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Cette thèse réunit en trois articles mon intérêt éclectique pour la théorie des graphes. Le premier problème étudié est la conjecture de Erdos-Faber-Lovász: La réunion de k graphes complets distincts, ayant chacun k sommets, qui ont deux-à-deux au plus un sommet en commun peut être proprement coloriée en k couleurs. Cette conjecture se caractérise par le peu de résultats publiés. Nous prouvons qu’une nouvelle classe de graphes, construite de manière inductive, satisfait la conjecture. Le résultat consistera à présenter une classe qui ne présente pas les limitations courantes d’uniformité ou de régularité. Le deuxième problème considère une conjecture concernant la couverture des arêtes d’un graphe: Si G est un graphe simple avec alpha(G) = 2, alors le nombre minimum de cliques nécessaires pour couvrir l’ensemble des arêtes de G (noté ecc(G)) est au plus n, le nombre de sommets de G. La meilleure borne connue satisfaite par ecc(G) pour tous les graphes avec nombre d’indépendance de deux est le minimum de n + delta(G) et 2n − omega(racine (n log n)), où delta(G) est le plus petit nombre de voisins d’un sommet de G. Notre objectif a été d’obtenir la borne ecc(G), This thesis represents in three articles my eclectic interest for graph theory. The first problem is the conjecture of Erdos-Faber-Lovász: If k complete graphs, each having k vertices, have the property that every pair of distinct complete graphs have at most one vertex in common, then the vertices of the resulting graph can be properly coloured by using k colours. This conjecture is notable in that only a handful of classes of EFL graphs are proved to satisfy the conjecture. We prove that the Erdos-Faber-Lovász Conjecture holds for a new class of graphs, and we do so by an inductive argument. Furthermore, graphs in this class have no restrictions on the number of complete graphs to which a vertex belongs or on the number of vertices of a certain type that a complete graph must contain. The second problem addresses a conjecture concerning the covering of the edges of a graph: The minimal number of cliques necessary to cover all the edges of a simple graph G is denoted by ecc(G). If alpha(G) = 2, then ecc(G)

Let be the total domination number and let be the total domination subdivision number of a graph G with no isolated vertex. In this paper, we show that for some classes of graphs G, which partially solve the conjecture presented by Favaron et al.

The Randić index of a graph G, denoted by R(G), is defined as the sum of 1/d(u)d(v) for all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this note, we show that R(L(T))>n4 for any tree T of order n≥3. A number of relevant conjectures are proposed.

A Conflict-Free Open Neighborhood coloring, abbreviated CFON^* coloring, of a graph G = (V,E) using k colors is an assignment of colors from a set of k colors to a subset of vertices of V(G) such that every vertex sees some color exactly once in its open neighborhood. The minimum k for which G has a CFON^* coloring using k colors is called the CFON^* chromatic number of G, denoted by χ_{ON}^*(G). The analogous notion for closed neighborhood is called CFCN^* coloring and the analogous parameter is denoted by χ_{CN}^*(G). The problem of deciding whether a given graph admits a CFON^* (or CFCN^*) coloring that uses k colors is NP-complete. Below, we describe briefly the main results of this paper. - For k ≥ 3, we show that if G is a K_{1,k}-free graph then χ_{ON}^*(G) = O(k²log Δ), where Δ denotes the maximum degree of G. Dębski and Przybyło in [J. Graph Theory, 2021] had shown that if G is a line graph, then χ_{CN}^*(G) = O(log Δ). As an open question, they had asked if their result could be extended to claw-free (K_{1,3}-free) graphs, which are a superclass of line graphs. Since it is known that the CFCN^* chromatic number of a graph is at most twice its CFON^* chromatic number, our result positively answers the open question posed by Dębski and Przybyło. - We show that if the minimum degree of any vertex in G is Ω(Δ/{log^ε Δ}) for some ε ≥ 0, then χ_{ON}^*(G) = O(log^{1+ε}Δ). This is a generalization of the result given by Dębski and Przybyło in the same paper where they showed that if the minimum degree of any vertex in G is Ω(Δ), then χ_{ON}^*(G)= O(logΔ). - We give a polynomial time algorithm to compute χ_{ON}^*(G) for interval graphs G. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether the CFON^* chromatic number can be computed in polynomial time on interval graphs. - We explore biconvex graphs, a subclass of bipartite graphs and give a polynomial time algorithm to compute their CFON^* chromatic number. This is interesting as Abel et al. [SIDMA, 2018] had shown that it is NP-complete to decide whether a planar bipartite graph G has χ_{ON}^*(G) = k where k ∈ {1, 2, 3}., LIPIcs, Vol. 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), pages 19:1-19:14

