Circular-perfect Graphs Research Articles

On circular-perfect graphs: A survey

Circular-perfect graphs form a natural superclass of perfect graphs, introduced by Zhu almost 20 years ago: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1.In this paper, we survey the results about circular-perfect graphs obtained in the two last decades, with a focus on whether the fascinating properties of perfect graphs can be extended to this superclass. We recall the deep algorithmic properties of perfect graphs and the underlying polyhedral graph theory, in order to outline in this setting the main graph parameters that are also related to circular-perfect graphs. Finally, we give a list of open problems.

On the theta number of powers of cycle graphs

We give a closed formula for Lovasz's theta number of the powers of cycle graphs C k d?1 and of their complements, the circular complete graphs K k/d . As a consequence, we establish that the circular chromatic number of a circular perfect graph is computable in polynomial time. We also derive an asymptotic estimate for the theta number of C k d .

Polynomial time computability of some graph parameters for superclasses of perfect graphs

A main result in combinatorial optimisation is that clique and chromatic number of a perfect graph are computable in polynomial time (Grotschel et al., 1981). The circular-clique and circular-chromatic number are well-studied refinements of these graph parameters, and circular-perfect graphs form the corresponding superclass of perfect graphs. So far, it is unknown whether clique, circular-clique, circular-chromatic and chromatic numbers of a circular-perfect graph are computable in polynomial time. In this paper, we show the polynomial time computability of these graph parameters for some classes of circular-perfect graphs with the help of polyhedral arguments.

Clique and chromatic number of circular-perfect graphs

Otto-von-Guericke Universit¨at MagdeburgInstitut fur Mathematische Optimierung, Universit¨ ¨atsplatz 239106 Magdeburg, GermanyAbstractA main result of combinatorial optimization is that clique and chromatic number of a per-fect graph are computable in polynomial time (Gr¨otschel, Lov asz and Schrijver 1981).´Circular-perfect graphs form a well-studied superclass of perfect graphs. We extend theabove result for perfect graphs by showing that clique and chromatic number of a circular-perfect graph are computable in polynomial time as well. The results strongly rely uponLov´asz’s Theta function.Keywords: circular-perfect graph, polytope, circular-clique

On the polynomial time computability of the circular-chromatic number for some superclasses of perfect graphs

Abstract A main result in combinatorial optimization is that clique and chromatic number of a perfect graph are computable in polynomial time (Grotschel, Lovasz and Schrijver 1981). The circular-clique and circular-chromatic number are well-studied refinements of these graph parameters, and circular-perfect graphs form the corresponding superclass of perfect graphs. So far, it is unknown whether the (weighted) circular-clique and circular-chromatic number of a circular-perfect graph are computable in polynomial time. In this paper, we show the polynomial time computability of these two graph parameters for some super-classes of perfect graphs with the help of polyhedral arguments.

Triangle-free strongly circular-perfect graphs

Zhu [X. Zhu, Circular-perfect graphs, J. Graph Theory 48 (2005) 186–209] introduced circular-perfect graphs as a superclass of the well-known perfect graphs and as an important χ -bound class of graphs with the smallest non-trivial χ -binding function χ ( G ) ≤ ω ( G ) + 1 . Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs [M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, The strong perfect graph theorem, Ann. Math. (in press)]; in particular, perfect graphs are closed under complementation [L. Lovász, Normal hypergraphs and the weak perfect graph conjecture, Discrete Math. 2 (1972) 253–267]. To the contrary, circular-perfect graphs are not closed under complementation and the list of forbidden subgraphs is unknown. We study strongly circular-perfect graphs: a circular-perfect graph is strongly circular-perfect if its complement is circular-perfect as well. This subclass entails perfect graphs, odd holes, and odd antiholes. As the main result, we fully characterize the triangle-free strongly circular-perfect graphs, and prove that, for this graph class, both the stable set problem and the recognition problem can be solved in polynomial time. Moreover, we address the characterization of strongly circular-perfect graphs by means of forbidden subgraphs. Results from [A. Pêcher, A. Wagler, On classes of minimal circular-imperfect graphs, Discrete Math. (in press)] suggest that formulating a corresponding conjecture for circular-perfect graphs is difficult; it is even unknown which triangle-free graphs are minimal circular-imperfect. We present the complete list of all triangle-free minimal not strongly circular-perfect graphs.

On classes of minimal circular-imperfect graphs

Circular-perfect graphs form a natural superclass of perfect graphs: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ -bound graphs with the smallest non-trivial χ -binding function χ ( G ) ⩽ ω ( G ) + 1 . The Strong Perfect Graph Conjecture, recently settled by Chudnovsky et al. [The strong perfect graph theorem, Ann. of Math. 164 (2006) 51–229], provides a characterization of perfect graphs by means of forbidden subgraphs. It is, therefore, natural to ask for an analogous conjecture for circular-perfect graphs, that is for a characterization of all minimal circular-imperfect graphs. At present, not many minimal circular-imperfect graphs are known. This paper studies the circular-(im)perfection of some families of graphs: normalized circular cliques, partitionable graphs, planar graphs, and complete joins. We thereby exhibit classes of minimal circular-imperfect graphs, namely, certain partitionable webs, a subclass of planar graphs, and odd wheels and odd antiwheels. As those classes appear to be very different from a structural point of view, we infer that formulating an appropriate conjecture for circular-perfect graphs, as analogue to the Strong Perfect Graph Theorem, seems to be difficult.

On minimally circular-imperfect graphs

A circular-perfect graph is a graph of which each induced subgraph has the same circular chromatic number as its circular clique number. In this paper, (1) we prove a lower bound on the order of minimally circular-imperfect graphs, and characterize those that attain the bound; (2) we prove that if G is a claw-free minimally circular-imperfect graph such that ω c ( G - x ) > ω ( G - x ) for some x ∈ V ( G ) , then G = K ( 2 k + 1 ) / 2 + x for an integer k; and (3) we also characterize all minimally circular-imperfect line graphs.

Circular-imperfection of triangle-free graphs

Abstract Circular-perfect graphs form a natural superclass of the well-known perfect graphs by means of a more general coloring concept. For perfect graphs, a characterization by means of forbidden subgraphs was recently settled by Chudnovsky et al. [Chudnovsky, M., N. Robertson, P. Seymour, and R. Thomas, The Strong Perfect Graph Theorem, Annals of Mathematics 164 (2006) 51–229]. It is, therefore, natural to ask for an analogous characterization for circular-perfect graphs or, equivalently, for a characterization of all minimally circular-imperfect graphs. Our focus is the circular-(im)perfection of triangle-free graphs. We exhibit several different new infinite families of minimally circular-imperfect triangle-free graphs. This shows that a characterization of circular-perfect graphs by means of forbidden subgraphs is a difficult task, even if restricted to the class of triangle-free graphs. This is in contrary to the perfect case where it is long-time known that the only minimally imperfect triangle-free graphs are the odd holes [Tucker, A., Critical Perfect Graphs and Perfect 3-chromatic Graphs, J. Combin. Theory (B) 23 (1977) 143–149].

Claw-free circular-perfect graphs

Abstract The circular chromatic number of a graph is a well-studied refinement of the chromatic number. Circular-perfect graphs is a superclass of perfect graphs defined by means of this more general coloring concept. This paper studies claw-free circular-perfect graphs. A consequence of the strong perfect graph theorem is that minimal circular-imperfect graphs G have min { α ( G ) , ω ( G ) } = 2 . In contrast to this result, it is shown in [Z. Pan and X. Zhu. Minimal circular-imperfect graphs of large clique number and large independence number. manuscript, 2006] that minimal circular-imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. We prove that claw-free minimal circular-imperfect graphs G have min { α ( G ) , ω ( G ) } ⩽ 3 .

On strongly circular-perfectness

Abstract Perfect graphs constitute a well-studied graph class with a rich structure. Circular-perfect graphs as introduced by Zhu are a natural superclass of perfect graphs defined by means of a more general coloring concept and form an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ ( G ) ⩽ ω ( G ) + 1 . Apart from perfect graphs, circular-perfect graphs include all convex-round graphs and outerplanar graphs. A linear description of facets of stable set polytopes of circular-perfect graphs is still unknown, though convex-round graphs and outerplanar graphs are rank-perfect. In this paper, we exhibit an infinite class of non rank-perfect circular-perfect graphs. We introduce strongly circular-perfectness: a circular-perfect graph is said to be strongly circular-perfect if its complement is also circular-perfect. This subclass of circular-perfect graphs includes perfect graphs, odd holes and antiholes. We show that there are infinitely many non rank-perfect strongly circular-perfect graphs and fully characterize triangle-free minimal strongly circular-imperfect graphs: it turns out that these graphs are very close to odd holes.

Circular perfect graphs

AbstractFor 1 ≤ d ≤ k, let Kk/d be the graph with vertices 0, 1, …, k − 1, in which i ∼j if d ≤ |i − j| ≤ k − d. The circular chromatic number χc(G) of a graph G is the minimum of those k/d for which G admits a homomorphism to Kk/d. The circular clique number ωc(G) of G is the maximum of those k/d for which Kk/d admits a homomorphism to G. A graph G is circular perfect if for every induced subgraph H of G, we have χc(H) = ωc(H). In this paper, we prove that if G is circular perfect then for every vertex x of G, NG[x] is a perfect graph. Conversely, we prove that if for every vertex x of G, NG[x] is a perfect graph and G − N[x] is a bipartite graph with no induced P5 (the path with five vertices), then G is a circular perfect graph. In a companion paper, we apply the main result of this paper to prove an analog of Haj́os theorem for circular chromatic number for k/d ≥ 3. Namely, we shall design a few graph operations and prove that for any k/d ≥ 3, starting from the graph Kk/d, one can construct all graphs of circular chromatic number at least k/d by repeatedly applying these graph operations. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 186–209, 2005