Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs

Discrete Applied Mathematics

Rautenbach

Soulignac

et al. 2011

a b s t r a c tIn 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.

Section: (V) G Is a Ui Graph And For Some Ui Modelmentioning

confidence: 96%

Powers of cycles, powers of paths, and distance graphs

Discrete Applied Mathematics

Rautenbach

Soulignac

et al. 2011

“…If H is not a PCA graph, then it is not in K (HCA) [20]. Otherwise, a PCA model can be found using the algorithms in [26,15] in O(n + m) time and, if needed, it can be normalized in O(n) time as in [27]. [28], and testing if [17].…”

Section: Corollary 8 Let H Be a Graph Then H = K (G) For Some Nhca mentioning

confidence: 98%

“…Circular-arc graphs and their subclasses have been receiving much attention recently [13,14]. For CA, PCA, HCA, I, PI and PHCA graphs, there are several characterizations and linear time recognition algorithms which construct a model (see [15][16][17][18][19]14]). …”

Section: Introductionmentioning

confidence: 99%

The clique operator on circular-arc graphs

Discrete Applied Mathematics

Soulignac

Szwarcfiter

2010

a b s t r a c tA circular-arc graph G 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 graph K (G) of a graph G is the intersection graph of its cliques. The iterated clique graph K 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.

“…The emphasis is on linear time recognition algorithms, both for this class and subclasses of it. It is worth mentioning that in the last year, at least five papers appeared in conference proceedings, describing linear time algorithms for circular-arc graphs and subclasses [34,36,37,41,42]. The books by Spinrad [65] and by Golumbic [19] contain comprehensive accounts on the class.…”

Section: Introductionmentioning

confidence: 99%

Characterizations and recognition of circular-arc graphs and subclasses: A survey

Szwarcfiter

2009

Discrete Mathematics

a b s t r a c tCircular graphs are intersection graphs of arcs on a circle. These graphs are reported to have been studied since 1964, and they have been receiving considerable attention since a series of papers by Tucker in the 1970s. Various subclasses of circular-arc graphs have also been studied. Among these are the proper circular-arc graphs, unit circular-arc graphs, Helly circular-arc graphs and co-bipartite circular-arc graphs. Several characterizations and recognition algorithms have been formulated for circular-arc graphs and its subclasses. In particular, it should be mentioned that linear time algorithms are known for all these classes of graphs. In the present paper, we survey these characterizations and recognition algorithms, with emphasis on the linear time algorithms.