Circular-arc Graphs Research Articles

Modification problems toward proper (Helly) circular-arc graphs

We present a 9k⋅nO(1)-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [Computer Science Review 2023]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs.

Graph Bipartization and Via Minimization for Intersection Graphs

The graph bipartization problem, arising from via minimization in VLSI design and related areas, consists in finding a vertex subset [Formula: see text] of graph [Formula: see text] such that the induced subgraph [Formula: see text] is bipartite and [Formula: see text] is maximized. The problem has been proved to be NP-hard even for planar graphs and cubic graphs. On the other hand, the study of polynomial-time algorithms for typical graph classes is significant in both theoretical and applied aspects. This paper focuses on several intersection graph classes, such as line graphs, circular-arc graphs, and directed path graphs. For the line graphs, we show the NP-hardness results in general and present the polynomial-time algorithms for special cases. For circular-arc graphs and directed path graphs, we propose algorithms that improve on the previously known ones.

Maximum Bipartite Subgraphs of Geometric Intersection Graphs

We study the Maximum Bipartite Subgraph ([Formula: see text]) problem, which is defined as follows. Given a set [Formula: see text] of [Formula: see text] geometric objects in the plane, we want to compute a maximum-size subset [Formula: see text] such that the intersection graph of the objects in [Formula: see text] is bipartite. We first give an [Formula: see text]-time algorithm that computes an almost optimal solution for the problem on circular-arc graphs. We show that the [Formula: see text] problem is [Formula: see text]-hard on geometric graphs for which the maximum independent set is [Formula: see text]-hard (hence, it is [Formula: see text]-hard even on unit squares and unit disks). On the other hand, we give a [Formula: see text] for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks, and unit-height axis parallel rectangles. Additionally, we prove that the Maximum Triangle-free Subgraph ([Formula: see text]) problem is NP-hard for axis-parallel rectangles. Here the objective is the same as that of the [Formula: see text] except the intersection graph induced by the set [Formula: see text] needs to be triangle-free only (instead of being bipartite).

Partitioning subclasses of chordal graphs with few deletions

In the (Vertex)k-Way Cut problem, input is an undirected graph G, an integer s, and the goal is to find a subset S of edges (vertices) of size at most s, such that G−S has at least k connected components. Downey et al. [Electr. Notes Theor. Comput. Sci. 2003] showed that k-Way Cut is W[1]-hard parameterized by k. However, Kawarabayashi and Thorup [FOCS 2011] showed that the problem is fixed-parameter tractable (FPT) in general graphs with respect to the parameter s and provided a O(ssO(s)n2) time algorithm, where n denotes the number of vertices in G. The best-known algorithm for this problem runs in time sO(s)nO(1) given by Lokshtanov et al. [ACM Tran. of Algo. 2021]. On the other hand, Vertexk-Way Cut is W[1]-hard with respect to either of the parameters, k or s or k+s. These algorithmic results motivate us to look at the problems on special classes of graphs. In this paper, we consider the (Vertex)k-Way Cut problem on subclasses of chordal graphs and obtain the following results.•We first give a sub-exponential FPT algorithm for k-Way Cut running in time 2O(slog⁡s)nO(1) on chordal graphs.•It is “known” that Vertexk-Way Cut is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k+s. We complement this hardness result by designing polynomial-time algorithms for Vertexk-Way Cut on interval graphs, circular-arc graphs and permutation graphs.

