Circular-arc Graphs Research Articles

English

English

Interval k-Graphs and Orders

An interval k-graph is the intersection graph of a family of intervals of the real line partitioned into k classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes. In this paper we study the cocomparability interval k-graphs; that is, the interval k-graphs whose complements have a transitive orientation and are therefore the incomparability graphs of strict partial orders. For brevity we call these orders interval k-orders. We characterize the kind of interval representations a cocomparability interval k-graph must have, and identify the structure that guarantees an order is an interval k-order. The case k = 2 is peculiar: cocomparability interval 2-graphs (equivalently proper- or unit-interval bigraphs, bipartite permutation graphs, and complements of proper circular-arc graphs to name a few) have been characterized in many ways, but we show that analogous characterizations do not hold if k > 2. We characterize the cocomparability interval 3-graphs via one forbidden subgraph and hence interval 3-orders via one forbidden suborder.

Formula omitted]-perfectly orientable [formula omitted]-minor-free and outerplanar graphs

A graph G is said to be 1-perfectly orientable if it has an orientation D such that for every vertex v∈V(G), the out-neighborhood of v in D is a clique in G. In 1982, Skrien posed the problem of characterizing the class of 1-perfectly orientable graphs. This graph class forms a common generalization of the classes of chordal and circular arc graphs; however, while polynomially recognizable via a reduction to 2-SAT, no structural characterization of this intriguing class of graphs is known. Based on a reduction of the study of 1-perfectly orientable graphs to the biconnected case, we characterize, both in terms of forbidden induced minors and in terms of composition theorems, the classes of 1-perfectly orientable K4-minor-free graphs and of 1-perfectly orientable outerplanar graphs. As part of our approach, we introduce a class of graphs defined similarly as the class of 2-trees and relate the classes of graphs under consideration to two other graph classes closed under induced minors studied in the literature: cyclically orientable graphs and graphs of separability at most 2.

The Induced Separation Dimension of a Graph

A linear ordering of the vertices of a graph G separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges $$\{a,b\}$$ and $$\{c,d\}$$ of G strongly independent if the set of endpoints $$\{a,b,c,d\}$$ induces a $$2K_2$$ in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$ of linear orders of V(G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in $$\mathcal {L}$$ . For each $$k \in \mathbb {N}$$ , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$ . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$ or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in $${\text {ISD}}(1)$$ which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for $${\text {ISD}}(1)$$ is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$ . We then briefly study $${\text {ISD}}(2)$$ and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is $$\Theta (\lg n)$$ and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every $$k \ge 2$$ , the recognition problem for $${\text {ISD}}(k)$$ is NP-complete even on AT-free graphs.

Completing orientations of partially oriented graphs

AbstractWe initiate a general study of what we call orientation completion problems. For a fixed class of oriented graphs, the orientation completion problem asks whether a given partially oriented graph P can be completed to an oriented graph in by orienting the (nonoriented) edges in P. Orientation completion problems commonly generalize several existing problems including recognition of certain classes of graphs and digraphs as well as extending representations of certain geometrically representable graphs.We study orientation completion problems for various classes of oriented graphs, including k‐arc‐strong oriented graphs, k‐strong oriented graphs, quasi‐transitive‐oriented graphs, local tournaments, acyclic local tournaments, locally transitive tournaments, locally transitive local tournaments, in‐tournaments, and oriented graphs that have directed cycle factors. We show that the orientation completion problem for each of these classes is either polynomial time solvable or NP‐complete. We also show that some of the NP‐complete problems become polynomial time solvable when the input‐oriented graphs satisfy certain extra conditions. Our results imply that the representation extension problems for proper interval graphs and for proper circular arc graphs are polynomial time solvable. The latter generalizes a previous result.

Completing colored graphs to meet a target property

We consider the problem of deciding whether a k-colored graph can be completed to have a given property. We establish that, when k is not fixed, the completion problems for (Helly) circular-arc graphs, even (Helly) proper or (Helly) unit circular-arc graphs, are NP-complete. When k is fixed, in the case of completion to a (Helly) circular-arc graph, (Helly) proper circular-arc graph, or (Helly) unit circular-arc graph we fully classify the complexities of the problems. We also show that deciding whether a 3-colored graph can be completed to be strongly chordal can be done in O(n2) time. As a corollary of our results, the sandwich problems for Helly circular-arc graphs, Helly proper circular-arc graphs, and Helly unit circular-arc graphs are NP-complete.

Graphs and digraphs represented by intervals and circular arcs

In this paper, we have shown that a graph is a circular arc graph if and only if the corresponding symmetric digraph with loops is a circular arc digraph. We characterize circular arc digraphs and circular arc graphs using circular ordering of their edges. We also characterize a circular arc graph as a union of an interval graph and a threshold graph. Finally, we characterize proper interval bigraphs and proper circular arc bigraphs using two linear orderings of their vertex set.