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is $NP$-complete even when $G$ is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is $NP$-complete for cubic graphs.

Strongly perfect graphs have been studied by several authors (e.g. Berge and Duchet (1984) [1], Ravindra (1984) [12] and Wang (2006) [14]). In a series of two papers, the current paper being the first one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [14] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.

Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the clique number of $G$? Erd\H{o}s and Ne\v{s}et\v{r}il asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erd\H{o}s and Ne\v{s}et\v{r}il., Comment: 19 pages; v2 corrects for a subtlety in the original derivation of Thm 1; v3 accepted to Canadian Journal of Mathematics

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ form a strictly increasing sequence converging to $-1/2$. Then we formulate the problem of determining all the graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G), 24 pages, 6 figures

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds., Comment: 24 pages, Discussiones Mathematicae Graph Theory, Special Issue in honor of Slobodan K. Simi\'c, accepted for publication

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved.

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Gilles and Trotter (7) by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1) valued coefficinets (rank facets have onnly coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already obseerved by Cook (3) and Shepperd (18)

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.

In [6], Edmonds provided the first complete description of the polyhedron associated with a combinatorial optimization problem: the matching polytope. As the matching problem is equivalent to the stable set problem on line graphs, many researchers tried to generalize Edmonds' result by considering the stable set problem on a superclass of line graphs: the claw-free graphs. However, as testified also by Grotschel, Lovasz, and Schrijver [14], "in spite of considerable efforts, no decent system of inequalities describing STAB(G) for claw-free graphs is known". Here, we provide an explicit linear description of the stable set polytope of claw-free graphs with stability number at least four and with no 1-join.

We propose an algorithm for solving the maximum weighted stable set problem on claw-free graphs that runs in O (| V |(| E | + | V | log| V |))-time, drastically improving the previous best known complexity bound. This algorithm is based on a novel decomposition theorem for claw-free graphs, which is also introduced in the present article. Despite being weaker than the structural results for claw-free graphs given by Chudnovsky and Seymour [2005, 2008a, 2008b] our decomposition theorem is, on the other hand, algorithmic, that is, it is coupled with an O (| V || E |)-time algorithm that actually produces the decomposition.

Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2] , fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins. A graph is W -perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W -perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs.

In this paper we show how to solve the Maximum Weight Stable Set Problem in a claw-free graph G(V, E) with α(G)≤3 in time O(|E|log|V|). More precisely, in time O(|E|) we check whether α(G)≤3 or produce a stable set with cardinality at least 4; moreover, if α(G)≤3 we produce in time O(|E|log|V|) a maximum weight stable set of G. This improves the bound of O(|E||V|) due to Faenza, Oriolo and Stauffer.

Let G be an undirected graph. An edge of G dominates itself and all edges adjacent to it. A subset $$E'$$ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of $$E'$$ . We say that $$E'$$ is a perfect edge dominating set of G, if every edge not in $$E'$$ is dominated by exactly one edge of $$E'$$ . The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most $$d \ge 3$$ and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a $$P_5$$ -free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a $$P_5$$ .

Strongly perfect graphs have been studied by several authors (e.g., Berge and Duchet (1984) [1], Ravindra (1984) [7] and Wang (2006) [8]). In a series of two papers, the current paper being the second one, we investigate a fractional relaxation of strong perfection. Motivated by a wireless networking problem, we consider claw-free graphs that are fractionally strongly perfect in the complement. We obtain a forbidden induced subgraph characterization and display graph-theoretic properties of such graphs. It turns out that the forbidden induced subgraphs that characterize claw-free graphs that are fractionally strongly perfect in the complement are precisely the cycle of length 6, all cycles of length at least 8, four particular graphs, and a collection of graphs that are constructed by taking two graphs, each a copy of one of three particular graphs, and joining them in a certain way by a path of arbitrary length. Wang (2006) [8] gave a characterization of strongly perfect claw-free graphs. As a corollary of the results in this paper, we obtain a characterization of claw-free graphs whose complements are strongly perfect.

We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that Dominating Set on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that Dominating Set is not fixed-parameter tractable on the slightly larger class of graphs that exclude K_{1,4} as an induced subgraph (K_{1,4}-free graphs). We show that our algorithmization can also be used to show that the related Connected Dominating Set problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that Connected Dominating Set has no polynomial kernel on claw-free graphs and is not fixed-parameter tractable on K_{1,4}-free graphs. Combined, our results provide a dichotomy for Dominating Set and Connected Dominating Set on K_{1,L}-free graphs and show that the problem is fixed-parameter tractable if and only if L, Revised some proofs compared to v2. Significantly expanded proofs and several additional results compared to v1

Let G be a graph and let D"6(G)={[email protected]?V(G)|d"G(v)=6}. In this paper we prove that: (i) If G is a 6-connected claw-free graph and if |D"6(G)|@?74 or G[D"6(G)] contains at most 8 vertex disjoint K"4's, then G is Hamiltonian; (ii) If G is a 6-connected line graph and if |D"6(G)|@?54 or G[D"6(G)] contains at most 5 vertex disjoint K"4's, then G is Hamilton-connected.

We study parallel knock-out schemes for graphs. These schemes proceed in rounds in each of which each surviving vertex simultaneously eliminates one of its surviving neighbours; a graph is reducible if such a scheme can eliminate every vertex in the graph. We resolve the square-root conjecture, first posed at MFCS 2004, by showing that for a reducible graph $G$, the minimum number of required rounds is $O(\sqrt{n})$; in fact, our result is stronger than the conjecture as we show that the minimum number of required rounds is $O(\sqrt{\alpha})$, where $\alpha$ is the independence number of $G$. This upper bound is tight. We also show that for reducible $K_{1,p}$-free graphs at most $p-1$ rounds are required. It is already known that the problem of whether a given graph is reducible is NP-complete. For claw-free graphs, however, we show that this problem can be solved in polynomial time. We also pinpoint a relationship with (locally bijective) graph homomorphisms.

For any permutation @p of the vertex set of a graph G, the generalized prism @pG is obtained by joining two copies of G by the matching {u@p(u):u@?V(G)}. Denote the domination number of G by @c(G). If @c(@pG)=@c(G) for all @p, then G is called a universal fixer. The edgeless graphs are the only known universal fixers, and are conjectured to be the only universal fixers. We prove that claw-free graphs are not universal fixers.

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In earlier papers of this series we proved that every claw-free graph either belongs to one of several basic classes that we described explicitly, or admits one of a few kinds of decomposition. In this paper we convert this “decomposition” theorem into a theorem describing the global structure of claw-free graphs.

A graph is claw-free if no vertex has three pairwise nonadjacent neighbours. In this series of papers we give a structural description of all claw-free graphs. In this paper, we achieve a major part of that goal; we prove that every claw-free graph either belongs to one of a few basic classes, or admits a decomposition in a useful way.

