, 2. a characterization of the class of P^-free graphs,

3. the Pfc-hitting set problem.

We now give a brief overview of the content of each chapter. Chapter 2 - Background

We give basic définitions used in the following chapters. On the complexity side, we briefly recall essential notions such as complexity classes, L- reductions and approximation algorithms, and we give elementary définitions involving graphs and hypergraphs. We then consider three famous problems involving hitting sets in graphs; the vertex cover problem, the dominating set problem and the feedback vertex set problem.

Chapter 3 - The price of connectivity and other prices

This chapter is completely devoted to giving the context of our first topic. We briefly recall related notions in mathematics and computer science such as the compétitive ratio, the price of anarchy and the price of stability. Then we State some of the previously known results on the price of connectivity;

• for the vertex cover problem, initiated by Cardinal and Levy [41,88], • for the dominating set problem, based on our Master thesis [30], • for the feedback vertex set problem, studied by Belmonte, van ’t Hof,

Kamihski and Paulusma [16,17], Grigoriev and Sitters [70], and Schwei- tzer and Schweitzer [112].

Finally, we summarize several other works comparing different variants of the domination invariant.

Chapter 4 — The price of connectivity for vertex cover

We continue the study of the price of connectivity for the vertex cover prob lem, initiated by Cardinal and Levy [41,88]. We investigate both complexity and structural aspects.

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number of this graph. More precisely, computing the price of connectivity is a 02-complete problem.

Secondly, we consider an analogous problem in which the price of connec tivity for ail induced subgraphs of G is bounded by a fixed constant r. This problem is clearly different from the previous one, at least for small values of 7’. Indeed, we find a characterization in ternis of a finite list of forbidden induced subgraphs when r 6 [1,3/2]. In order to find such results for other constants r, we define a PoC-critical graph as one that appears in the list of minimal forbidden induced subgraphs for some threshold. Towards this goal, we define the restricted subclass of PoC-strongly-critical graphs. Every PoC-critical chordal graphs is also PoC-strongly-critical. Moreover, we also characterize this class by spécial trees.

Besides, we answer the following natural question: for which rational number r is there a graph whose price of connectivity is exactly r?

Chapter 5 — The price of connectivity for domination

This chapter is split into two sections and extends the work started in our Master thesis [30].

For the vertex cover problem, we prove that, given a graph G and a constant r, deciding whether the price of connectivity of G is at most r is also 02-complete. For a fixed constant r, the following decision problem is different from the previous one: given a graph G. is the price of connectivity for every induced subgraph at most r? Similarly to the vertex cover problem, for r € [1, 3/2], we characterize the class of graphs with a ‘yes’-answer to the previous problem. Among other results, we prove that for any (Pgi C6)-free graph, the différence between the connected domination number and the domination number is at most 1.

We also introduce likewise the notion of PoC-critical graphs and PoC- strongly-critical graphs for the dominating set problem.

Finally, for ail rational number r S [1,3), we construct a graph whose price of connectivity is exactly r.

Chapter 6 - A characterization of P^-free graphs

This chapter introduces our second topic and gives a structural application of connected dominating sets. The class of P/;-free graphs can be characterized in terms of connected dominating sets: being Pk-hee is équivalent to the fact that every connected induced subgraph admits a connected dominating set which is either isomorphic to G^ or P/c_2-free.

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7

1.1 OUTLINE . , ;

Chapter 7 — 2-coIorability of hypergraphs

We give an application of our P^-free graph characterization for the 2-colorability of hypergraphs: the 2-colorability problem can be solved in polynomial time for hypergraphs with Py-îvee incidence graph. This generalizes previous results of van ’t Hof and Paulusma [122] who proved this for hypergraphs with Pg-free incidence graph. Before giving the proof, we give a brief survey of the 2-colorability problem in hypergraphs by giving a detailed account of known sufEcient conditions and by mentioning some related complexity results.

Chapter 8 - The P/j-hitting set problem

Our third and last topic concerns the JP-hitting set problem, where Jif is the set of ail paths on k vertices in a graph G. This problem is as hard as the vertex cover problem, for finding exact solutions but also in the sense of approximation algorithms. For k = 3, there exists a polynomial-time 2-approximation algorithm [120], inspired from the primal-dual method for the feedback vertex set problem [12,15,46]. At a higher level, the connection between the two problems is explained by the fact that graphs not containing P3 as a subgraph are very restricted forests. Unfortunately, for fc ^ 4, graphs containing no P/j as a subgraph may hâve cycles. This makes it more difhcult to adapt the primal-dual algorithm to the P^-hitting set problem for k ^ 4.

A ^-approximation algorithm can trivially be obtained by taking ail ver tices in an inclusion-wise maximal packing of vertex-disjoint subgraphs each isomorphic to Pfc. However, nothing better than a fc-approximation algo rithm is known for the general problem in graphs when /c ^ 4.

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Chapter 1. Introduction

.2 Contributions

• The results of Chapter 4 were obtained in collaboration with Jean Cardinal, Samuel Fiorini and Oliver Schaudt. They were presented at the 11^^ Cologne-Twente Workshop on Graphs and Combinatorial

Optimization [33] and also at GraphDay@Mons and Young Women in Discrète Mathematics. The results hâve been published in Discrète

Mathematics & Theoretical Computer Science [32].

• The results of Chapter 5, except those in the last two subsections, were obtained with Oliver Schaudt, and presented at the 12*^ Gologne-

Twente Workshop on Graphs and Combinatorial Optimization [37]. Be-

sides, they hâve been published in Discrète Applied Mathematics [36]. • The characterization of P^-free graphs and its application to the 2-

colorability of hypergraphs (Chapter 6 and 7) were originally presented at 40*^ International Workshop on Graph-Theoretic Concepts in Com

puter Science [35] (WG2014). The paper received a“best paper award”

in WG2014 and resulted in a publication in Algorithmica [34]. This is also joint work with Oliver Schaudt.