Bonds Intersecting Long Paths in \(k\) -Connected Graphs

In 1966, Gallai asked whether there is a vertex which passes through all longest paths of a connected graph. Although this has been verified for some special classes of graphs such as outerplanar graphs, circular arc graphs, and series-parallel graphs, the answer is negative for general graphs. In this paper, we prove among other results that if we replace the vertex by a bond, then the answer is affirmative. A bond of a graph is a minimal nonempty edge-cut. In particular, in any 2-connected graph, the set of all edges incident to a vertex is a bond, called a vertex-bond. Clearly, for a 2-connected graph, a path passes through a vertex if and only if it meets the vertex-bond with respect to . Therefore, a very natural approach to Gallai’s question is to study whether there is a bond meeting all longest paths. Let denote the length of a longest path of connected graphs. We show that there is a bond meeting all paths of length at least and for any 2- and 3-connected graph, respectively. For any -connected graph , we show that there is a bond meeting all paths of length at least , where if is even and if is odd. Our results also provide analogs of the results on bonds meeting long cycles given in [P.-L. Wu, Combin. Probab. Comput., 6 (1997), pp. 107–113; and S. McGuinness, Combinatorica, 25 (2005), pp. 439–450].

Compact representation of interval graphs and circular-arc graphs of bounded degree and chromatic number

In this paper we initiate the study of designing parameterized compact data structures for an interval graph G with n vertices. First, we show that when the maximum degree of G is bounded by Δ, we show that the space requirement of our data structure is at least (16nlog2⁡Δ−O(n))-bit, which is one of the main contributions of this work. Next, by straightforward modifications of the result of Acan et al. (2021) [5], we obtain an (nlog2⁡Δ+O(n))-bit data structure supporting the standard navigational queries i.e., degree, adjacency, and neighborhood optimally. Hence, we provide a compact representation of interval graphs with bounded degree for the first time in literature. Note that this upper bound result takes less space than their (nlog2⁡n+O(n))-bit upper bound when Δ=O(nϵ), for any 0<ϵ<1. Next, we consider the interval graphs with bounded chromatic number ▪, and design a ▪-bit data structure with efficient query times. Unlike the previous upper bound, this data structure is completely new and doesn't follow from the result of Acan et al. (2021) [5]. Moreover, this takes less space than their data structure when ▪. Finally, we provide parameterized compact data structures for circular-arc graphs as well with bounded degree or bounded chromatic number condition.

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.

Essential obstacles to Helly circular-arc graphs

A Helly circular-arc graph is the intersection graph of a set of arcs on a circle having the Helly property. We introduce essential obstacles, which are a refinement of the notion of obstacles, and prove that essential obstacles are precisely the minimal forbidden induced circular-arc subgraphs for the class of Helly circular-arc graphs. We show that it is possible to find in linear time, in any given obstacle, some minimal forbidden induced subgraph for the class of Helly circular-arc graphs contained as an induced subgraph. Moreover, relying on an existing linear-time algorithm for finding induced obstacles in circular-arc graphs, we conclude that it is possible to find in linear time an induced essential obstacle in any circular-arc graph that is not a Helly circular-arc graph. The problem of finding a forbidden induced subgraph characterization, not restricted only to circular-arc graphs, for the class of Helly circular-arc graphs remains unresolved. As a partial answer to this problem, we find the minimal forbidden induced subgraph characterization for the class of Helly circular-arc graphs restricted to graphs containing no induced claw and no induced 5-wheel. Furthermore, we show that there is a linear-time algorithm for finding, in any given graph that is not a Helly circular-arc graph, an induced subgraph isomorphic to claw, 5-wheel, or some minimal forbidden induced subgraph for the class of Helly circular-arc graphs.

Describing hereditary properties by forbidden circular orderings

Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is k-colourable if and only if V(G) admits a linear ordering ≤ with no vertices v1≤⋯≤vk+1 such that vivi+1∈E(G) for every i∈{1,⋯,k}. In this paper, we study such characterizations when the extra structure is a circular ordering of the vertex set. We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than k. In fact, every description by finitely many forbidden circularly ordered graphs can be translated to a description by finitely many forbidden linearly ordered graphs. Nevertheless, our observations underscore the fact that in many cases the circular order descriptions are nicer and more natural.

The balanced connected subgraph problem for geometric intersection graphs

We study the ▪ (shortly, ▪) problem on geometric intersection graphs such as interval, circular-arc, permutation, unit-disk, outer-string graphs, etc. We are given a ▪ graph G=(V,E) and an integer k. Each vertex in V is colored with either “▪” or “▪”. The BCS problem seeks an induced connected subgraph H of size at least k in G such that H is ▪, i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On the one hand, we prove that the BCS problem is NP-complete on the unit-disk, outer-string, grid, and unit-square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on the interval, circular-arc, and permutation graphs. Our algorithm for interval and circular-arc graphs solves the more general problem of computing a minimum cardinality ▪ in the same classes of graphs, which may be of independent interest.

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time 2^{O(rw^3)}cdot n^{4}, 2^{O(q^2log (q))}cdot n^{4}, and n^{O(k^2)} where rw is the rank-width, q the {mathbb {Q}}-rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an XP algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

Avoidable vertices and edges in graphs: Existence, characterization, and applications

A vertex in a graph is simplicial if its neighborhood forms a clique. We consider three generalizations of the concept of simplicial vertices: avoidable vertices (also known as OCF-vertices), simplicial paths, and their common generalization avoidable paths. In an earlier version of this paper, published as an extended abstract in the proceedings of the 16th International Symposium on Algorithms and Data Structures (WADS 2019), we presented a general conjecture on the existence of avoidable paths. The conjecture was recently proved by Bonamy et al. (2020). The conjecture—now a theorem—implies a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence of avoidable vertices and a result due to Chvátal, Sritharan, and Rusu from 2002 on the existence of simplicial paths in graphs excluding all sufficiently long induced cycles. In turn, both of these results generalize Dirac’s classical result on the existence of simplicial vertices in chordal graphs.We prove that every graph with at least two edges has two avoidable edges, which strengthens the statement of the theorem of Bonamy et al. for the case of paths of length one. We point out a close relationship between avoidable vertices in a graph and its minimal triangulations, and identify new algorithmic uses of avoidable vertices, leading to new polynomially solvable cases of the maximum weight clique problem in two classes of graphs simultaneously generalizing chordal graphs and circular-arc graphs. Finally, we observe that the results about the existence of avoidable vertices and edges have interesting consequences for highly symmetric graphs: in a vertex-transitive graph every induced two-edge path closes to an induced cycle, while in an edge-transitive graph every three-edge path closes to a cycle and every induced three-edge path closes to an induced cycle.

A certifying and dynamic algorithm for the recognition of proper circular-arc graphs

We present a dynamic algorithm for the recognition of proper circular-arc (PCA) graphs, that supports the insertion and removal of vertices (together with its incident edges). The main feature of the algorithm is that it outputs a minimally non-PCA induced subgraph when the insertion of a vertex fails. Each operation cost O(log⁡n+d) time, where n is the number vertices and d is the degree of the modified vertex. When removals are disallowed, each insertion is processed in O(d) time. The algorithm also provides two constant-time operations to query if the dynamic graph is proper Helly (PHCA) or proper interval (PIG). When the dynamic graph is not PHCA (resp. PIG), a minimally non-PHCA (resp. non-PIG) induced subgraph is obtained.

