Circle Graphs Research Articles

The Classical Ramsey Number R(3,4)>8 Using the Weighted Graph

At present, research on Ramsey Numbers has expanded to a wider scope, not only between 2 complete graphs that are complementary to each other but also a combination of complete graphs, circle graphs, star graphs, wheel graphs, and others. While the classic Ramsey number still leaves problems that need to be solved. Ramsey number R(3,4) > 8. This means that m=8 is the largest integer such that K_(8) which contains components of a red graph G and a complete blue graph G which is still possible not to get K_3 in graph G and not get a blue K_4 in graph G .The graph K_(8) has a total of 28 edges. There are as many as the combination (28,3) red edge pairs that need to be avoided so as not to get any red K_3 edge pairs. And there are as many as the combination (28,4) edge pairs blue. That needs to be avoided in order not to get a pair of blue K_4 edges. Determining the coloring of the of the graph directly is certainly very difficult, especially if the Ramsey number is getting bigger. It's like looking for a needle in a haystack. Need to use a special method in order to solve this problem. The weighting graph method, where each edge is given weight with a certain value, is able to solve this problem. The weighting graph method is able to display the graph K_8 in the form of a G matrix with the order of 8×8.

Algorithmic results in secure total dominating sets on graphs

Given a graph G=(V,E), a (total) dominating set of G is a subset D⊆V such that each vertex in V∖D (respectively, V) is adjacent to at least one vertex in D. A (total) dominating set D of G is a secure (total) dominating set if for each v∈V∖D there is a vertex u∈D adjacent to v such that (D∖{u})∪{v} is also a (total) dominating set of G. The minimum cardinality of a secure (total) dominating set of G is called the secure (total) domination number of G. The secure (total) domination problem is to find a minimum secure (total) dominating set of any graph.In this paper, we study the secure total domination problem. We show that the problem restricted to trees is solvable in linear-time, the decision version of the problem is NP-complete for circle graphs, the complexity of the problem differs from that of the secure domination problem, and the problem is APX-complete for graphs of degree at most 4. Furthermore, we show that the optimization version of the problem on bipartite graphs cannot be approximated in polynomial time within (1−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME(|V|O(log⁡log⁡|V|)).

Strong edge coloring of circle graphs

A strong clique in a graph G is a set S of edges of G such that every two edges in S are either adjacent or joined by an edge of G. We say that G is a circle graph if vertices of G correspond to chords of a circle and G contains an edge uv if and only if chords corresponding to u and v intersect.We show that the maximum size of a strong clique in a circle graph with maximum degree Δ is at most 54Δ2. This bound is tight for even Δ and correct up to a smaller order error term for odd Δ.This result supports a conjecture of Erdős and Nešetřil from 1985, which states that the strong chromatic index of a graph with maximum degree Δ is at most 54Δ2. It also confirms a conjecture of Faudree, Gyárfás, Schelp and Tuza from 1990 for circle graphs.

New Formulation for Coloring Circle Graphs And its Application to Capacitated Stowage Stack Minimization

A circle graph is a graph in which the adjacency of vertices can be represented as the intersection of chords of a circle. The problem of calculating the chromatic number is known to be NP-complete, even on circle graphs. In this paper, we propose a new integer linear programming formulation for a coloring problem on circle graphs. We also show that the linear relaxation problem of our formulation finds the fractional chromatic number of a given circle graph. As a byproduct, our formulation gives a polynomial-sized linear programming formulation for calculating the fractional chromatic number of a circle graph. We also extend our result to a formulation for a capacitated stowage stack minimization problem.

A circle criterion for a generalized cross graph in terms of minimal excluded minors

Geelen and Oum described classes of minimal excluded pivot-minors for a simple graph to be a circle graph and for a delta-matroid to be Eulerian. Pivot-equivalence classes of circle simple graphs and delta-matroids arise in the investigation of Eulerian cycles on cross graphs (4-valent graphs with cross structure). The results established by Geelen and Oum rely on some lemmas in their work, which are shown below to be not quite correct. We consider generalized cross graphs, which arise in the description of rotating circuits on cross graphs. For such graphs we derive a circle criterion: we reproduce and augment the arguments due to Geelen and Oum, and we improve some incorrectly formulated statements. As a result, we obtain the same list of 166 inequivalent graphs, the minimal excluded minors for a generalized cross graph to be a circle graph. Bibliography: 14 titles.

Computing maximum independent set on outerstring graphs and their relatives

A graph G with n vertices is called an outerstring graph if it has an intersection representation with a set of n curves inside a disk such that one endpoint of every curve is attached to the boundary of the disk. Given an outerstring graph representation of G with s segments, a Maximum Independent Set (MIS) of G can be computed in O(s3) time (Keil et al. (2017) [22]).We examine the fine-grained complexity of the MIS problem on some well-known outerstring representations (e.g., line segments, L-shapes, etc.), where the strings are of constant size. We show that computing MIS on grounded segment and grounded square-L representations is at least as hard as computing MIS on circle graph representations. Note that no O(n2−δ)-time algorithm, δ>0, is known for computing MIS on circle graphs. For the grounded string representations, where the strings are y-monotone simple polygonal paths of constant length with segments at integral coordinates, we solve MIS in O(n2) time and show this to be the best possible under the Strong Exponential Time Hypothesis. For the intersection graph of nL-shapes in the plane, we give a (4⋅log⁡OPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4⋅log⁡n)-approximation algorithm of Biedl and Derka (WADS 2017).

The complexity of the vertex-minor problem

A graph H is a vertex-minor of a graph G if it can be reached from G by the successive application of local complementations and vertex deletions. Vertex-minors have been the subject of intense study in graph theory over the last decades and have found applications in other fields such as quantum information theory. Therefore it is natural to consider the computational complexity of deciding whether a given graph G has a vertex-minor isomorphic to another graph H. Here we prove that this decision problem is NP-complete, even when restricting H and G to be circle graphs, a class of graphs that has a natural relation to vertex-minors.

A characterization of circle graphs in terms of total unimodularity

A graph G has an associated multimatroid Z3(G), which is equivalent to the isotropic system of G studied by Bouchet. In previous work it was shown that G is a circle graph if and only if for every field F, the rank function of Z3(G) can be extended to the rank function of an F-representable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this total-unimodularity analogue.

Approximation algorithms for maximum weight k-coverings of graphs by packings

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

Forbidden induced subgraph characterization of circle graphs within split graphs

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known, not even restricted to split graphs (which are the graphs whose vertex set can be partitioned into a clique and a stable set). In this work, we give a characterization by minimal forbidden induced subgraphs of circle graphs, restricted to split graphs.

Graph isomorphism restricted by lists

The complexity of graph isomorphism (GraphIso ) is a famous problem in computer science. For graphs G and H, it asks whether they are the same up to a relabeling of vertices. In 1981, Lubiw proved that list restricted graph isomorphism (ListIso ) is NP -complete: for each u∈V(G), we are given a list L(u)⊆V(H) of possible images of u. After 35 years, we revive the study of this problem and consider which results for GraphIso can be modified to solve ListIso. We prove: 1) Under certain conditions, GI -completeness of a class of graphs implies NP -completeness of ListIso. 2) Several combinatorial algorithms for GraphIso can be modified to solve ListIso: for trees, planar graphs, interval graphs, circle graphs, permutation graphs, and bounded treewidth graphs. 3) ListIso is NP -complete for cubic colored graphs with sizes of color classes bounded by 8 with all lists of size at most 3.

Circle graphs are quadratically χ‐bounded

We prove that the chromatic number of a circle graph with clique number $\omega$ is at most $7\omega^2$.

The Grid Theorem for vertex-minors

We prove that, for each circle graph H, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to H.

Algorithmic aspects of k-part degree restricted domination in graphs

The concept of network is predominantly used in several applications of computer communication networks. It is also a fact that the dominating set acts as a virtual backbone in a communication network. These networks are vulnerable to breakdown due to various causes, including traffic congestion. In such an environment, it is necessary to regulate the traffic so that these vulnerabilities could be reasonably controlled. Motivated by this, [Formula: see text]-part degree restricted domination is defined as follows. For a positive integer [Formula: see text], a dominating set [Formula: see text] of a graph [Formula: see text] is said to be a [Formula: see text]-part degree restricted dominating set ([Formula: see text]-DRD set) if for all [Formula: see text], there exists a set [Formula: see text] such that [Formula: see text] and [Formula: see text]. The minimum cardinality of a [Formula: see text]-DRD set of a graph [Formula: see text] is called the [Formula: see text]-part degree restricted domination number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we present a polynomial time reduction that proves the NP -completeness of the [Formula: see text]-part degree restricted domination problem for bipartite graphs, chordal graphs, undirected path graphs, chordal bipartite graphs, circle graphs, planar graphs and split graphs. We propose a polynomial time algorithm to compute a minimum [Formula: see text]-DRD set of a tree and minimal [Formula: see text]-DRD set of a graph.

On semi-transitive orientability of Kneser graphs and their complements

An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v0→v1→⋯→vk either there is no edge between v0 and vk, or vi→vj is an edge for all 0≤i<j≤k. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs include several important classes of graphs such as 3-colourable graphs, comparability graphs, and circle graphs, and they are precisely the class of word-representable graphs studied extensively in the literature.In this paper, we study semi-transitive orientability of the celebrated Kneser graph K(n,k), which is the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. We show that K(n,k) is not semi-transitive for any even integers k≥2 and n≥3k and for any odd integers k≥3 and n≥3k+3. On the other hand, for m∈{2k,2k+1}, K(n,k) is semi-transitive. Also, if K(p,q) is not semi-transitive, then K(n,k) is not semi-transitive for any k≥q and n≥kqp. Moreover, we show computationally that K(8,3) is not semi-transitive, which results in K(n,k) being not semi-transitive for any k≥3 and n≥8⋅k3. A certain subgraph S of K(8,3) presented by us and K(8,3) itself are the first explicit examples of triangle-free non-semi-transitive graphs, whose existence was established via Erdős’ theorem by Halldórsson, Kitaev and Pyatkin in Halldórsson et al. (2011). Finally, the complement graph K(n,k)¯ of K(n,k) is not semi-transitive if and only if n>2k.