• The work on the P/j-hitting set problem (Chapter 8) was done in col laboration with Jean Cardinal, Mathieu Chapelle, Samuel Fiorini and Gwenaël Joret. It was presented at the 9*^ International colloquium

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Chapter 2 Background

In this section, we give a brief introduction of basic définitions and graph- theoretic concepts. Since ail relevant définitions are listed in the index of this thesis, any person familiar with graph theory can skip the current chapter and proceed to Chapter 3. We use the notations of Diestel [50]. A good introduction to the theory of computation is given by Sipser [114].

2.1 Computational complexity

In computational complexity theory, we distinguish two main classes of prob- lems: decision problems and optimization problems. A decision problem is a problem where the expected answer is “yes” or “no”, whereas an optimization problem involves a set of feasible solutions, and the expected answer is one whose value of the objective function is optimal. An optimization problem is a minimization (resp. maximization) problem when its objective function must be minimized (resp. maximized). Note that the value of the objec tive function for a feasible solution is its objective value. Besides, for each optimization problem, there exists a corresponding decision problem where the inputs are the input of the optimization problem and some constant. Then the form of the corresponding decision problem is: “Does there exist a feasible solution whose objective value is bounded by the constant?”.

For instance, the set cover problem is an optimization problem. An instance of the set cover problem is an ordered pair ([/, 5), where U is called the universe and 5 is a family of subsets of U whose union equals the universe. A set cover, a feasible solution, is a subfamily of S whose union equals the universe. The objective function is the size of the subfamily. The set cover problem is a minimization problem. For some inputs {U, S) and some constant k, we can consider the decision problem: “Does there exist a set cover of U with at most k subsets?”.

Some problems can be solved efficiently by an algorithm. The compu tational complexity theory classifies problems into classes according to the

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10 Chapter 2. Background

effectiveness of their resolution. We présent three classes encountered in this thesis.

2.1.1 Classes P and N P

The classes P and N P are two of the most fundamental complexity classes. A decision problem is in P if it can be solved by a deterministic algorithm in polynomial time, whereas the class N P contains ail decision probleras solvable by a non-deterministic algorithm in polynomial time. In practice, the former concerns efhciently solvable or tractable problems. However, the latter ensures that checking the feasibility of a ‘yes’-instance can be done by a deterministic algorithm in polynomial time. We note the set of decision problems that can be solved by a deterministic algorithm in a running time

0{f{n)) by DTIME{f{n)). Consequently, P =[j DTIME{v!^).

ken

For instance, primality testing is in P [1], whereas the decision problem for set cover is in NP.

A major unsolved problem in computer science is whether P ^ NP. If

P ^ NP, some N P problems would be harder to compute than to verify.

One significant advance on this question came by distinguishing certain problems in NP\ if there exists a polynomial-time algorithm solving any of these problems, ail problems in N P would be solvable in polynomial time. We call these problems N P-complété. The set cover problem is NP- complete.

An A^P-complete problem is a problem in N P which is NP-hard: at least as hard as the hardest problems in N P. Formally, a problem V\ is

NP-hard if for every problem P2 in d^P, there exists a polynomial-time

réduction from V2 to V\. A polynomial-time réduction from V2 to 'Pi is a polynomial-time algorithm transforming an input x to problem V2 into an input y to problem V\ such that x is a ‘yes’-instance of V2 if and only if y is a ‘yes’-instance of V\. We dénoté the réduction from problem V2 to

V\ by V2 Pi- In practice, proving the A^P-completeness of a problem P

consists of proving that P G N P and V P for one A^P-complete problem P', because of the transitivity of the polynomial-time réduction.

2.1.2 Class ©2

To compare the difïiculties of problems, we introduce the concept of an oracle

machine', this is an algorithm with a black box, the oracle, which is able

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2.2 Approximation algorithms 11

The class ©2, sometimes denoted by pNP[iog]^ is defined as the class of decision problems solvable in polynomial time by a deterininistic algorithm that allows using 0{\ogn) many queries to an AP-oracle, where n is the size of the input. Clearly, NP is included in 02.

A Q^-complete problem is in 02 and is 02-hard. Similarly to an NP- complete problem, a problem V is ©^-complété if P € ©2 and V V for one ©2-complete problem V. Spakowski and Vogel [116] proved the ©2- completeness of the following decision problem: “given two graphs Gi and

G2, is t(Gi) ^ t(G2)?”, where t(G) is the minimum number of vertices of G meeting ail the edges of G, i.e. the minimum number of vertices of a vertex cover.

2.2 Approximation algorithms

Because of such intractability concerns, alternative methods were developed for optimization problems: heuristics are algorithms designed for finding an approximate solution when an exact solution is out of reach. Approxi

mation algorithms are polynomial-time heuristics whose solution is a good

approximation of the optimal one(s). The guarantee of the quality of this approximation is measured by the performance ratio. Given a feasible solu tion X provided by an approximation algorithm, the performance ratio q of a minimization problem (resp. maximization problem) is an upper (resp. lower) bound on the ratio j^opt) ■ ^ Oif{OPT) (resp. af{OPT) ^ f{x)), where / is the objective function and OPT is an optimal solution. In this case, we say that the optimization problem is a-approximable or that the algorithm is an a-approximation.

For instance, the set cover problem is if (n)-approximable [47], where — X^fc=i 1/^ is the harmonie number. This is achieved by using the greedy algorithm: iteratively choose a set that contains the largest number of uncovered éléments.

By analogy to polynomial-time réductions in the case of decision prob lems, an L-reduction compares two optimization problems V1.V2 with ob jective functions /i,/2 and is defined by a quadruple (g.h.lS.'f) as follows:

• for every instance x oiVi, g computes in polynomial time an instance

g{x) of V2,

• for every feasible solution y to g{x), h computes a feasible solution h{y)

of X in polynomial time, •

• for every instance x of V\, f2{OPTg(^^'^) ^ ^f\{OPTx), where OPTx (resp. OPTgf^x)) is an optimal solution for the instance x (resp. g{x)). • for every feasible solution y to g{x),

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In this case, V\ is said to be L-reducible to V2 (denoted by Vi ^2)- Note that the constants /3 and 7 allow to preserve a good performance ratio. For instance, \îV\ V2 with 0 = j = 1 and V2 is a-approximable, then Vi is also a-approximable.

2.3 Graphs and hypergraphs

A graph is an ordered pair G = (F, E) where V and E are finite sets and E is a set of subsets of V containing exactly two éléments of V. We suppose that V and E are disjoint. The éléments of V are called vertices and those of E edges. lî uv € E with vertices u and v, then u and v are called the

endpoints of the edge uv. When the vertex or edge set of a graph G are not

specified, we dénoté by V{G) its vertex set and E{G) its edge set. Given a graph G, the number of vertices |F(G)| is its order.

Two edges e and / are adjacent if e fl / 7^ 0. Two vertices u and v are

adjacent if E contains the edge uv. In this case, we say that u and v are neighbors. The neighborhood Nc{v) (sometimes called the open neighbor- hood) of a vertex v is the set of ail its neighbors. If we add the vertex v

to this set, we obtain the closed neighborhood Ng[v] of u in G. A private

neighbor of a vertex v with respect to a vertex set 5 is a vertex u ^ S such

that Ng{u) n 5 = {u}. The degree of a vertex v, denoted by dciv), is the number of its neighbors. A degree-0 vertex is isolated whereas a degree-1 vertex is pendent . A graph whose vertices ail hâve the same degree d is

d-regular. We say that G is cubic if it is 3-regular. The maximum degree

(resp. minimum degree) of a graph G, denoted by A(G) (resp. <^(G)), is the maximum (resp. the minimum) degree of ail vertices of G. We omit the graph G from the previous notations, for instance N{v),N[v],d{v), if there is no possible confusion.

A path in a graph G is a sequence U1U2U3 ■ ■ - Vk oi k distinct vertices with 6 E for any 1 ^ i < A:. The length of the path V\V2V^ ■ • -Vk is the number of its edges (i.e. k~l) and the path V1V2V3 ■ ■ - Vk connects vertices and Vk- An induced path is a path V1V2V3 ■ ■ - Vk such that Vi is not adjacent to Vj for any j ^ {i — 1, f -H 1}. By abuse of notation, the graph formed by a sequence V1V2V3 ■ ■ - Vk is also called a path and is denoted by Pk- The

distance between two vertices u and v, denoted by d{u,v), is the minimum

length of a path connecting them. The distance between two vertex sets X and Y is min(d(x,y)|x £ X,y € Y}. The diameter of a graph is the largest distance between any pair of vertices. A vertex is central (or a center) in G if its largest distance from any other vertex is minimum. This distance is the radius of G.

We say that H is a subgraph of G if V{H) Ç V{G) and E{H) Ç E{G).

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2.3 Graphs and hypergraphs 13

is denoted by G[X], Moreover, an induced subgraph of G is a subgraph of

G which is induced by some set X Ç V{G). The complément of a graph G,

denoted by G, is the graph with V{G) = V{G) such that two vertices are adjacent in G if and only if they are not adjacent in G. Of course, G = G for every graph G. Let X be a subset of V{G). G — X dénotés the subgraph induced by V{G) \ X. In particular, when X is reduced to the vertex u, we dénoté it by G — v. Let T be a subset of E{G). The graph G — T is the graph such that V{G — Y) = V{G) and E{G — T) = E{G) — Y, especially if Y is reduced to the edge e, G - e is the graph obtained by rernoving from G the edge e.

A graph G is connected if there exists a path connecting each pair of vertices from G. Otherwise the graph G is disconnected. The connected

components of a graph are the inclusion-wise maximal connected subgraphs.

A set of vertices X is a cutset of G if the number of connected components of G — X is different from that of G, while a vertex u is a cutvertex of G if the set {n} is a cutset of G. The disjoint union of some graphs Gi,..., Gfc with disjoint vertex and edge sets is the graph G with V{G) = \J^^iV{Gi) and E{G) = \J^^-^E{Gi) and is denoted by Gi + G2 + • • • + Gfc. We dénoté by kG the disjoint union of k copies of G.

A cycle is a path viV2Vs ■ ■ -Vk where v\ is adjacent to v^- The length of the cycle V1V2V3 ■■ - Vk is also its number of edges, i.e. k. An induced cycle is a cycle G = viV2V^ • • -Vk such that rernoving any edge from G results in an induced path. By abuse of notation, the graph formed by this sequence

V1V2V3 ■ ■ - Vk is also called a cycle and is denoted by Ck- An acyclic graph,

which is one not containing any cycle as a subgraph, is called a forest. A connected forest is called a tree and any subgraph of a tree is a subtree. The

internai vertices of a tree are those with degree at least two. A linear forest

is a forest where each connected component is a path. A spanning tree T of a graph G is a subgraph of G which is a tree with V(T) = V(G). A maximum

leaf spanning tree is a spanning tree with a maximum number of leaves.