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by @c"t(G). A graph is claw-free if it does not contain K"1","3 as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G@?{C"3,C"5, C"6, C"1"0}, then @c"t(G)=

The problem of finding a complete linear description of the stable set polytope of claw-free graphs has been a major topic in combinatorial optimization in the last decades and it is still not completely solved. While it is known that this linear description contains facet defining inequalities with arbitrarily many and arbitrarily high coefficients, the set of claw-free graphs whose stable set polytope is described only by inequalities with { 0 , 1 , 2 } -valued coefficients is considerably large. In fact Galluccio et?al. (2014) proved that this set contains almost all claw-free graphs with stability number greater than three plus some of the building blocks with stability number smaller than or equal to three, identified by Chudnovsky and Seymour in their Structure Theorem.Here we show that another important class of claw-free graphs with stability number three belongs to this set: the class of icosahedral graphs, named S 1 by Chudnovsky and Seymour (2008). In particular, we prove that the stable set polytope of icosahedral graphs is described by: rank, lifted 5-wheel and lifted wedge inequalities, and all these linear inequalities have coefficients in { 0 , 1 , 2 } .

Let G be a connected, locally connected, claw-free graph of order n and x,y be two vertices of G. In this paper, we prove that if for any 2-cut S of G, S∩{x,y}=∅, then each (x,y)-path of length less than n-1 in G is extendable, that is, for any path P joining x and y of length h(

A graph G is called quasi-claw-free if it satisfies the property: d(x,y)=2 ⇒ there exists u∈N(x)∩N(y) such that N[u]⊆N[x]∪N[y]. Let G be a 2-connected quasi-claw-free graph of order n. If δ(G)⩾n/4, then G is hamiltonian or G∈F, where F is a family of nonhamiltonian graphs of connectivity 2.

We prove that every 3-chromatic claw-free perfect graph is 3-choosable.

In one of fundamental works in combinatorial optimization, Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets in its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs are a subclass of claw-free graphs, and as for claw-free graphs, there exists a polynomial algorithm for finding a maximum weighted stable set in such graphs, but we do not have a complete characterization of their stable set polytope (SSP). In the paper, we introduce a class of inequalities, called clique-family inequalities, which are valid for the SSP of any graph and match the odd set inequalities defined by Edmonds for the matching polytope. This class of inequalities unifies all the known (non-trivial) facet inducing inequalities for the SSP of a quasi-line graph. We, therefore, conjecture that all the non-trivial facets of the SSP of a quasi-line graph belong to this class. We show that the conjecture is indeed correct for the subclasses of quasi-line graphs for which we have a complete description of the SSP. We discuss some approaches for solving the conjecture and a related problem.

A generalized (i,j,k)-net Ni,j,k is the graph obtained by identifying each of the vertices of a triangle with an endvertex of one of three vertex-disjoint paths of lengths i,j,k. We prove that every 2-connected claw-free N1,2,j-free graph of diameter at least max{7,2j} (j⩾2) is hamiltonian.

A graph G is quasi claw-free if it satisfies the property: d(x, y) = 2 ⇒ there exists u ϵ N(x) ∩ N(y) such that N[u]⊂ N[x]∪ N[y]. This property is satisfied if in particular u does not center a claw (induced K1,3). Many known results on claw-free graphs, dealing with matching and hamiltonicity are extended to the larger class of quasi-claw-free graphs.

A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation., 12 pages, 2 figures

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In this article the Lovász–Plummer clique reduction is extended to the weighted case and used to find a maximum weight stable set in a claw-free graph GG with nn nodes in O(n2(n2+L(n)))O(n2(n2+L(n))) time, where L(n)L(n) is the complexity of finding a maximum weight augmenting path in a line graph HH with nn nodes. The best algorithm known to date to solve the latter problem is Gabow’s maximum weight matching algorithm (applied to the root graph of HH) which has a complexity of O(n2logn)O(n2logn). It follows that our algorithm can produce a maximum weight stable set in a claw-free graph in O(n4logn)O(n4logn) time.

