在有限维偏序的(不可比)图的极小支配集计数

Automorphism Groups of Comparability Graphs

Comparability graphs are graphs which have transitive orientations. The dimension of a poset is the least number of linear orders whose intersection gives this poset. The dimension ${\rm dim}(X)$ of a comparability graph $X$ is the dimension of any transitive orientation of X, and by $k$-DIM we denote the class of comparability graphs $X$ with ${\rm dim}(X) \le k$. It is known that the complements of comparability graphs are exactly function graphs and permutation graphs equal 2-DIM. In this paper, we characterize the automorphism groups of permutation graphs similarly to Jordan's characteriza…

Better bounds for poset dimension and boxicity

We prove that the dimension of every poset whose comparability graph has maximum degree $Δ$ is at most $Δ\log^{1+o(1)} Δ$. This result improves on a 30-year old bound of Füredi and Kahn, and is within a $\log^{o(1)}Δ$ factor of optimal. We prove this result via the notion of boxicity. The "boxicity" of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $Δ$ has boxicity at most $Δ\log^{1+o(1)} Δ$, which is also within a $\log^{o(1)}Δ$ factor of optimal. We also show that the maximum …

偏序集是易于测试的

Posets are easily testable

针对偏序集单调类的属性测试问题，通过证明多项式消除引理和简单分类，提出使用 $O(\varepsilon^{-1})$ 个样本的测试方法，证明了偏序集单调类是易于测试的。

枚举双子模多面体所有极点的首个高效算法

A first efficient algorithm for enumerating all the extreme points of a bisubmodular polyhedron

针对双子模多面体极点枚举难题，提出基于反向搜索与有符号偏序集的高效算法，以 O(n^4|V|) 时间和 O(n^2) 空间避免冗余搜索，并推广了基多面体极点枚举。

Containment Graphs, Posets, and Related Classes of Graphs

In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \in V $ a set $S_i \in Z$ such that $v_i v_j \in E$ if and only if $S_i \subset S_j$ or $S_j \subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \in V $ we can assign a set $S_i \in Z$ such that $v_i < v_j$ if and only if $S_i \subset S_j$. Obviously, …

On the Representation Number of Bipartite Graphs

A word-representable graph is a simple graph $G$ which can be represented by a word $w$ over the vertices of $G$ such that any two vertices are adjacent in $G$ if and only if they alternate in $w$. It is known that the class of comparability graphs -- the graphs which admit a transitive orientation -- is precisely the class of graphs that can be represented by a concatenation of permutations of vertices. The class of bipartite graphs is a subclass of comparability graphs. While it is an open problem to determine the representation number of comparability graphs, it was conjectured that the rep…