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter II: algorithms

This is the second and last chapter of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version (${\rm R{\small EP}}$), a proper circular-arc (PCA) model $\mathcal M$ is given and the goal is to obtain an equivalent UCA model $\mathcal U$. In the bounded version (${\rm B{\small OUND}R{\small EP}}$), $\mathcal M$ is given together with some lower and upper bounds that the beginning points of $\mathcal U$ must satisfy. In the minimal version (${\rm M{\small IN}UCA}$), the circumference of the circle and the length of the arcs in $\mathcal U$ must be simultaneously as small as possible, while the separation of the extremes is greater than a given threshold. In this chapter we take advantage of the theoretical framework developed in Chapter I to design efficient algorithms for these problems. We show a linear-time algorithm with negative certification for ${\rm R{\small EP}}$, that can also be implemented to run in logspace. We develop algorithms for different versions of ${\rm B{\small OUND}R{\small EP}}$ that run in linear space and quadratic time. Regarding ${\rm M{\small IN}UCA}$, we first show that the previous linear-time algorithm for ${\rm M{\small IN}UIG}$ (i.e., ${\rm M{\small IN}UCA}$ on UIG models) fails to provide a minimal model for some input graphs. We fix this algorithm but, unfortunately, it runs in linear space and quadratic time. Then, we apply the algorithms for ${\rm M{\small IN}UIG}$ and ${\rm M{\small IN}UCA}$ (Chapter I) to find the minimum powers of paths and cycles that contain given UIG and UCA models, respectively.

Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter I: theory

This is the first of two chapters of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version, a proper circular-arc (PCA) model ${\mathcal{M}}$ is given and the goal is to obtain an equivalent UCA model ${\mathcal{U}}$. In the bounded version, ${\mathcal{M}}$ is given together with some lower and upper bounds that the beginning points of ${\mathcal{U}}$ must satisfy. In the minimal version, we have to find a minimal model equivalent to ${\mathcal{M}}$, in which the circumference of the circle and length of the arcs must be simultaneously as small as possible. In this chapter we motivate these problems from an historical perspective, and we develop the theoretical framework required for the algorithms in Chapter II. We present new characterizations of those PCA models that have equivalent UCA models, and of those UCA models with a circle of circumference ${c}$ and the arcs of length $\ell$. We also prove that every UCA model is equivalent to a minimal one. We remark that all our results are of an algorithmic nature and can be readily employed to solve the problems at hand, even though these algorithms are not as efficient as those in Chapter II.

Interval Routing Schemes for Circular-Arc Graphs

Interval routing is a space efficient method to realize a distributed routing function. In this paper we show that every circular-arc graph allows a shortest path strict 2-interval routing scheme, i.e., by introducing a global order on the vertices and assigning at most two (strict) intervals in this order to the ends of every edge allows to depict a routing function that implies exclusively shortest paths. Since circular-arc graphs do not allow shortest path 1-interval routing schemes in general, the result implies that the class of circular-arc graphs has strict compactness 2, which was a hitherto open question. Additionally, we show that the constructed 2-interval routing scheme is a 1-interval routing scheme with at most one additional interval assigned at each vertex and we outline an algorithm to calculate the routing scheme for circular-arc graphs in 𝒪(n2) time, where n is the number of vertices.

Formula omitted]-perfectly orientable graphs and graph products

A graph G is said to be 1-perfectly orientable (1-p.o. for short) if it admits an orientation such that the out-neighborhood of every vertex is a clique in G. The class of 1-p.o. graphs forms a common generalization of the classes of chordal and circular arc graphs. Even though 1-p.o. graphs can be recognized in polynomial time, no structural characterization of 1-p.o. graphs is known. In this paper we consider the four standard graph products: the Cartesian product, the strong product, the direct product, and the lexicographic product. For each of them, we characterize when a nontrivial product of two graphs is 1-p.o.

Circular-arc hypergraphs: Rigidity via connectedness

A circular-arc hypergraphH is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set V(H) such that every hyperedge is an arc of consecutive vertices. We give a criterion for the uniqueness of an arc ordering in terms of connectedness properties of H. This generalizes the relationship between rigidity and connectedness disclosed by Chen and Yesha (1991) in the case of interval hypergraphs. Moreover, we state sufficient conditions for the uniqueness of tight arc orderings where, for any two hyperedges A and B such that A⊆B≠V(H), the corresponding arcs must share a common endpoint. We notice that these conditions are obeyed for the closed neighborhood hypergraphs of proper circular-arc graphs, implying for them the known rigidity results that were originally obtained using the theory of local tournament graph orientations.

