Proper Circular-arc Graphs Research Articles

Hadwiger’s conjecture for proper circular arc graphs

Circular arc graphs are graphs whose vertices can be represented as arcs on a circle such that any two vertices are adjacent if and only if their corresponding arcs intersect. Proper circular arc graphs are graphs which have a circular arc representation where no arc is completely contained in any other arc. Hadwiger’s conjecture states that if a graph G has chromatic number k , then a complete graph with k vertices is a minor of G . We prove Hadwiger’s conjecture for proper circular arc graphs.

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

Circular 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.

Recognizing and representing proper interval graphs in parallel using merging and sorting

We present a parallel algorithm for recognizing and representing a proper interval graph in O ( log 2 n ) time with O ( m + n ) processors on the CREW PRAM, where m and n are the number of edges and vertices in the graph. The algorithm uses sorting to compute a weak linear ordering of the vertices, from which an interval representation is easily obtained. It is simple, uses no complex data structures, and extends ideas from an optimal sequential algorithm for recognizing and representing a proper interval graph [X. Deng, P. Hell, J. Huang, Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs, SIAM J. Comput. 25 (2) (1996) 390–403].

Locally connected spanning trees in strongly chordal graphs and proper circular-arc graphs

A locally connected spanning tree of a graph G is a spanning tree T of G such that the set of all neighbors of v in T induces a connected subgraph of G for every v ∈ V ( G ) . The purpose of this paper is to give linear-time algorithms for finding locally connected spanning trees on strongly chordal graphs and proper circular-arc graphs, respectively.

Polynomial time recognition of unit circular-arc graphs

We present an efficient algorithm for recognizing unit circular-arc (UCA) graphs, based on a characterization theorem for UCA graphs proved by Tucker in the seventies. Given a proper circular-arc (PCA) graph G, the algorithm starts from a PCA model for G, removes all its circle-covering pairs of arcs and determines whether G is a UCA graph. We also give an O ( N ) time bound for Tucker's 3 / 2 -approximation algorithm for coloring circular-arc graphs with N vertices, when a circular-arc model is given.

Interval bigraphs and circular arc graphs

AbstractWe prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two, which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidal‐triple‐free graphs, permutation graphs, and co‐comparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of Müller. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 313–327, 2004

Some new results on circle graphs

Abstract: The intersection graph of a set of chords on a circle is called a circle graph. This class of graphs admits some interesting subclasses, such as: Helly circle graphs, clique-Helly circle graphs, unit circle graphs and proper circular-arc graphs. In this paper, we prove some inclusion relations among these subclasses. A necessary condition for a graph being a Helly circle graph is shown and we conjecture that this condition is sufficient too. All possible intersections among these subclasses are analized. The number of regions generated is 10. We show a minimal example belonging to each one of them. Finally, some properties about minimal forbidden subgraphs for circle graphs are proved.

The round-up property of the fractional chromatic number for proper circular arc graphs

Let G = (V, E) be a graph and let k be a nonnegative integer. A vector c ∈ ℤ is called k-colorable iff there exists a coloring of G with k colors that assigns exactly c(v) colors to vertex v ∈ V. Denote by χ (G) and χf(G) the chromatic number and fractional chromatic number, respectively. We prove that χ(G) = ⌈χf(G)⌉ holds for every proper circular arc graph G. For this purpose, a more general round-up property is characterized by means of a polyhedral description of all k-colorable vectors. Both round-up properties are equivalent for proper circular arc graphs. The polyhedral description is established and, as a by-product, a known coloring algorithm is generalized to multicolorings. The round-up properties do not hold for the larger classes of circular arc graphs and circle graphs, unless P = NP. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 256–267, 2000

The round-up property of the fractional chromatic number for proper circular arc graphs

The round-up property of the fractional chromatic number for proper circular arc graphs

Convex-Round and Concave-Round Graphs

We introduce two new classes of graphs which we call convex-round, respectively concave-round graphs. Convex-round (concave-round) graphs are those graphs whose vertices can be circularly enumerated so that the (closed) neighborhood of each vertex forms an interval in the enumeration. Hence the two classes transform into each other by taking complements. We show that both classes of graphs have nice structural properties. We observe that the class of concave-round graphs properly contains the class of proper circular arc graphs and, by a result of Tucker [ Pacific J. Math., 39 (1971), pp. 535--545], is properly contained in the class of general circular arc graphs. We point out that convex-round and concave-round graphs can be recognized in O(n+m) time (here n denotes the number of vertices and m the number of edges of the graph in question). We show that the chromatic number of a graph which is convex-round (concave-round) can be found in time O(n+m) (O(n2 )). We describe optimal O(n+m) time algorithms for finding a maximum clique, a maximum matching, and a Hamiltonian cycle (if one exists) for the class of convex-round graphs. Finally, we pose a number of open problems and conjectures concerning the structure and algorithmic properties of the two new classes and a related third class of graphs.

Efficient parallel recognition of some circular arc graphs, II

Based on Tucker's work, we present an accurate proof of the characterization of proper circular arc graphs and obtain the first efficient parallel algorithm which not only recognizes proper circular arc graphs but also constructs proper circular arc representations. The algorithm runs inO(log2 n) time withO(n 3) processors on a Common CRCW PRAM. The sequential algorithm can be implemented to run inO(n 2) time and is optimal if the input graph is given as an adjacency matrix, so to speak.

Radius versus diameter in cocomparability and intersection graphs

Whereas in general graphs the only connection between radius and diameter can be expressed by the inequality r( G)⩽diam( G)⩽2 r( G), in this paper it is shown that much sharper inequalities of type 2 r( G)-diam( G)⩽ a are valid inside the classes of cocomparability graphs and trapezoid graphs, with constants a = 3,2. For circular-arc graphs and proper circular-arc graphs we obtain diam( G) - r( G)⩽ b, with constants b=2,1.

On powers of circular arc graphs and proper circular arc graphs

In this paper it is shown that the class K of circular arc graphs is closed under the operation of power; this means, G ϵ K implies G k ϵ K for any positive integer k Furthermore, G k ϵ K implies G k+2 ϵ K and if diam( G k ) ⩾ 4 then G k ϵ K implies G k+1 ϵ K . Another result on the powers of directed graphs is that if G k belongs to the class of in-, out-, or local tournaments, then so does G k+1 . This result is used to prove the closedness of proper circular arc graphs under the power operation.

A Linear Algorithm for Maximum Weight Cliques in Proper Circular Arc Graphs

We present an $O(n)$algorithm to find a maximum clique in a proper circular arc graph. We assume that the input graph is represented by a sorted simple family of circular arcs or by an equivalent representation. In Deng, Hell, and Huang [SIAM J. Comput., 25 (1996), pp. 390–403], we gave an $O(m + n)$ algorithm to find such a representation for a proper circular arc graph given by its adjacency lists. As an application we also give an $O(n)$ algorithm for q-coloring proper circular arc graphs for a fixed q. (Such an algorithm was first given by Teng and Tucker.) Finally we indicate how our algorithm can be modified to find a maximum weight clique in a weighted graph, also in time $O(n)$.

Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs

Our main result is a linear-time (that is, time $O(m + n)$) algorithm to recognize and represent proper circular-arc graphs. The best previous algorithm, due to A. Tucker, has time complexity $O(n^2 )$. We take advantage of the fact that (among connected graphs) proper circular-arc graphs are precisely the graphs orientable as local tournaments, and we use a new characterization of local tournaments. The algorithm depends on repeated representation of portions of the input graph as proper interval graphs. Thus we also find it useful to give a new linear-time algorithm to represent proper interval graphs. This latter algorithm also depends on an orientation characterization of proper interval graphs. It is conceptually simple and does not use complex data structures. As a byproduct of the correctness proof of the algorithm, we also obtain a new proof of a characterization of proper interval graphs by forbidden subgraphs.

Lexicographic orientation and representation algorithms for comparability graphs, proper circular arc graphs, and proper interval graphs

AbstractWe introduce a simple new technique which allows us to solve several problems that can be formulated as seeking a suitable orientation of a given undirected graph. In particular, we use this technique to recognize and transitively orient comparability graphs, to recognize and represent proper circular arc graphs, and to recognize and represent proper interval graphs. As a consequence, we derive and represent proper interval graphs. As a consequence, we derive simple new proofs of a theorem of Ghouila‐Houri and a theorem of Skrien. Our algorithms are conceptually simpler than (and often of comparable efficiency to) the existing algorithms for these problems. © 1995 John Wiley & Sons, Inc.

On chordal proper circular arc graphs

Wegner proved that a chordal graph is a proper interval graph if and only if it is claw-free, net-free and tent-free. We observe that in fact the characterization can be refined to say that a chordal graph is a proper interval graph if and only if it is claw-free, net-free and not a multiple of the tent. This observation implies that a chordal graph is a proper circular arc graph if and only if it is claw-free and net-free. A further implication of this result is that a chordal graph can be oriented as a local tournament - an oriented raph in which the predecessors as well as the successors of any vertex induce a tournament - if and only if it is claw-free and net-free.

On the structure of locally semicomplete digraphs

In this paper we continue the study, started by J. Bang-Jensen (1989), of locally semicomplete digraphs, a generalization of tournaments, to which many well-known tournament results extend. The underlying undirected graphs of the locally semicomplete digraphs are precisely the proper circular-arc graphs. We give new results on the structure of locally semicomplete digraphs, as well as several examples of properties of tournaments and semicomplete digraphs that do not extend to the class of locally semicomplete digraphs.

Efficient reduction for path problems on circular-arc graphs

An efficient reduction process for path problems on circular-arc graphs is introduced. For the parity path problem, this reduction gives anO(n+m) algorithm for proper circular-arc graphs, and anO(n+m) algorithm for general circular-arc graphs. This reduction also gives anO(n+m) algorithm for the two path problem on circular-arc graphs.

Dominating sets and domatic number of circular arc graphs

A dominating set of a graph G = ( N, E) is a subset S of nodes such that every node is either in S or adjacent to a node which is in S. The domatic number of G is the size of a maximum cardinality partition of N into dominating sets. The problems of finding a minimum cardinality dominating set and the domatic number are both NP-complete even for special classes of graphs. In the present paper we give an O( n∣ E∣) time algorithm that finds a minimum cardinality dominating set when G is a circular arc graph (intersection graph of arcs on a circle). The domatic number problem is solved in O( n 2 log n) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs.