Circle Graph Isomorphism in Almost Linear Time

Highly unbreakable graph with a fixed excluded minor are almost rigid

A set $X \subseteq V(G)$ in a graph $G$ is $(q,k)$-unbreakable if every separation $(A,B)$ of order at most $k$ in $G$ satisfies $|A \cap X| \leq q$ or $|B \cap X| \leq q$. In this paper, we prove the following result: If a graph $G$ excludes a fixed complete graph $K_h$ as a minor and satisfies certain unbreakability guarantees, then $G$ is almost rigid in the following sense: the vertices of $G$ can be partitioned in an isomorphism-invariant way into a part inducing a graph of bounded treewidth and a part that admits a small isomorphism-invariant family of labelings. This result is the key i…

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.

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…

圆图可以在线性时间内被识别

Circle graphs can be recognized in linear time

本文提出了一种基于PC树数据结构的线性时间圆图识别算法，取代了传统的并查集方法。

有限阿贝尔群同构测试与基构造的几乎紧界

Almost Tight Bounds for Isomorphism Testing and Basis Construction in Finite Abelian Groups

本文研究已知阶有限阿贝尔群的同构判定与基构造，提出随机化同构判定算法达到 $\tilde O(n^{1/4})$ 复杂度，并证明基构造需 $\Omega(\sqrt n)$ 加法，从而分离两者的最优复杂度。

Revisiting the Graph Isomorphism Problem with Semidefinite Programming

It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for this problem are either not efficient or not exact. In this paper, we present a new algorithm which solves this equivalent formulation via semidefinite programming. Specifically, we show that the problem of determining whether the auxiliary graph contains a clique of specific order can be formulated as a semidefinite programming problem, and can thus be (almo…