Kristina Vušković

Balanced matrices

A 0,+/-1 matrix is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of 4. This definition was introduced by Truemper and generalizes the notion of balanced 0,1 matrix introduced by Berge. In this tutorial, we survey what is currently known about these matrices: polyhedral results, combinatorial and structural theorems, and recognition algorithms.

Even and odd holes in cap-free graphs

In [J Combin Theory Ser B, 26 (1979), 205–216], Jaeger showed that every graph with 2 edge-disjoint spanning trees admits a nowhere-zero 4-flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with |A| = 4, then every graph with 2 edge-disjoint spanning trees is A-connected. As graphs with 2 edge-disjoint spanning trees are all collapsible, we in this note improve the latter result by showing that, if A is an abelian group with |A| = 4, then every collapsible graph is A-connected. This allows us to prove the following generalization of Jaegers theorem: Let G be a graph with 2 edge-disjoint spanning trees and let M be an edge cut of G with |M| ≥ 4. Then either any partial nowhere-zero 4-flow on M can be extended to a nowhere-zero 4-flow of the whole graph G, or G can be contracted to one of three configurations, including the wheel of 5 vertices, in which cases certain partial nowhere-zero 4-flows on M cannot be extended. Our results also improve a theorem of Catlin in [J Graph Theory, 13 (1989), 465–483].

A polynomial algorithm for recognizing perfect graphs

We present a polynomial algorithm for recognizing whether a graph is perfect, thus settling a long standing open question. The algorithm uses a decomposition theorem of Conforti, Cornuejols and Vuskovic. Another polynomial algorithm for recognizing perfect graphs, which does not use decomposition, was obtained simultaneously by Chudnovsky and Seymour. Both algorithms need a first phase developed jointly by Chudnovsky, Cornuejols, Liu, Seymour and Vuskovic.

Even-hole-free graphs that do not contain diamonds: A structure theorem and its consequences

In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e., chordless cycles of even length) and diamonds (i.e., a graph obtained from a clique of size 4 by removing an edge). We say that such graphs are (even-hole, diamond)-free. For this class of graphs we first obtain a decomposition theorem, using clique cutsets, bisimplicial cutsets (which is a special type of a star cutset) and 2-joins. This decomposition theorem is then used to prove that every graph that is (even-hole, diamond)-free contains a simplicial extreme (i.e., a vertex that is either of degree 2 or whose neighborhood induces a clique). This characterization implies that for every (even-hole, diamond)-free graph G, @g(G)=<@w(G)+1 (where @g denotes the chromatic number and @w the size of a largest clique). In other words, the class of (even-hole, diamond)-free graphs is a @g-bounded family of graphs with the Vizing bound for the chromatic number. The existence of simplicial extremes also shows that (even-hole, diamond)-free graphs are @b-perfect, which implies a polynomial time coloring algorithm, by coloring greedily on a particular, easily constructable, ordering of vertices. Note that the class of (even-hole, diamond)-free graphs can also be recognized in polynomial time.

Square-free perfect graphs

We prove that square-free perfect graphs are bipartite graphs or line graphs of bipartite graphs or have a 2-join or a star cutset.

The graph sandwich problem for 1-join composition is NP-complete

A graph is a 1-join composition if its vertex set can be partitioned into four nonempty sets AL, AR , SL and SR such that: every vertex of AL is adjacent to every vertex of AR; no vertex of SL is adjacent to vertex of AR∪SR; no vertex of SR is adjacent to a vertex of AL∪SL. The graph sandwich problem for 1-join composition is defined as follows: Given a vertex set V, a forced edge set E1, and a forbidden edge set E3, is there a graph G= (V,E) such that E1 ⊆ E and E ∩ E3 = φ, which is a 1-join composition graph? We prove that the graph sandwich problem for 1-join composition is NP-complete. This result stands in contrast to the case where SL = φ (SR = φ), namely, the graph sandwich problem for homogeneous set, which has a polynomial-time solution.

Triangulated neighborhoods in even-hole-free graphs

An even-hole-free graph is a graph that does not contain, as an induced subgraph, a chordless cycle of even length. A graph is triangulated if it does not contain any chordless cycle of length greater than three, as an induced subgraph. We prove that every even-hole-free graph has a node whose neighborhood is triangulated. This implies that in an even-hole-free graph, with n nodes and m edges, there are at most n+2m maximal cliques. It also yields an O(n^2m) algorithm that generates all maximal cliques of an even-hole-free graph. In fact these results are obtained for a larger class of graphs that contains even-hole-free graphs.

Combinatorial optimization with 2-joins

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.

Balanced 0, ±1 Matrices I. Decomposition

A 0, ±1 matrix is balanced if, in every square submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This paper extends the decomposition of balanced 0, 1 matrices obtained by Conforti, Cornuejols, and Rao (1999, J. Combin. Theory Ser. B77, 292?406) to the class of balanced 0, ±1 matrices. As a consequence, we obtain a polynomial time algorithm for recognizing balanced 0, ±1 matrices.

Vertex elimination orderings for hereditary graph classes

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vuskovic.