Henri Thuillier

On graphs without P 5 and P¯ 5

Abstract We extend results due to Blazsik et al. (1993) on graphs with no induced C 4 and 2 K 2 to the self-complementary class of ( P 5 , P 5 )- free graphs. Moreover, we obtain an O(ω 2 ) γ -binding function for this last class of graphs, answering thus partially a question of A. Gyarfas.

ON GRAPHS WITH LIMITED NUMBER OF P4-PARTNERS

The study of graphs containing few P4s generated an important number of results related to perfection, recognition, optimization problems (see [12], [15], [8]). We define here a new, larger class of graphs and show that the indicated problems may be efficiently solved on this class too (thus generalizing some of the previous results). Namely, we give a linear time recognition algorithm for this class and we note that the optimization problems concerning the clique number, stability number, chromatic number and clique cover number are solvable in linear time.

On ( K q , k ) vertex stable graphs with minimum size

Abstract A graph G is a ( K q , k ) vertex stable graph if it contains a K q after deleting any subset of k vertices. We give a characterization of ( K q , k ) vertex stable graphs with minimum size for q = 3 , 4 , 5 .

An O ( n ) time algorithm for maximum matching in P 4 -tidy graphs

Abstract The P 4 -tidy graphs were introduced by I. Rusu to generalize some already known classes of graphs with “few” induced P 4 s. In this paper, we extend to P 4 -tidy graphs a linear time algorithm of C.-H. Yang and M.-S. Yu for finding a maximum matching in a cograph G (given a parse tree associated to G ).

The Strong Perfect Graph Conjecture: 40 years of attempts, and its resolution

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

Scattering number and modular decomposition

Abstract The scattering number of a graph G equals max { c ( G ⧹ S ) − | S | S is a cutset of G } where c ( G ⧹ S ) denotes the number of connected components in G ⧹ S . Jung (1978) has given for any graph having no induced path on four vertices ( P 4 -free graph) a correspondence between the value of its scattering number and the existence of Hamiltonian paths or Hamiltonian cycles. Hochstattler and Tinhofer (to appear) studied the Hamiltonicity of P 4 -sparse graphs introduced by Hoang (1985). In this paper, using modular decomposition, we show that the results of Jung and Hochstattler and Tinhofer can be generalized to a subclass of the family of semi- P 4 -sparse graphs introduced in Fouquet and Giakoumakis (to appear).

On isomorphic linear partitions in cubic graphs

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. It is well known that la(G)=2 when G is a cubic graph and Wormald [N. Wormald, Problem 13, Ars Combinatoria 23(A) (1987) 332-334] conjectured that if |V(G)|=0 (mod 4), then it is always possible to find a linear partition in two isomorphic linear forests. Here, we give some new results concerning this conjecture.

Decomposition of 3-connected cubic graphs

Abstract We solve a conjecture of Foulds and Robinson (1979) on decomposable triangulations in the plane, in the more general context of a decomposition theory of cubic 3-connected graphs. The decomposition gives us a natural way to obtain some known results about specific homeomorphic subgraphs and the extremal diameter of 3-connected cubic graphs.

On removable edges in 3-connected cubic graphs

A removable edge in a 3-connected cubic graph G is an edge e=uv such that the cubic graph obtained from G-{u,v} by adding an edge between the two neighbours of u distinct from v and an edge between the two neighbours of v disctinct from u is still 3-connected. Li and Wu [1] showed that a spanning tree in a 3-connected cubic graph avoids at least two removable edges, and Kang, Li and Wu [2] showed that a spanning tree contains at least two removable edges. We show here how to obtain these results easily from the structure of the sets of non removable edges and we give a characterization of the extremal graphs for these two results. [1] WU Jichang and LI Xueliang, Removable edges outside a spanning tree of a 3-connected 3-regular graph, Journal of Mathematical Study, 36(3), 2003, 223-229. [2] KANG Haiyan, WU Jichang and LI Guojun, Removable edges of a spanning tree in 3-connected 3-regular graphs, LNCS, 4613, 2007, 337-345.

On Minimum (Kq, K) Stable Graphs

Abstract A graph G is a (Kq, k) stable graph (q ≥ 3) if it contains a Kq after deleting any subset of k vertices (k ≥ 0). Andrzej ˙ Zak in the paper On (Kq; k)-stable graphs, ( doi:/10.1002/jgt.21705) has proved a conjecture of Dudek, Szyma´nski and Zwonek stating that for sufficiently large k the number of edges of a minimum (Kq, k) stable graph is (2q − 3)(k + 1) and that such a graph is isomorphic to sK2q−2 + tK2q−3 where s and t are integers such that s(q − 1) + t(q − 2) − 1 = k. We have proved (Fouquet et al. On (Kq, k) stable graphs with small k, Elektron. J. Combin. 19 (2012) #P50) that for q ≥ 5 and k ≤ q 2 +1 the graph Kq+k is the unique minimum (Kq, k) stable graph. In the present paper we are interested in the (Kq, k(q)) stable graphs of minimum size where k(q) is the maximum value for which for every nonnegative integer k <k(q) the only (Kq, k) stable graph of minimum size is Kq+k and by determining the exact value of k(q).