Maria Chudnovsky

Packing Non-Zero A -Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊂V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Maders S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k −2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem.

Claw-free graphs. IV. Decomposition theorem

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.

Progress on perfect graphs

Abstract. A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph. The class of perfect graphs is important for several reasons. For instance, many problems of interest in practice but intractable in general can be solved efficiently when restricted to the class of perfect graphs. Also, the question of when a certain class of linear programs always have an integer solution can be answered in terms of perfection of an associated graph. In the first part of the paper we survey the main aspects of perfect graphs and their relevance. In the second part we outline our recent proof of the Strong Perfect Graph Conjecture of Berge from 1961, the following: a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such an odd cycle.

The three-in-a-tree problem

We show that there is a polynomial time algorithm that, given three vertices of a graph, tests whether there is an induced subgraph that is a tree, containing the three vertices. (Indeed, there is an explicit construction of the cases when there is no such tree.) As a consequence, we show that there is a polynomial time algorithm to test whether a graph contains a “theta” as an induced subgraph (this was an open question of interest) and an alternative way to test whether a graph contains a “pyramid” (a fundamental step in checking whether a graph is perfect).

The Erdős--Hajnal conjecture for bull-free graphs

The bull is a graph consisting of a triangle and two pendant edges. A graphs is called bull-free if no induced subgraph of it is a bull. In this paper we prove that every bull-free graph on n vertices contains either a clique or a stable set of size n^1^4, thus settling the Erdos-Hajnal conjecture [P. Erdos, A. Hajnal, Ramsey-type theorems, Discrete Appl. Math. 25 (1989) 37-52] for the bull.

Bisimplicial vertices in even-hole-free graphs

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In this paper we prove that every even-hole-free graph has a bisimplicial vertex, which was originally conjectured by Reed.

Cycles in dense digraphs

AbstractLet G be a digraph (without parallel edges) such that every directed cycle has length at least four; let β(G) denote the size of the smallest subset X ⊆ E(G) such that G∖X has no directed cycles, and let γ(G) be the number of unordered pairs {u, v} of vertices such that u, v are nonadjacent in G. It is easy to see that if γ(G) = 0 then β(G) = 0; what can we say about β(G) if γ(G) is bounded?We prove that in general β(G) ≤ γ(G). We conjecture that in fact β(G) ≤ ½γ(G) (this would be best possible if true), and prove this conjecture in two special cases: when V(G) is the union of two cliqueswhen the vertices of G can be arranged in a circle such that if distinct u, v, w are in clockwise order and uw is a (directed) edge, then so are both uv, vw.

The structure of bull-free graphs II and III—A summary

The bull is a graph consisting of a triangle and two pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is a summary of the last two papers [2,3] in a series [1–3] (Chudnovsky, 2012). The goal of the series is to give a complete description of all bull-free graphs. We call a bull-free graph elementary if it does not contain an induced three-edge-path P such that some vertex c∉V(P) is complete to V(P), and some vertex a∉V(P) is anticomplete to V(P). Here we prove that every elementary graph either belongs to one of a few basic classes, or admits a certain decomposition, and then uses this result together with the results of [1] (this issue) to give an explicit description of the structure of all bull-free graphs.

The structure of bull-free graphs I-Three-edge-paths with centers and anticenters

The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q, and vertices c@?V(P) and a@?V(Q), such that c is adjacent to every vertex of V(P) and a has no neighbor in V(Q). One of the theorems in this paper, namely 1.2, is used in Chudnovsky and Safra (2008) [9] in order to prove that every bull-free graph on n vertices contains either a clique or a stable set of size n^1^4, thus settling the Erdos-Hajnal conjecture (Erdos and Hajnal, 1989) [17] for the bull.

A well-quasi-order for tournaments

A digraph H is immersed in a digraph G if the vertices of H are mapped to (distinct) vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths are pairwise edge-disjoint. For graphs the same relation (using paths instead of directed paths) is a well-quasi-order; that is, in every infinite set of graphs some one of them is immersed in some other. The same is not true for digraphs in general; but we show it is true for tournaments (a tournament is a directed complete graph).