Decomposing Berge graphs

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no old hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved a stronger theorem by restricting the allowed decompositions and another theorem where some decompositions were restricted while other decompositions were extended. We prove here a theorem stronger than all those previously known results. Our proof uses at an essential step one of the theorems of Chudnovsky.Perfect graph ; Berge graph ; 2-join ; even skew partition ; decomposition.

EMU enlargement towards CEEC's : risks of sector-based and geographic asymmetric shocks

The future membership of CEECs in the eurozone involves the risk of external asymmetric shocks, due to too strong a dependence on one sector or one customer country. By defining two indicators - sector-based and geographic - of exposure to shocks, taking into account the symmetry of the export structures of a candidate state with the EMU and the opening rate of the candidate state, we are able to draw up a classification of countries according to the fulfillment of Kenen's criterion, revised and transposed to the geographic variety of exports. The results, compared with those of the two countries of the EMU which are most sensitive to sector-based and geographic shocks (Finland and Ireland), testify to a generally pronounced exposure to shocks. An inventory of the pairs country / branch and country / destination at the origin of shocks with strong macroeconomic impact shows that Bulgaria and Slovakia, and even more Estonia and Latvia, are exposed to major risks. Thus, these two small economies would be well advised to create a cyclical stabilization fund before envisaging joining the EMU. Against the thesis of the endogeneity of the OCAs (optimum currency areas), this recommendation remains valid, insofar as the increase in the share of intra-industry trade between the EMU and a candidate state does not necessarily entail a convergence of multilateral sector-based export structures.Central and Eastern European countries; external asymmetric shock; Kenen's criterion; optimum currency area; symmetry of export structures

Decomposing Berge graphs and detecting balanced skew partitions

We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an O(n9)-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive : it certifies that a graph has a balanced skew partition if it has one. It relies on a new decomposition theorem for Berge graphs, that is more precise than the previously known theorems and implies them easily. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the decomposition theorems of Chudnovsky.Perfect graph, Berge graph, 2-join, balanced skew partition, decomposition, detection, recognition.

A new decomposition theorem for Berge graphs

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class or has some kind of decomposition. Then, Chudnovsky proved a stronger theorem by restricting the allowed decompositions. We prove here a stronger theorem by restricting again the allowed decompositions. Motivation for this new theorem will be given in a work in preparation.Graph, Berge, decomposition, 2-join, skew partition.

Equistarable graphs and counterexamples to three conjectures on equistable graphs

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight $1$. In $1994$, Mahadev et al.~introduced a subclass of equistable graphs, called strongly equistable graphs, as graphs such that for every $c \le 1$ and every non-empty subset $T$ of vertices that is not a maximal stable set, there exist positive vertex weights such that every maximal stable set is of total weight $1$ and the total weight of $T$ does not equal $c$. Mahadev et al. conjectured that every equistable graph is strongly equistable. General partition graphs are the intersection graphs of set systems over a finite ground set $U$ such that every maximal stable set of the graph corresponds to a partition of $U$. In $2009$, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin's conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed by Miklavi\v{c} and Milani\v{c} in $2011$, and states that every equistable graph has a clique intersecting all maximal stable sets. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle-free graphs