On H-Topological Intersection Graphs

Biró et al. (Discrete. Math 100(1–3):267–279, 1992) introduced the concept of H-graphs, intersection graphs of connected subgraphs of a subdivision of a graph H. They are related to and generalize many important classes of geometric intersection graphs, e.g., interval graphs, circular-arc graphs, split graphs, and chordal graphs. Our paper starts a new line of research in the area of geometric intersection graphs by studying several classical computational problems on H-graphs: recognition, graph isomorphism, dominating set, clique, and colorability. We negatively answer the 25-year-old question of Biró, Hujter, and Tuza which asks whether H-graphs can be recognized in polynomial time, for a fixed graph H. We prove that it is \(\textsf {NP}\)-complete if H contains the diamond graph as a minor. On the positive side, we provide a polynomial-time algorithm recognizing T-graphs, for each fixed tree T. For the special case when T is a star \(S_d\) of degree d, we have an \(\mathcal{O}(n^{3.5})\)-time algorithm. We give \(\textsf {FPT}\)- and \(\textsf {XP}\)-time algorithms solving the minimum dominating set problem on \(S_d\)-graphs and H-graphs, parametrized by d and the size of H, respectively. The algorithm for H-graphs adapts to an \(\textsf {XP}\)-time algorithm for the independent set and the independent dominating set problems on H-graphs. If H contains the double-triangle as a minor, we prove that the graph isomorphism problem is GI-complete and that the clique problem is APX-hard. On the positive side, we show that the clique problem can be solved in polynomial time if H is a cactus graph. Also, when a graph has a Helly H-representation, the clique problem is polynomial-time solvable. Further, we show that both the k-clique and the list k-coloring problems are solvable in FPT-time on H-graphs, parameterized by k and the treewidth of H. In fact, these results apply to classes of graphs with treewidth bounded by a function of the clique number. We observe that H-graphs have at most \(n^{O(\Vert H\Vert )}\) minimal separators which allows us to apply the meta-algorithmic framework of Fomin, Todinca, and Villanger (2015) to show that for each fixed t, finding a maximum induced subgraph of treewidth t can be done in polynomial time. In the case when H is a cactus, we improve the bound to \(O(\Vert H\Vert n^2)\).

Some Properties on Degrees of Dominating Sets for Circular-Arc Graph

Some Properties on Degrees of Dominating Sets for Circular-Arc Graph

L(2,1,1)-Labeling of Circular-Arc Graphs

Graph labeling problem has been broadly studied in recent past for its wide applications, in mobile communication system for frequency assignment, radar, circuit design, X-ray crystallography, coding theory, etc. An L211-labeling (L211L) of a graph G = (V, E) is a function γ : V → Z∗ such that |γ(u) − γ(v)| ≥ 2, if d(u, v) = 1 and |γ(u) − γ(v)| ≥ 1, if d(u, v) = 1 or 2, where Z∗ be the set of non-negative integers and d(u, v) represents the distance between the nodes u and v. The L211L numbers of a graph G, are denoted by λ2,1,1(G) which is the difference between largest and smallest labels used in L211L. In this article, for circular-arc graph (CAG) G we have proved that λ2,1,1(G) ≤ 6∆ − 4, where ∆ represents the degree of the graph. Beside this we have designed a polynomial time algorithm to label a CAG satisfying the conditions of L211L. The time complexity of the algorithm is O(n∆2), where n is the number of nodes of the graph G.

$L(p,q)$-labeling of graphs with interval representations

We provide upper bounds on the $L(p,q)$-labeling number of graphs which have interval (or circular-arc) representations via simple greedy algorithms. We prove that there exists an $L(p,q)$-labeling with span at most $\max\{2(p+q-1)\Delta-4q+2, (2p-1)\mu+(2q-1)\Delta-2q+1\}$ for interval $k$-graphs, $\max\{p,q\}\Delta$ for interval graphs, $3\max\{p,q\}\Delta+p$ for circular-arc graphs, $2(p+q-1)\Delta-2q+1$ for permutation graphs and $(2p-1)\Delta+(2q-1)(\mu-1)$ for cointerval graphs. In particular, these improve existing bounds on $L(p,q)$-labeling of interval graphs and $L(2,1)$-labeling of permutation graphs. Furthermore, we provide upper bounds on the coloring of the squares of aforementioned classes.

Latest Algorithms on Particular Graph Classes

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

Locally definable vertex set properties are efficiently enumerable

We propose a general framework that allows for the study of enumeration of vertex set properties in graphs. We prove that when such a property is locally definable with respect to some order on the set of vertices, then it can be enumerated with linear delay. Our method consists in reducing the considered enumeration problem to the enumeration of paths in directed acyclic graphs. We then apply this general method to enumerate minimal connected dominating sets and maximal irredundant sets in interval graphs and in permutation graphs, as well as maximal irredundant sets in circular-arc graphs and in circular permutation graphs, with linear delay.

Disconnected cuts in claw-free graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. This problem is known to be NP-hard on general graphs. We prove that it is polynomial-time solvable on claw-free graphs, answering a question of Ito et al. (TCS 2011). The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.