Conflict-Free Coloring of String Graphs

Conflict-free coloring (in short, CF-coloring) of a graph $$G = (V,E)$$ is a coloring of V such that the punctured neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane: (i) we provide a general upper bound of $$O(\chi (G)^2 \log n)$$ on the CF-chromatic number of any string graph G with n vertices in terms of the classical chromatic number $$\chi (G)$$ . This result stands in contrast to general graphs where the CF-chromatic number can be $$\varOmega (\sqrt{n})$$ already for bipartite graphs. (ii) For some central classes of string graphs, the CF-chromatic number is as large as $$\varTheta (\sqrt{n})$$ , which was shown to be the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of $$\varTheta (\log n)$$ with respect to the relaxed notion of k-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most k times), for a small constant k. (iii) We obtain a general upper bound on the k-CF chromatic number of arbitrary hypergraphs (i.e., the number of colors needed to color the vertices, such that in each hyperedge there is a color that appears at most k times): any hypergraph with m hyperedges can be k-CF colored with $${\widetilde{O}}\bigl (m^{\frac{1}{k+1}}\bigr )$$ colors. This bound, which extends a bound of Pach and Tardos (Comb Probab Comput 18(5):819–834, 2009), is tight for some string graphs, up to a logarithmic factor. (iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of $$\varTheta (\log n)$$ on the CF-chromatic number of circle graphs, and an upper bound of $$O(\log ^{3} n)$$ for a wider class of string graphs that contains circle graphs, namely, intersection graphs of grounded L-shapes.

Bipartite complements of circle graphs

Using an algebraic characterization of circle graphs, Bouchet proved in 1999 that if a bipartite graph G is the complement of a circle graph, then G is a circle graph. We give an elementary proof of this result.

A Characterization of Circle Graphs in Terms of Multimatroid Representations

The isotropic matroid $M[IAS(G)]$ of a looped simple graph $G$ is a binary matroid equivalent to the isotropic system of $G$. In general, $M[IAS(G)]$ is not regular, so it cannot be represented over fields of characteristic $\neq 2$. The ground set of $M[IAS(G)]$ is denoted $W(G)$; it is partitioned into 3-element subsets corresponding to the vertices of $G$. When the rank function of $M[IAS(G)]$ is restricted to subtransversals of this partition, the resulting structure is a multimatroid denoted $\mathcal{Z}_{3}(G)$. In this paper we prove that $G$ is a circle graph if and only if for every field $\mathbb{F}$, there is an $\mathbb{F}$-representable matroid with ground set $W(G)$, which defines $\mathcal{Z}_{3}(G)$ by restriction. We connect this characterization with several other circle graph characterizations that have appeared in the literature.

Polymorphisms of C-reactive protein and growth arrest-specific gene 6 modulate ischemic stroke susceptibility through gene–gene interactions in Chinese Han population

This study explored whether inflammatory gene including C-reactive protein (CRP) and growth arrest-specific gene 6 (GAS6) are related to the etiology of ischemic stroke (IS), and to verify their potential interaction upon susceptibility to IS in Chinese population. A total of 236 patients with IS and 291 non-IS subjects were enrolled. Five single nucleotide polymorphisms (SNPs) in two candidate genes (CRP and GAS6) for IS were examined and determined by using polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP) and detected by DNA sequencing. We evaluated the association between SNPs and the risk of IS and evaluated their potential gene–gene interactions by using the generalized multifactor dimensionality reduction (GMDR), interaction dendrogram, entropy analysis and Interaction circle graph. A significant gene–gene interaction was then selected. We further tested the best model by logistic regression models. The results showed no significant difference between the IS (or its subtypes) group and the control group in the single-locus of all the analyzed variants. GMDR analysis indicated that the CRP SNPs rs3093059 and the GAS6 SNPs rs7400722 shared strong synergism in LAA analysis (Testing accuracy = 59.66%, CVC = 10, p = 0.0107). Dendrogram and circle graph analysis indicated a more-than-additive effect between the two loci. Furthermore, a recessive model of the two-locus interactions was confirmed by multivariate logistic regression model (OR = 1.940, 95% CI: 1.068–3.525, p = 0.030). The results indicate that CRP and GAS6 genes may fully participate in affecting the susceptibility to IS by gene–gene interaction, and may be conducive to provide a novel area for IS research.

Minimizing crossings in constrained two-sided circular graph layouts.

Circular graph layout is a popular drawing style, in which vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts, which draw some edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all vertices placed on a line (the spine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal $k$-plane set of exteriorly drawn edges for $k \ge 1$, extending the previously studied case $k=0$. We show that this relates to finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary $k$, present an efficient algorithm for $k=1$, and generalize it to an explicit XP-time algorithm for any fixed $k$. For the practically interesting case $k=1$ we implemented our algorithm and present experimental results that confirm its applicability.