On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid

Golumbic, Lipshteyn and Stern [12] proved that every graph can be represented as the edge intersection graph of paths on a grid (EPG graph), i.e., one can associate with each vertex of the graph a nontrivial path on a rectangular grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. For a nonnegative integer k, Bk-EPG graphs are defined as EPG graphs admitting a model in which each path has at most k bends. Circular-arc graphs are intersection graphs of open arcs of a circle. It is easy to see that every circular-arc graph is a B4-EPG graph, by embedding the circle into a rectangle of the grid. In this paper, we prove that circular-arc graphs are B3-EPG, and that there exist circular-arc graphs which are not B2-EPG. If we restrict ourselves to rectangular representations (i.e., the union of the paths used in the model is contained in the boundary of a rectangle of the grid), we obtain EPR (edge intersection of paths in a rectangle) representations. We may define Bk-EPR graphs, k≥0, the same way as Bk-EPG graphs. Circular-arc graphs are clearly B4-EPR graphs and we will show that there exist circular-arc graphs that are not B3-EPR graphs. We also show that normal circular-arc graphs are B2-EPR graphs and that there exist normal circular-arc graphs that are not B1-EPR graphs. Finally, we characterize B1-EPR graphs by a family of minimal forbidden induced subgraphs, and show that they form a subclass of normal Helly circular-arc graphs.

Partial Characterizations of 1‐Perfectly Orientable Graphs

AbstractWe study the class of 1‐perfectly orientable graphs, that is, graphs having an orientation in which every out‐neighborhood induces a tournament. 1‐perfectly orientable graphs form a common generalization of chordal graphs and circular arc graphs. Even though they can be recognized in polynomial time, little is known about their structure. In this article, we develop several results on 1‐perfectly orientable graphs. In particular, we (i) give a characterization of 1‐perfectly orientable graphs in terms of edge clique covers, (ii) identify several graph transformations preserving the class of 1‐perfectly orientable graphs, (iii) exhibit an infinite family of minimal forbidden induced minors for the class of 1‐perfectly orientable graphs, and (iv) characterize the class of 1‐perfectly orientable graphs within the classes of cographs and of cobipartite graphs. The class of 1‐perfectly orientable cobipartite graphs coincides with the class of cobipartite circular arc graphs.

Induced disjoint paths in circular-arc graphs in linear time

The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤i<j≤k. We present a linear-time algorithm that solves Induced Disjoint Paths and finds the corresponding paths (if they exist) on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (si,ti) are not necessarily distinct. In both cases, if a representation of the graph is given, then the algorithms run in O(n+k) time.

List H-Coloring a Graph by Removing Few Vertices

In the deletion version of the list homomorphism problem, we are given graphs G and H, a list $$L(v)\subseteq V(H)$$ for each vertex $$v\in V(G)$$ , and an integer k. The task is to decide whether there exists a set $$W \subseteq V(G)$$ of size at most k such that there is a homomorphism from $$G {\setminus } W$$ to H respecting the lists. We show that DL-Hom( $${H}$$ ), parameterized by k and |H|, is fixed-parameter tractable for any $$(P_6,C_6)$$ -free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom( $${H}$$ ) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

Solving the canonical representation and Star System Problems for proper circular-arc graphs in logspace

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concave-round graphs, which still possess (not necessarily proper) circular-arc models, we show that a canonical circular-arc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circular-arc models of circular-arc hypergraphs. This class of hypergraphs corresponds to matrices with the circular ones property, which play an important role in computational genomics. Our results imply that there is a logspace algorithm that decides whether a given matrix has this property.Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circular-arc, concave-round, and co-convex graphs.Note that solving a problem in logspace implies that it is solvable by a parallel algorithm of the class AC1. For the problems under consideration, at most AC2 algorithms were known earlier.

On the isomorphism problem for Helly circular-arc graphs

The isomorphism problem is known to be efficiently solvable for interval graphs, while for the larger class of circular-arc graphs its complexity status stays open. We consider the intermediate class of intersection graphs for families of circular arcs that satisfy the Helly property. We solve the isomorphism problem for this class in logarithmic space. If an input graph has a Helly circular-arc model, then our algorithm constructs it canonically, which means that the models constructed for isomorphic graphs are equal.

To Find the Comparison of Non-Split Domination Number, The Average Distance and The Diameter of Circular-Arc Graphs

To Find the Comparison of Non-Split Domination Number, The Average Distance and The Diameter of Circular-Arc Graphs

An Algorithm for the Feedback Vertex Set Problem on a Normal Helly Circular-Arc Graph

The feedback vertex set (FVS) problem is to find the set of vertices of minimum cardinality whose removal renders the graph acyclic. The FVS problem has applications in several areas such as combinatorial circuit design, synchronous systems, computer systems, and very-large-scale integration (VLSI) circuits. The FVS problem is known to be NP-hard for simple graphs, but polynomi-al-time algorithms have been found for special classes of graphs. The intersection graph of a collection of arcs on a circle is called a circular-arc graph. A normal Helly circular-arc graph is a proper subclass of the set of circular-arc graphs. In this paper, we present an algorithm that takes time to solve the FVS problem in a normal Helly circular-arc graph with n vertices and m edges.