Two graphs G and H are isomorphic if there is a bijection (p : V{G) —>

V {H) such that uv is an edge of G if and only if f{u)(f){v) is an edge of H, for

every u,v E V{G). If two graphs G and H are isomorphic, we write G H. A (graph) parameter is a fonction / from the set of ail graphs to the natural numbers N. A graph parameter / is an invariant if /(G) = f(H) whenever the graphs G and H are isomorphic.

A clique in a graph is a vertex subset X whose vertices are pairwise adjacent. When the vertex set of a graph G is a clique, we say that G is

complété. The complété graph on n vertices is denoted by

Kn-An independent set of a graph G is a vertex set X such that 110 two of its éléments are adjacent. We call the independence number of a graph G the maximum size of an independent set. It is denoted by a(G).

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14 Chaptbr 2. Background

each part is an independent set. When k = 2, we usually obtain bipartite graphs. The complété bipartite graph, where one block is of size n and the other of size m with ail possible edges between those two blocks, is denoted by The graph is a claw while, for every fc ^ 0, the graph is a star.

A matching is a set of edges such that no two edges share an endpoint. If M is a matching such that every vertex is incident to some edge of M, then M is called a perfect matching.

The graph G is said to be H-free if no induced subgraph of G is isomor- phic to H. Furthermore, we say that G is (Hi,..., He)-îvee if G is Hi-îree for every i E {1,

A huge number of graph classes hâve been studied in literature. Below, we list a few of them:

1. A chordal graph, or triangulated graph, is a graph in which ail induced cycles hâve length 3.

2. A planar graph is a graph that can be embedded in the plane i.e. it can be drawn in in such a way that its edges are internally disjoint. 3. A split graph is a graph whose vertices can be partitioned into a clique

and an independent set.

4. A Moore graph is a d-regular graph with diameter k whose number of vertices is exactly 1 + dY^^Zai^ ~ !)*■ Notice that the number of vertices of any o?-regular graph with diameter k is upper bounded by

l + dYtoid-iy.

5. A cograph is a P4-free graph.

6. A trivially perfect graph is a (P4,C4)-free graph.

A hypergraph H is an ordered pair {V,E) where V is a finite set, called the ground set of vertices, and P is a finite set of subsets of V. The éléments of E are called hyperedges. The rank of a hypergraph H is the maximum size of its hyperedges. If ail hyperedges hâve the same size k, the hypergraph is said to be k-uniform. A graph is a 2-uniform hypergraph. The degree d//(u) of a vertex v is the number of hyperedges that contain it. H is k-regular if every vertex has exactly degree k. The (vertex-hyperedge) incidence graph of a hypergraph El = {V,E) is the bipartite graph G with vertex set F U P and edge set {ve \ v E V,e E E,v E e}. A hypergraph H = {V,E) is connected if there is no bipartition A (J B of F such that for ail e E E,

either e Ç A or e Ç P. For a hypergraph H = {V,E), a k-coloring is a map c: F —> {l,...,fc} such that every hyperedge of size at least 2 is not

monochromatic, i.e. every such hyperedge contains at least two vertices of

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2.4 ^-HITTING SET PROBLBMS 15

2.4 c^-hitting set problems

Given a graph G and a collection of subgraphs of G, we define an Jif-hitting

set as a set X of vertices such that for every H G V{H) G X ^ An

^-hitting set is minimum if its size is minimum, The M’-hitting number, denoted by t_^{G), is the size of a minimum Jf'-hitting set. For a connected

graph G, an .^-hitting set of G inducing a connected subgraph is called a

connected J^-hitting set. Assume that G is disconnected, i.e. G admits the

connected components Gi,. .. ,Gfc. Let s ^ k he the number of connected components G such that T^y/^{C) ^ 0. Then a connected J^-hitting set of G is an .;??’-hitting set whose number of connected components is exactly

s. Also, a connected Jif-hitting set whose size is minimum is a minimum connected Jif-hitting set. The size of such a set, denoted by r^^c(G), is

called the connected Jif-hitting number. An (resp. connected) cj?^-hitting set is minimal if noue of its proper subsets is an (resp. connected) J^-hitting set. The (resp. connected) ^-hitting set problem consists of finding a minimum (resp. connected) J^-hitting set. In this thesis, we are interested in three particular collections Jif whose corresponding .:?f’-hitting set problems are the vertex cover problem, the dominating set problem and the feedback vertex set problem.

The following table describes définitions and notations for these three problems, given a graph G.

Problems Vertex cover problem

Dominating set problem

Feedback vertex set problem set vertex cover set dominating set feedback vertex set

Jif ail edges ail stars induced by a closed neighborhood

ail cycles

T.J)e{G) r{G) 7(G) P{G)

r.x’.c{G) Tc{G) 7c(G) Pc{G)

(connected) (connected) (connected) (connected) J^-hitting vertex cover domination feedback vertex

number number number number

A” is a vertex cover (resp. feedback vertex set) of G if and only if V{G)\X is an independent set (resp. a forest) in G. An alternative définition of dominating sets is the following one: a dominating set of a graph G is a vertex set D such that every vertex not in D lias a neighbor in D, i.e. U,;eDAG[u] = V{G).

