Francisco J. Soulignac

Normal Helly circular-arc graphs and its subclasses

A Helly circular-arc model M=(C,A) is a circle C together with a Helly family A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how these classes of graphs relate with straight and round digraphs.

Arboricity, h-index, and dynamic algorithms

We propose a new data structure for manipulating graphs, called h-graph, which is particularly suited for designing dynamic algorithms. The structure itself is simple, consisting basically of a triple of elements, for each vertex of the graph. The overall size of all triples is O(n+m), for a graph with n vertices and m edges. We describe algorithms for performing the basic operations related to dynamic applications, as insertions and deletions of vertices or edges, and adjacency queries. The data structure employs a technique first described by Chiba and Nishizeki [Chiba, Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput. 14 (1) (1985) 210-223], and relies on the arboricity of graphs. Using the proposed data structure, we describe several dynamic algorithms for solving problems as listing the cliques of a given size, recognizing diamond-free graphs, and finding simple, simplicial and dominated vertices. These algorithms are the first of their kind to be proposed in the literature. In fact, the dynamic algorithms for the above problems lead directly to new static algorithms, and using the data structure we also design new static algorithms for the problems of counting subgraphs of size 4, recognizing cop-win graphs and recognizing strongly chordal graphs. The complexities of all of the proposed static algorithms improve over the complexities of the so far existing algorithms, for graphs of low arboricity. In addition, for the problems of counting subgraphs of size 4 and recognizing diamond-free graphs, the improvement is general.

The clique operator on circular-arc graphs

A circular-arc graphG is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graphK(G) of a graph G is the intersection graph of its cliques. The iterated clique graphK^i(G) of G is defined by K^0(G)=G and K^i^+^1(G)=K(K^i(G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K-converges, if it is K-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.

Powers of cycles, powers of paths, and distance graphs

In 1988, Golumbic and Hammer characterized the powers of cycles, relating them to circular arc graphs. We extend their results and propose several further structural characterizations for both powers of cycles and powers of paths. The characterizations lead to linear-time recognition algorithms of these classes of graphs. Furthermore, as a generalization of powers of cycles, powers of paths, and even of the well-known circulant graphs, we consider distance graphs. While the colorings of these graphs have been intensively studied, the recognition problem has been so far neglected. We propose polynomial-time recognition algorithms for these graphs under additional restrictions.

Proper Helly circular-arc graphs

A circular-arc model M=(C,A) is a circle C together with a collection A of arcs of C. If no arc is contained in any other then M is a proper circular-arc model, if every arc has the same length then M is a unit circular-arc model and if A satisfies the Helly Property then M is a Helly circular-arc model. A (proper) (unit) (Helly) circular-arc graph is the intersection graph of the arcs of a (proper) (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear time recognition algorithms have been described both for the general class and for some of its subclasses. In this article we study the circular-arc graphs which admit a model which is simultaneously proper and Helly. We describe characterizations for this class, including one by forbidden induced subgraphs. These characterizations lead to linear time certifying algorithms for recognizing such graphs. Furthermore, we extend the results to graphs which admit a model which is simultaneously unit and Helly, also leading to characterizations and a linear time certifying algorithm.

On cliques of Helly Circular-arc Graphs

Abstract A circular-arc graph is the intersection graph of a set of arcs on the circle. It is a Helly circular-arc graph if it has a Helly model, where every maximal clique is the set of arcs that traverse some clique point on the circle. A clique model is a Helly model that identifies one clique point for each maximal clique. A Helly circular-arc graph is proper if it has a Helly model where no arc is a subset of another. In this paper, we show that the clique intersection graphs of Helly circular-arc graphs are related to the proper Helly circular-arc graphs. This yields the first polynomial (linear) time recognition algorithm for the clique graphs of Helly circular-arc graphs. Next, we develop an O ( n ) time algorithm to obtain a clique model of Helly model, improving the previous O ( n 2 ) bound. This gives a linear-time algorithm to find a proper Helly model for the clique graph of a Helly circular-arc graph. As an application, we find a maximum weighted clique of a Helly circular-arc graph in linear time.

A Simple Linear Time Algorithm for the Isomorphism Problem on Proper Circular-Arc Graphs

A circular-arc model

Partial characterizations of coordinated graphs: line graphs and complements of forests

{\mathcal {M}} =(C,\mathcal{A})

Fully Dynamic Recognition of Proper Circular-Arc Graphs

is a circle Ctogether with a collection

Short Models for Unit Interval Graphs

\mathcal{A}