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

Abstract

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.

Helly Circular-Arc Graph Isomorphism Is in Logspace

“…Our interest is in intersection matrices that describe the intersection types between the arcs of a CA system. The following notation was introduced in [LS09]. Definition 2.1.…”

Two cases of polynomial-time solvability for the coloring problem

“…A proper circular-arc graph is a circular-arc graph that has an intersection model in which no arc properly contains another. A minimal set of forbidden induced subgraphs for the class of proper circular-arc graphs is known and completely described by Lin and Szwarcfiter 2009. It is constituted by C 3 + P 1 , C 5 + P 1 , C 6 , C 7 + P 1 , C 8 , C 9 + P 1 , C 10 , .…”

Interval Routing Schemes for Circular-Arc Graphs

“…A difference between interval graphs and circular-arc graphs, that is worth mentioning, is that the maximal cliques of interval graphs can be associated to points of the "interval model" and therefore an interval graph can have no more maximal cliques than vertices. In contrast, circular-arc graphs may contain maximal cliques that do not correspond to points of some arc model [17]. In fact, just as in arbitrary graphs, the number of maximal cliques in circular-arc graphs can grow exponentially in the size of the graph [26].…”

“…A unit circulararc graph is a circular-arc graph that has an arc model in which the arcs have unit length. For a survey on circular-arc graphs see [17] and for the definition of further special graph classes see [3]. Figure 1 illustrates a strict circular-arc graph.…”