论文标题
列举数字凸的循环和笛卡尔产品的功率集和完整图
Enumerating the Digitally Convex Sets of Powers of Cycles and Cartesian Products of Paths and Complete Graphs
论文作者
论文摘要
给定有限集$ v $，一个$ \ mathscr {c} $，是$ v $的子集的集合，其中包含空套和集合$ v $，并且在交叉点下关闭。 $ \ mathscr {c} $的元素称为凸集。最初建议作为处理数字图像的工具的数字凸度定义如下：subset $ s \ subseteq v（g）$在数字上是在数字上凸出的，对于每一个$ v \ in v（g）$，我们都有$ n [v] \ subseteq n [s] $ in In s $中的$ n [v] \ subseteq n [s] $ v。至少$ k $的循环二进制字符串的数量表示为$ k \ geq 2 $的线性复发关系。这些环状二进制字符串与$（k-1）^{th} $的数字凸集之间建立了两者。得出了两个完整图的笛卡尔产品的数字凸组数量的封闭公式。在两条路径的笛卡尔产品的数字凸集，$ p_n \ square p_m $和某些类型的$ n \ times m $二进制阵列之间建立了双线。
Given a finite set $V$, a convexity $\mathscr{C}$, is a collection of subsets of $V$ that contains both the empty set and the set $V$ and is closed under intersections. The elements of $\mathscr{C}$ are called convex sets. The digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset $S\subseteq V(G)$ is digitally convex if, for every $v\in V(G)$, we have $N[v]\subseteq N[S]$ implies $v\in S$. The number of cyclic binary strings with blocks of length at least $k$ is expressed as a linear recurrence relation for $k\geq 2$. A bijection is established between these cyclic binary strings and the digitally convex sets of the $(k-1)^{th}$ power of a cycle. A closed formula for the number of digitally convex sets of the Cartesian product of two complete graphs is derived. A bijection is established between the digitally convex sets of the Cartesian product of two paths, $P_n \square P_m$, and certain types of $n \times m$ binary arrays.