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2.4.1 The vertex cover problem

The vertex cover problem has been widely studied in literature and is one of the 21 N'P-complete problems identified by Karp [83] in 1972. Moreover, Garey, Johnson and Stockmeyer [63] proved that it remains 77P-complete in cubic graphs. As is well known, the vertex cover problem admits a 2- approximation algorithm, by the vertex set of an inclusion-wise maximal matching or by the internai vertex set of a depth-first search tree [109[, and better performance ratio can be achieved; 2 — ~ o(l)) [74[. Dinur and Safra [52] proved that the vertex cover problem is A^P-hard to approximate to within a factor of 1.3606, unless P = NP. Moreover, it is widely believed that this problem is hard to approximate to within 2—e under the Unique Games Conjecture [85], unless P = N P. For more explanations about the Unique Games Conjecture, see [84],

For the connected version, Fernau and Manlove [58] showed that the connected problem is not approximable within a performance ratio of 1.3606 —^ for any 5 > 0, unless P = NP. Furthermore, EscofRer, Gourvès and Mon- not [57] proved that this problem is polynomial in chordal graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomially solvable, especially in bipartite graphs.

2.4.2 The dominating set problem

Like the vertex cover problem, the dominating set problem has been inten- sively studied in literature. The dominating set problem is A^P-complete [62, p. 190] by a réduction from the vertex cover problem. Moreover, Kami [79, pp. 108-109] described a pair of polynomial-time L-reductions between the dominating set problem and the set cover problem, which préservés the performance ratio. In other words, if there exists a polynomial-time a- approximation algorithm for the dominating set problem, then the réduction gives a polynomial-time a-approximation algorithm for the set cover prob lem and vice versa. Therefore, the dominating set problem is (1 J- ln(n))- approximable by the greedy algorithm [47]. Raz and Safra [104] showed that no polynomial-time approximation algorithm can run within a ratio better than cln(n) for some c > 0 unless P = NP, for the set cover problem, hence also for the dominating set problem. Recently, Alon, Moshkovitz and Safra [4] proved a similar resuit with higher values of c, for instance when c = 0.2267. In terms of exact algorithms, Fomin, Kratsch and Woegin- ger [60] designed exponential algorithms whose computational complexity is in 0(1.93782”) whereas Grandoni [69] developed one with a better perfor mance in 0(1.8021”). Later, Fomin, Grandoni and Kratsch [80, pp. 284-286] deduced a brandi & reduce algorithm in 0(1.52626”).

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2.4 ^-HITTING SET PROBLEMS 17

Cheng [21] explained the usefulness of connected dominating sets in this context. From a theoretical point df view, the connected dominating set problem is équivalent to finding a maximum leaf spanning tree. Garey and Johnson [62, pp. 206] explained that the last problem is AiP-complete. Therefore, it is the saine for the connected dominating set problem. Guha and Khuller ]71] designed two approximation algorithms with ratio 4 + 21n(A) and 3 + ln(A) (where A is the maximum degree of G) for the connected dominating set problem, and they proved that there is no ap proximation algorithm with performance ratio pH{A) for p < 1 unless

N P Ç where H is the Harmonie function. While

Ruan, Du, Jia, Wu, Li and Ko ]108] developed an approximation algorithm with performance ratio 2 -h ln(A), Du, Graham, Pardalos, Wan, Wu and Zhao [54] showed that there exists an approximation algorithm with perfor mance ratio a(l -t- ln(A — 1)), for any a > 1.

2.4.3 The feedback vertex set problem

Karp [83], and more generally Lewis and Yannakakis [89], proved the NP- completeness of the feedback vertex set problem. Moreover, approximation algorithms [12,15,46] with performance ratio 2, for instance by the primal- dual method, or exact exponential algorithm ]59] in 0(1.7548”) were de signed. However, Guruswami and Lee [73] proved recently the strong NP- hardness of approximation resuit for a variant: under the Unique Games Conjecture, for any integer k ^ 3 and e > 0, it is hard to find a (/c — e)- approximate solution to the problem of intersecting every cycle of length at most k.

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Chapter 3 The price of connectivity and

other priées

The price of connectivity, abbreviated by PoC, expresses the interdependence of the connected version of a graph invariant and the original invariant. Many authors studied other priées by comparing a graph invariant with some variants of this invariant or by comparing varions variants of one graph invariant. This chapter retraces related works and is split into three sections. The first one mentions fanions prices in mathematics or cornpnter science, in different areas compared to graph theory. The second one is dedicated to the price of connectivity, especially on the vertex cover problem, the dominating set problem and the feedback vertex set problem. The last one investigates other prices involving domination nnmbers.

3.1 Famous prices

Comparisons between related parameters of discrète strnctnres are nbiqni- tons in mathematics and cornpnter science. We présent a brief snrvey of some popnlar ones.

In cornpnter science, alongside the performance ratio for approximation algorithms, the compétitive ratio deals with on-line algorithms in a theory starting with the work of Sleator and Tarjan [115]. An on-line algorithm is one that reçoives a seqnence of reqnests and performs an immédiate action in response to each reqnest. The novelty of their paper [115] lies in a new measnre of performance, the compétitive ratio for on-line algorithms. The

compétitive ratio of an algorithm is defined as the worst-case ratio between

its cost and that of a hypothetical ofîline algorithm which knows the en- tire seqnence of reqnests in advance and chooses its actions optimally. An algorithm is compétitive if its compétitive ratio is bonnded. Compétitive algorithms are nsed to overcome nncertainties abont the fntnre, in the case of on-line reqnests from a server. Many anthors [7, 9, 10,18, 25, 82, 86, 95]

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20 Chapter 3. The price of connectivity and other pricbs

developed compétitive algorithms and proved upper and lower bounds on the compétitive ratios achievable by on-line algorithms.

In applied mathematics, game theory [8,38,97,98] has been used to study a wide variety of human and animal behaviors. The applications are man- ifold: modeling, economy, business, political science, biology, computer sci ence, logic, philosophy, ...In this context, many authors [2,6,43-45,87,90,

106,107] established two famous notions; the price of anarchy and the price of stability. A good introduction on the game theory is given by Nisan, Roughgarden, Tardos and Vazirani [98].

First of ail, the price of anarchy of a game is a concept that measures how the efficiency of a game dégradés due to selfish behavior of its players, in other words, the ratio between the worst welfare function value of one of its Nash equilibria and that of an optimal outcome. Notice that if the price of anarchy is doser to 1, choosing an arbitrary Nash equilibrium as a solution is relevant since the welfare function evaluated in any Nash equilibrium seems a good approximation to the optimal value. Some authors [43-45,87,106,107] attempt to bound the price of anarchy in particular cases. Unfortunately, a game with multiple Nash equilibria has a large price of anarchy even if only one of its equilibria is highly inefficient.

Secondly, the price of stability is a measure of inefficiency designed to differentiate between games in which ail equilibria are inefficient and those in which some equilibrium is inefficient. Formally, the price of stability of a game is the ratio between the best welfare function value of one of its Nash equilibria and that of an optimal outcome. Of course, in a game with a unique equilibrium, its price of anarchy and price of stability are identical. In general, the price of stability is relevant for games in which there is some objective authority that can partly influence the players, and can help them converge to a good Nash equilibrium. As the case of the price of anarchy, bounding the price of stability is a challenge raised by several authors [2,6,90]. Obviously, for a game with multiple equilibria, its price of stability is at least as close to 1 as its price of anarchy, and it can be much doser.

3.2 Price of connectivity

3.2.1 Vertex cover problem

The price of connectivity has been introduced by Cardinal and Levy [41, 88] for the vertex cover problem and is deflned by the ratio between the connected vertex cover number and the vertex cover number r.

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3.2 Price of connectivity 21

Observation 3.1. For every graph G it holds that Tc(G) ^ 2t(G) — 1. As an immédiate conséquence of Observation 3.1, the following inequality holds for every graph G (with at least one edge):

1 ^ Tc{G)/t{G) < 2. (3.1) Hence the price of connectivity for the vertex cover problem of any graph lies in the interval [1,2). Note that the upper bound in (3.1) is asymptotically sharp in the class of paths Pk and in the class of cycles Cfc on k vertices, in the sense that

lim Tc{Pk)/T{Pk) = 2 = lim Tc(Cfc)/r(C'fc).

«—>•00 /c—>oo

First of ail. Cardinal and Levy [41,88] showed that for the vertex cover problem, the price of connectivity in dense graphs is bounded by a constant depending on the graph density.

Theorem 3.2 (Cardinal, Levy [41,88]). Let G be a graph with at least eÇ^)

edges. Then its price of connectivity for the vertex cover problem is at most ^s+0{l).

Moreover, they proved that this theorem is tight for the family of graphs

Gx,y with y — x a, multiple of 3, defined as follows; Gx,y is the graph composed

of a clique of size x and {y — x)/3 paths on 3 vertices, ail endpoints of which are totally joined to the clique. Figure 3.1 is an illustration of G4.10 and

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Ge.is-22 Chapter 3. The price of connectivity and othbr prices

3.2.2 Dominating set problem

First of ail, Duchet and Meyniel [55] observed that for every graph G it holds that 7c(G) ^ 3^(0) — 2. As an immédiate conséquence, every graph

G satisfies

1 ^ 7c(G)/7(G) < 3, (3.2)

that is, the price of connectivity for dominating set problem of a graph G,

7c(G)/7(G), is strictly bounded by 3.

Observation 3.3. It holds that

lim ^c{Pk)/l{Pk) = 3 = lim jc{Ck)/7{Ck)- (3.3)

«—>oo k—^oc

In particular, the upper bound (3.2) is asymptotically sharp in the class of paths and in the class of cycles.

Moreover, Zverovich [126] found a characterization of a particular class of graphs. Each graph of this particular class has the price of connectivity for the dominating set problem equal to 1, like ail induced subgraphs. The following theorem is our starting point in Chapter 5.

Theorem 3.4 (Zverovich [126]). The following assertions are équivalent for

every graph G.

(i) For every induced subgraph H of G it holds that = 'j{H).

(a) G is (P5, C^)-free.

Even though the class of (P5, G5)-free graphs is recognized in polynomial time, the dominating set problem restricted to this class is AfP-complete, as proved by Bertossi [19] and by Corneil and Perl [48]. By the previous theorem, the connected dominating set problem is also A^F-complete.

During her Master thesis, Camby [30] studied the price of connectivity for the dominating set problem. She proved that for an arbitrary constant (5, there exists a sequence of graphs with minimum degree <5 and price of connectivity approaching 3 for the dominating set problem. This infinité family of graphs G* ^ is defined as follows. Consider n disjoint A'(5+i and n — 1 disjoint We place alternately these cliques along a path and we add edges as follows. In each clique iF<5+i, choose two distinct vertices, say

U, V. and add ail possible edges between u and the previous clique and also between v and the following clique Ks-\. Notice that 5(G* ^) = ô. The graph G4 3 is illustrated by Figure 3.2.

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Figure 3.2: Graph G43.

Theorem 3.5 (Camby [30]). Let n ^ 4. Consider the function f defined by

Then f is decreasing.

As a resuit, for the dominating set problem, the meiximum value of the price of connectivity for graphs of order n is attained by a graph with mini mum degree 1.

Besides, Camby found upper bounds when the minimum degree is pro- portional to the order of the graph.

Theorem 3.6 (Camby [30]). Let n ^ 4 and G be a graph of order n with

minimum degree at least n/2. Then ydG)h{G) < 2. Moreover, if 5{G) = n/2 ^ 3 then yc{G)/y{G) ^ 2 —

The last bound may be not tight. However, Camby constructed by induc tion a graph G of order 25{G) whose price of connectivity for the dominating set problem is equal to 3/2. The basic case is illustrated by Figure 3.3. We conjecture that for any graph G with d{G) = n/2, the price of connectivity for the dominating set problem is bounded by 3/2.

Figure 3.3: Craph of order 8 with minimum degree 4 and price of connectivity for the dominating set problem 3/2.

Theorem 3.7 (Camby [30]). Let n ^ 4 and 5 G Nq. Let G be a graph of

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24 Chapter 3. The price of connectivity and other prices

Furtherraore, Camby studied the price of connectivity for the dominating set problem in some particular classes of graphs.

Theorem 3.8 (Camby [30]). Let G be a graph.

• If G is a cograph, i.e. a P^-free graph, then 7c(G)/7(G) = 1.

• If the diameter of G is 2 then 7c(G)/7(G) < 2.

However, we conjecture that the upper bound on the price of connectivity for the dominating set problem for graphs of diameter 2 is 3/2. Moreover, there exists a sequence of graphs of order n + 3 with diameter 2 whose price of connectivity for the dominating set problem is exactly 3/2. An instance of these graphs, obtained by subdividing an arbitrary edge of A'2.n, is depicted in Figure 3.4.

Figure 3.4: Graph obtained by subdividing one edge from

A'2po-3.2.3 Feedback vertex set problem

Belmonte, van ’t Hof, Kamihski and Paulusma [16,17] studied the price of connectivity for feedback vertex set problem, defined as the ratio between the connected feedback vertex number pc and the feedback vertex number p.

In general, the price of connectivity can be arbitrarily large for the feed back vertex set problem, for instance for butterflies. A butterfly is a graph consisting of two disjoint cycles on i and j vertices that are connected to each other by a path of length k. We dénoté it by The butterfly

84^8,2 is illustrated by Figure 3.5. Notice that for the vertex cover problem

and the dominating set problem, the price of connectivity for any graph is bounded by a fixed constant. At the opposite, the price of connectivity for the feedback vertex set problem can be arbitrarily large since its exact value for the butterfly is {k + 2)/2.

Observe that butterflies are planar. However, Grigoriev and Sitters [70] showed that the price of connectivity for feedback vertex set problem is at most 11 for planar graphs of minimum degree at least 3. Later, Schweitzer and Schweitzer [112] improved this upper bound down to 5, which is tight.

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Figure 3.5: Butterfly

B4^g.2-Theorem 3.9 (Belmonte et al. [17)). Let H be a graph.

• There is a constant dfj such that Pc{G) ^ df{p{G) for every connected

H-free graph G if and only if H is a linear forest.

• There is a constant ch such that pc{G) ^ p{G)+ch for every connected H-free graph G if and only if H is an induced subgraph of P^ + sP\ or sP^ for some integer s.

• Pc{G) = p{G) for every connected H-free graph G if and only if H is

an induced subgraph of P3.

• For every constant en, there is a H-free graph G with Pc(G) > ef{p{G)

if and only if H contains a cycle as a subgraph or a vertex of degree at least 3.

Afterwards, Belmonte et al. [16] generalized their results when the list of forbidden induced subgraphs is flnite. Let us introduce the following deflnition to explain the following resuit.

Let i,j ^ 3 be two integers, let Jif be a flnite family of graphs, and let 77 = 1 + 2max//g,^ \V{H)\. The family covers the pair {i,j) if

contains an induced subgraph of the butterfly A graph H covers the

pair {i,j) if the family {H} covers (i,j).

The following theorem states that the price of connectivity for feedback vertex set problem in the class of J^f-bee graphs is bounded by a constant

if and only if the forbidden induced subgraphs in .Jif prevent arbitrarily large butterflies from appearing as induced subgraphs.

Theorem 3.10 (Belmonte et al. [16]). Let -JiL be a flnite family of graphs.

Then the price of connectivity for feedback vertex set problem in -free graphs is upper bounded by a constant if and only if JP covers the pair {i,j) for every i,j ^ 3.

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simple when the complété graph bas only 3 vertices and is denoted by L„.

A simple lollipop is eaten if it is the subgraph of L„ obtained by removing the edge to hâve ho degree-2 vertex in the clique. An eaten simple lollipop is denoted by These graphs are illustrated in Figure 3.6. Notice that Lq is isomorphic to C3, the cycle on 3 vertices.

Figure 3.6: On the left, the simple lollipop Le and on the right, the eaten simple lollipop

Is-Theorem 3.11 (Belmonte et al. [16]). Let H\ and H2 be two graphs, and

let Then the price of connectivity for feedback vertex set problem in J^-free graphs is upper bounded by a constant gjf if and only if there exist integers n ^ 0 and r ^ 1 such that one of the following conditions holds:

- H\ or H2 is a linear forest,

- Hi and H2 are induced subgraphs of Ln and 2/^, respectively, - H\ and H2 are induced subgraphs of 2Ln and Ir, respectively, where 2G is the disjoint union of two copies of G.

Clearly, Theorem 3.11 generalizes Theorem 3.9, as the class of iî-free graphs is équivalent to the class of {H, H}-îree graphs. They pointed out that any graph H that is an induced subgraph of both Ln for some n ^ 0 and of 2Ir for some r ^ 1 is a linear forest.

3.3 Other prices

We can also examine other pair of invariants in graph theory. For instance, Fulman [61] and Zverovich [127] investigated the ratio between the indepen- dence number and the upper domination number, which is the maximum size of an inclusion-wise minimal dominating set. However, we stay focused on our topic: pair of parameters involving the same invariant.

