Minimal Imperfect Graphs Research Articles

Discs in unbreakable graphs

We show that every vertex in an unbreakable graph is in a disc, where a disc is a chordless cycle, or the complement of a chordless cycle, with at least five vertices. A corollary is that every vertex in a minimal imperfect graph is in a disc.

Complete Multi-partite Cutsets in Minimal Imperfect Graphs

We show that a minimal imperfect graph G cannot contain a cutset C which induces a complete multi-partite graph unless C is a stable set and G is an odd hole. This generalizes a result of Tucker, who proved that the only minimal imperfect graphs containing stable cutsets are the odd holes. It is also a special case of a conjecture of Chvátal.

Locally perfect graphs

A new property of minimal imperfect graphs is given. This leads to a way to add a new vertex to a perfect graph so that the resulting graph remains perfect. It is shown that the same holds for strongly perfect graphs. Some colouring properties are also considered. In particular we define a new kind of colouring, the locally perfect colouring, and we show that the graphs having such a colouring for any subgraph are perfect. Such graphs are called locally perfect. The triangulated graphs, the parity graphs, and the perfect graphs without a clique with more than three vertices are shown to be locally perfect. Finally the graphs obtained by some rules of construction are studied.

A generalization of chvátal's star-cutset lemma

A graph is perfect if for each of its induced subgraphs H, the chromatic number χ( H) of H equals the largest number ω( H) or pairwise adjacent vertices in H. We present a new property of minimal imperfect graphs which implies Chvátals well-known Star-Cutset Lemma, and which gives a new characterization of perfect graphs.

A new conjecture about minimal imperfect graphs

H. Meyniel proved that in every minimal imperfect graph, every pair of vertices is joined by a chordless path containing an odd number of edges. We conjectured that in every minimal imperfect graph, every pair of vertices is joined by a path containing an even number of edges. We give an equivalent version of this new conjecture.

No antitwins in minimal imperfect graphs

It is customary to call vertices x and y twins if every vertex distinct from x and y is adjacent either to both of them or to neither of them. By analogy, we shall call vertices x and y antitwins if every vertex distinct from x and y is adjacent to precisely one of them. Lovász proved that no minimal imperfect graph has twins. The purpose of this note is to prove the analogous statement for antitwins.

Bull-free Berge graphs are perfect

Abull is the (self-complementary) graph with verticesa, b, c, d, e and edgesab, ac, bc, bd, ce; a graphG is calledBerge if neitherG not its complement contains a chordless cycle whose length is odd and at least five. We prove that bull-free Berge graphs are perfect; a part of our argument relies on a new property of minimal imperfect graphs.

A note on even pairs

We generalize a theorem of Meyniel about minimal imperfect graphs to partitionable graphs. The general result follows easily from a theorem of Bland, Huang, and Trotter.

Alternating orientation and alternating colouration of perfect graphs

We establish a property of minimal imperfect graphs, and use this property to generate two classes of perfect graphs. The first class contains all comparability graphs, all triangulated graphs, and two other classes of perfect graphs. The second class contains all triangulated graphs and all line-graphs of bipartite graphs.

Star-cutsets and perfect graphs

We first establish a certain property of minimal imperfect graphs and then use it to generate large classes of perfect graphs.

On stable set polyhedra for K1,3-free graphs

We give several classes of facets for the convex hull of incidence vectors of stable sets in a K 1,3-free graph, including facets with ( a, a + 1)-valued coefficients, where a = 1, 2, 3,…. These provide counterexamples to three recent conjectures concerning such facets. We also give a necessary and sufficient condition for a minimal imperfect graph to be an odd hole or an odd antihole and indicate that minimal imperfect K 1,3-free graphs satisfy the condition.

Graphical properties related to minimal imperfection

Say that graph G is partitionable if there exist integers α⩾2, ω⩾ 2, such that | V( G)| ≡ αω + 1 and for every υ ϵ V( G) there exist partitions of V( G)\\ υ into stable sets of size α and into eliques of size ω. An immediate consequence of Lovász' characterization of perfect graphs is that every minimal imperfect graph G is partitionable with α ≡ α ( G) and ω ≡ ω( G). Padberg has shown that in every minimal imperfect graph G the cliques and stable sets of maximum size satisfy a series of conditions that reflect extraordinary symmetry G. Among these conditions are: the number of cliques of size ω( G) is exactly |V( G)|; the number of stable sets of size α( G) is exactly |V( G)|: every vertex of G is contained in exactly ω( G) cliques of size ω( G) and α( G) stable sets of size α( G): for every clique Q (respectively, stable set S) of maximum size there is a unique stable set S (clique O) of maximum size such that Q∩ S ≡ Ø. Let C n k denote the graph whose vertices can be enumerated as υ 1,…,υ n in such a way that υ 1 and υ 1 are adjacent in G if and only if i and j differ by at most k, modulo n. Chvátal has shown that Berge's Strong Perfect graph Conjecture is equivalent to the conjecture that if G is minimal imperfect with α( G) ≡ α and ω( G) ≡ ω, then G has a spanning subgraph isomorphic to C αω+1 ω. Padberg's conditions are sufficiently restrictive to suggest the possibility of establishing the Strong Perfect Graph Conjecture by proving that any graph G satisfying these conditions must contain a spanning subgraph isomorphic to C αω+1 ω , where α( G) ≡ α and ω( G) ≡ ω. It is shown here, using only elementary linear algebra, that all partitionable graphs satisfy Padberg's conditions, as well as additional properties of the same spirit. Then examples are provided of partitionable graphs which contain no spanning subgraph isomorphic to C αω+1 ω , where α( G) ≡ α and ω( G) ≡ ω.

Combinatorial designs related to the strong perfect graph conjecture

When α, ω are positive integers, we set n = αω + 1 and look for zero-one matrices X, Y of size n × n such that XY= YX = J − I , JX = XJ = αJ , JY = YJ = ωJ. Simple solutions of these matrix equations are easy to find; we describe ways of constructing rather messy ones. Our investigations are motivated by an intimate relationship between the pairs X, Y and minimal imperfect graphs.