International audience; A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γpr (G). We establish bounds on Γpr (G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree δ. We show that Γpr (G) ≤ 4n/5 if δ = 1 and n ≥ 3, Γpr (G) ≤ 3n/4 if δ = 2 and n ≥ 6, and Γpr (G) ≤ 2n/3 if δ ≥ 3. All these bounds are sharp. Further, if n ≥ 6 the graphs G achieving the bound Γpr (G) = 4n/5 are characterized, while for n ≥ 9 the graphs G with δ = 2 achieving the bound Γpr (G) = 3n/4 are characterized.

We describe $Forb\{K_{1,3}, \overline {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $ \overline{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths or isolated vertices, and of the complements of these graphs. Considering the hypergraph ${\mathcal H} ^{(3)}(G)$ made of the $3$-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation., Contributions to Discrete Mathematics, Vol 6, No 1 (2011)

A connected even $[2,2s]$-factor of a graph $G$ is a connected factor with all vertices of degree $i(i=2,4,\ldots,2s)$, where $s\ge 1$ is an integer. In this paper, we show that every supereulerian $K_{1,s}$-free graph $(s\ge 2)$ contains a connected even $[2,2s-2]$-factor, hereby generalizing the result that every 4-connected claw-free graph has a connected $[2,4]$-factor by Broersma, Kriesell and Ryjacek.

In this paper, we introduce two powerful graph reductions for the maximum weighted stable set (mwss) in general graphs. We show that these reductions allow to reduce the mwss in claw-free graphs to the mwss in a class of quasi-line graphs, that we call bipolar-free. For this latter class, we provide a new algorithmic decomposition theorem running in polynomial time. We then exploit this decomposition result and our reduction tools again to transform the problem to either a single matching problem or a longest path computation in an acyclic auxiliary graph (in this latter part we use some results of Pulleyblank and Shepherd [10]). Putting all the pieces together, the main contribution of this paper is a new polynomial time algorithm for the mwss in claw-free graphs. A rough analysis of the complexity of this algorithm gives a time bound of O(n6), where n is the number of vertices in the graph, and which we hope can be improved by a finer analysis. Incidentally, we prove that the mwss problem can be solved efficiently for any class of graphs that admits a "suitable" decomposition into pieces where the mwss is easy.

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. © 2007 Elsevier B.V. All rights reserved. Fil:Bonomo, F. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Durán, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

International audience; The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, ...

We present a new graph composition that produces a graph G from a given graph H and a fixed graph B called gear and we study its polyhedral properties. This composition yields counterexamples to a conjecture on the facial structure of$STAB(G) when G is claw-free.

We consider the number of vertex independent sets i(G)In general, the problem of determining the value of i(G) is NP-complete. We present several upper and lower bounds for i(G) in terms of order, size or independence number. We obtain improved bounds for i(G) on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs. Udgivelsesdato: DEC We consider the number of vertex independent sets i(G)In general, the problem of determining the value of i(G) is NP-complete. We present several upper and lower bounds for i(G) in terms of order, size or independence number. We obtain improved bounds for i(G) on restricted graph classes such as the bipartite graphs, unicyclic graphs, regular graphs and claw-free graphs.

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea's conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea's conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of Ben Rebea's conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

The P_4-structure of a graph G is a hypergraph \textbfH on the same vertex set such that four vertices form a hyperedge in \textbfH whenever they induce a P_4 in G. We present a constructive algorithm which tests in polynomial time whether a given 4-uniform hypergraph is the P_4-structure of a claw-free graph and of (banner,chair,dart)-free graphs. The algorithm relies on new structural results for (banner,chair,dart)-free graphs which are based on the concept of p-connectedness. As a byproduct, we obtain a polynomial time criterion for perfectness for a large class of graphs properly containing claw-free graphs.

The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs