Fuzzy Circular Interval Graphs Research Articles

Counting kernels in directed graphs with arbitrary orientations

A kernel of a directed graph is a subset of vertices that is both independent and absorbing (every vertex not in the kernel has an out-neighbour in the kernel).Not all directed graphs contain kernels, and computing a kernel or deciding that none exist is NP-complete even on low-degree planar digraphs. The existing polynomial-time algorithms for this problem are all obtained by restricting both the undirected structure and the edge orientations of the input: for example, to chordal graphs without bidirectional edges (Pass-Lanneau, Igarashi and Meunier, Discrete Appl Math 2020) or to permutation graphs where each clique has a sink (Abbas and Saoula, 4OR 2005).By contrast, we count the kernels of a fuzzy circular interval graph in polynomial time, regardless of its edge orientations, and return a kernel when one exists. (Fuzzy circular interval graphs were introduced by Chudnovsky and Seymour in their structure theorem for claw-free graphs.)We also consider kernels on cographs, where we establish NP-hardness in general but linear-time solvability on the subclass of threshold graphs.

Graph classes and Ramsey numbers

For a graph class G and any two positive integers i and j, the Ramsey number RG(i,j) is the smallest positive integer such that every graph in G on at least RG(i,j) vertices has a clique of size i or an independent set of size j. For the class of all graphs, Ramsey numbers are notoriously hard to determine, and they are known only for very small values of i and j. Even if we restrict G to be the class of claw-free graphs, it is highly unlikely that a formula for determining RG(i,j) for all values of i and j will ever be found, as there are infinitely many nontrivial Ramsey numbers for claw-free graphs that are as difficult to determine as for arbitrary graphs. Motivated by this difficulty, we establish here exact formulas for all Ramsey numbers for three important subclasses of claw-free graphs: line graphs, long circular interval graphs, and fuzzy circular interval graphs. On the way to obtaining these results, we also establish all Ramsey numbers for the class of perfect graphs. Such positive results for graph classes are rare: a formula for determining RG(i,j) for all values of i and j, when G is the class of planar graphs, was obtained by Steinberg and Tovey (1993), and this seems to be the only previously known result of this kind. We complement our aforementioned results by giving exact formulas for determining all Ramsey numbers for several graph classes related to perfect graphs.

On the recognition of fuzzy circular interval graphs

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial time algorithm for recognizing such graphs, and more importantly for building a suitable model for these graphs.

Coloring fuzzy circular interval graphs

Given a graph G with nonnegative node labels w, a multiset of stable setsS1,…,Sk⊆V(G) such that each vertex v∈V(G) is contained in w(v) many of these stable sets is called a weighted coloring. The weighted coloring numberχw(G) is the smallest k such that there exist stable sets as above.We provide a polynomial time combinatorial algorithm that computes the weighted coloring number and the corresponding colorings for fuzzy circular interval graphs. The algorithm reduces the problem to the case of circular interval graphs, then making use of a coloring algorithm by Gijswijt.We also show that the stable set polytopes of fuzzy circular interval graphs have the integer decomposition property.

Coloring Fuzzy Circular Interval Graphs

Computing the weighted coloring number of graphs is a classical topic in combinatorics and graph theory. Recently these problems have again attracted a lot of attention for the class of quasi-line graphs and more specifically fuzzy circular interval graphs. The problem is NP-complete for quasi-line graphs. For the subclass of fuzzy circular interval graphs however, one can compute the weighted coloring number in polynomial time using recent results of Chudnovsky and Ovetsky and of King and Reed. Whether one could actually compute an optimal weighted coloring of a fuzzy circular interval graph in polynomial time however was still open. We provide a combinatorial algorithm that computes weighted colorings and the weighted coloring number for fuzzy circular interval graphs efficiently. The algorithm reduces the problem to the case of circular interval graphs, then making use of an algorithm by Gijswijt to compute integer decompositions.

Some classical combinatorial problems on circulant and claw-free graphs: the isomorphism and coloring problems on circulant graphs and the stable set problem on claw-free graphs

This is a summary of the author’s Ph.D. thesis supervised by Sara Nicoloso and Gianpaolo Oriolo and defended on 3 April 2008 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request. This work deals with three classical combinatorial problems, namely the isomorphism, the vertex-coloring and the stable set problem, restricted to two graph classes, namely circulant and claw-free graphs. In the first part (joint work with Sara Nicoloso), we derive a necessary and sufficient condition to test isomorphism of circulant graphs, and give simple algorithms to solve the vertex-coloring problem on this class of graphs. In the second part (joint work with Gianpaolo Oriolo and Gautier Stauffer), we propose a new combinatorial algorithm for the maximum weighted stable set problem in claw-free graphs, and devise a robust algorithm for the same problem in the subclass of fuzzy circular interval graphs, which also provides recognition when the stability number is greater than three.

The stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

Clique-circulants and the stable set polytope of fuzzy circular interval graphs

In this paper, we give a complete and explicit description of the rank facets of the stable set polytope for a class of claw-free graphs, recently introduced by Chudnovsky and Seymour (Proceedings of the Bristish Combinatorial Conference, 2005), called fuzzy circular interval graphs. The result builds upon the characterization of minimal rank facets for claw-free graphs by Galluccio and Sassano (J. Combinatorial Theory 69:1–38, 2005) and upon the introduction of a superclass of circulant graphs that are called clique-circulants. The new class of graphs is invetigated in depth. We characterize which clique-circulants C are facet producing, i.e. are such that $$\sum_{v\in V(C)} x_v\leq\alpha(C)$$ is a facet of STAB(C), thus extending a result of Trotter (Discrete Math. 12:373–388, 1975) for circulants. We show that a simple clique family inequality (Oriolo, Discrete Appl. Math. 132(2):185–201, 2004) may be associated with each clique-circulant $$C\subseteq G$$, when G is fuzzy circular interval. We show that these inequalities provide all the rank facets of STAB(G), if G is a fuzzy circular interval graph. Moreover we conjecture that, in this case, they also provide all the non-rank facets of STAB(G) and offer evidences for this conjecture.

On rank-perfect subclasses of near-bipartite graphs

Shepherd95 proved that the stable set polytopes of near-bipartite graphs are given by constraints associated with the complete join of antiwebs only. For antiwebs, the facet set reduces to rank constraints associated with single antiwebs by Wagler2004. We extend this result to a larger graph class, the complements of fuzzy circular interval graphs, recently introduced in ChudnovskySeymour2004.

Almost all webs are not rank-perfect

Graphs with circular symmetry, called webs, are crucial for describing the stable set polytopes of two larger graph classes, quasi-line graphs[8,12] and claw-free graphs [7,8]. Providing a complete linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] verified this conjecture recently for quasi-line graphs not belonging to the subclass of fuzzy circular interval graphs and showed that rank facets are required in this case only. Fuzzy circular interval graphs contain all webs and even the problem of finding all facets of their stable set polytopes is open. So far, it is only known that stable set polytopes of webs with clique number ≤3 have rank facets only [5,17] while there are examples with clique number ≥4 having non-rank facets [10_12,15].In this paper we prove, building on a construction for non-rank facets from [16], that the stable set polytopes of almost all webs with clique number ≥5 admit non-rank facets. This adds support to the belief that these graphs are indeed the core of Ben Rebea's conjecture. Finally, we present a conjecture how to construct all facets of the stable set polytopes of webs.