Fully dynamic recognition of proper circular-arc graphs

Testing isomorphism of circular-arc graphs -- Hsu's approach revisited

Circular-arc graphs are intersection graphs of arcs on the circle. The aim of our work is to present a polynomial time algorithm testing whether two circular-arc graphs are isomorphic. To accomplish our task we construct decomposition trees, which are the structures representing all normalized intersection models of circular-arc graphs. Normalized models reflect the neighbourhood relation in circular-arc graphs and can be seen as their canonical representations; in particular, every intersection model can be easily transformed into a normalized one. Our work adapts and appropriately extends th…

圆弧图的刻画：II. McConnell翻转

Characterization of Circular-arc Graphs: II. McConnell Flipping

本文对McConnell提出的圆弧图到区间图的翻转变换进行结构研究，针对 $C_{4}$-free 图实现了表示模式的完整刻画，从而能够识别所有非圆弧图的极小弦图。

Testing isomorphism of circular-arc graphs in polynomial time

A graph is said to be circular-arc if the vertices can be associated with arcs of a circle so that two vertices are adjacent if and only if the corresponding arcs overlap. It is proved that the isomorphism of circular-arc graphs can be tested by the Weisfeiler-Leman algorithm after individualization of two vertices.

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 hi…

Isomorphism of graph classes related to the circular-ones property

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, Γ-circular-arc graphs, proper circular-arc graphs and convex-round graphs.

关于圆弧图识别和同构问题的$\boldsymbol{\mathcal{O}(m\cdot n)}$算法的评论

Comments on "$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs"

本文指出徐文莲1995年提出的圆弧图$\mathcal{O}(m\cdot n)$识别算法、分解树构造及同构算法中，同构算法已被证明错误，另两个结果也存在缺陷。

Extending Partial Representations of Circular-Arc Graphs

The partial representation extension problem generalizes the recognition problem for classes of graphs defined in terms of vertex representations. We exhibit circular-arc graphs as the first example of a graph class where the recognition is polynomially solvable while the representation extension problem is NP-complete. In this setting, several arcs are predrawn and we ask whether this partial representation can be completed. We complement this hardness argument with tractability results of the representation extension problem on various subclasses of circular-arc graphs, most notably on all v…

A Note on Circular Arc Online Coloring using First Fit

In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.

Deciding Circular-Arc Graph Isomorphism in Parameterized Logspace

We compute a canonical circular-arc representation for a given circular-arc (CA) graph which implies solving the isomorphism and recognition problem for this class. To accomplish this we split the class of CA graphs into uniform and non-uniform ones and employ a generalized version of the argument given by Köbler et al (2013) that has been used to show that the subclass of Helly CA graphs can be canonized in logspace. For uniform CA graphs our approach works in logspace and in addition to that Helly CA graphs are a strict subset of uniform CA graphs. Thus our result is a generalization of the …

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 Hell…

计算单位圆弧交点数

Counting Unit Circular Arc Intersections

该研究提出了一种更高效的算法，用于计算平面中单位圆弧的交点数，针对小K情况进一步优化了时间复杂度。

圆弧图与 Helly 性质

Circular-arc graphs and the Helly property

论文研究圆弧图团的 Helly 性质，利用维护全部规范化模型的 PQM-tree 刻画三类团、给出判定算法，并证明 Helly Cliques 问题的 FPT 算法、ETH 下界及 O(k^6) 多项式核。

A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs

Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^…

Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs

In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborn\'ık, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm r…