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3.3 Other prices 27

3.3.1 Paired-domination versus total domination

Haynes and Slater [75] gave the following relation between the total domi nation number 7( and the paired-domination number 7p. A total dominating

set Ai is a vertex set such that every vertex, even in X, has a neighbor in X

whereas a paired-dominating set is a dominating set whose induced subgraph has a perfect matching. Moreover, the total domination number, resp. the

paired-domination number, is the minimum size of such a dominating set.

Theorem 3.12 (Haynes, Slater [75]). Let G be a graph with mmimum degree

at least 1. Then

7p(G) ^ 27t(G) - 2.

Therefore, for any graph G with minimum degree at least 1,

7t(G)

This bound is asymptotically sharp in the sense that

lp{o'{Ki^r)) ^____‘2_ __ ^ 2

7t(cr(ATi^r)) rl r-^+cc

where the corona cr{G) of a graph G is the graph obtained from G by attaching a pendent vertex to every vertex. Dorbec, Henning and Mc- Coy ]53] proved similar results for the ratio Tp/Ft between the upper paired- domination number and the upper total domination number, where the upper

paired-domination number Fp is the maximum size of an inclusion-wise min

imal paired-dominating set while the upper total domination number Ff is the maximum size of a minimal total dominating set.

Restricted to some classes of graphs, the upper bound on the ratio jp/jt could be smaller. Indeed, Brigham and Dutton [29] showed that for iFi_3- free graphs with minimum degree at least 1, the bound 7p/7t ^ 4/3 holds, whereas Schaudt [111] generalized the upper bound to the class of Xi r-free graphs.

MG)

Theorem 3.13 (Schaudt [111]). Let G be a Ki^r-free graph with minimum

degree at least 1, for some r ^ 3. Then MG) MG)

(36)

Unfortunately, the second upper bound is possibly not sharp. However, Schaudt enhanced the l2ist theorem by finding a class of graphs where both upper bounds are tight. It is the class of (C5, i/r)-free graphs. for some r ^ 3, where Hr is the graph obtained from Ki^r by subdividing each edge exactly once. As a conséquence of Theorem 3.13, Schaudt also obtained the following corollary.

Corollary 3.14 (Schaudt [111]). Let G be a graph with minimum degree at

least 1 and maximum degree A,

7p(G) ^ g 2

7t(G) " A + 1 and this bound is sharp. Moreover,

rp(G) ^. 2 rt(G)A + r

Besides, Schaudt established the following characterization on the ratio

Tp/Tt-Theorem 3.15 (Schaudt [111]). Let G be a graph with minimum degree at

least 1. The following assertions are équivalent.

(i) Jp{H) ^ Tt{H) for any induced subgraph H of G with minimum degree at least 1.

(iij G is {Gs, cr{K^), cr(P3))-free (see Figure 3.7).

Figure 3.7: Cs, cr(K3) and cr(P3).

Furthermore, similar to Theorem 3.13, Schaudt [111] proposed the fol lowing theorem for the ratio 7p/Ft.

Theorem 3.16 (Schaudt [111]). Let G be a cr{Ki^r)-ff'&e graph with mini

mum degree at least 1, for some r ^ 3. Then 7p(G)

(37)

3.3 Other prices 29

In particular, Schaudt found the following corollary.

Corollary 3.17 (Schaudt [111]). Let G be a connected graph with minimum

degree at least 1 and maximum degree A ^ 2 that is not isomorphic to C5, IpjG)

rt(G) ^2 _2 Â

and this bound is sharp.

3.3.2 Connected domination versus total domination

A graph is called perfect if the chromatic number of every induced subgraph equals the size of the largest clique of this subgraph, where the chromatic

number is the minimum value k such that the graph admits a fc-coloring.

More generally, we can define perfection with other parameters. For in stance, we can consider the domination number and the minimum size of a dominating set whose induced subgraph is indépendant. Zverovich and Zverovich [128] gave a minimal forbidden subgraph characterization of such domination perfect graphs. We recall that Zverovich [126] gave a charac terization of perfect graphs for the connected domination number and the domination number, which said that 7 = 7c for any induced subgraph is équivalent to being (C5, Pâ)-free. We define the clique-domination number

'fci as the minimum size of a dominating set whose induced subgraph is a

clique. Goddard and Henning [64] extended Zverovich’s resuit to total dom ination and clique-domination.

Theorem 3.18 (Goddard, Henning [64]). Let G be a graph. The following

assertions are équivalent.

(i) Every connected induced subgraph has a dominating clique.

(ii) 'y{H) = ^t{H) for any connected induced subgraph H with j{H) = 2. (iii) -y{H) = for any connected induced subgraph H with 'y(H) ^ 2. (iv) 7(/f) = jcl{H) for any connected induced subgraph H with 'y{H) ^ 2.

(v) 'y(H) = ^c{H) for any connected induced subgraph H with 'y(H) ^ 2. (vi) G is {Gô, Pô)-free.

Recently, Schaudt [110] studied the interdependence between the con nected domination number and the total domination number, as explained in the following theorem. In what follows, we define a connected graph as

non-trivial if it is not an isolated vertex.

Theorem 3.19 (Schaudt [110]). Let G be a graph. The following assertions

Disponible à / Available at permalink :

D 04084

CONSULTATION REFUSEE

ULB

Connecting hitting sets and hitting paths

in graphs

Eglantine CAMBY

Connecting hitting sets and hitting paths

in graphs

Eglantine CAMBY

Remerciements

Contents

Chapter 1

Introduction

1.1 Outline

.2 Contributions

Chapter 2

Background

2.1

Computational complexity

2.2 Approximation algorithms

2.3

Graphs and hypergraphs

2.4

c^-hitting set problems

Chapter 3

The price of connectivity and

other priées

3.1

Famous prices

3.2

Price of connectivity

3.3

Other prices