[论文解读] Upper bounds on the k-forcing number of a graph

David Amos, Yair Caro+2 more|arXiv (Cornell University)|Jan 23, 2014

Graph theory and applications参考文献 10被引用 10

一句话总结

本文建立了图的k-强迫数的新上界，推广了零强迫数（k=1）的情形。在最大度Δ和图阶n的条件下，推导出紧致的上界，得到一般图的Fₖ(G) ≤ (Δ−k+1)n / (Δ−k+1 + min{δ,k})，并对k-连通图得到改进的上界。关键结果为：当Δ ≥ 2时，零强迫数Z(G) ≤ (Δ−2)n+2 / (Δ−1)，解决了关于正则二分循环图零强迫数的一个开放问题。

ABSTRACT

Given a simple undirected graph $G$ and a positive integer $k$, the $k$-forcing number of $G$, denoted $F_k(G)$, is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most $k$ non-colored neighbors, then each of its non-colored neighbors becomes colored. When $k=1$, this is equivalent to the zero forcing number, usually denoted with $Z(G)$, a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the $k$-forcing number. Notable among these, we show that if $G$ is a graph with order $n \ge 2$ and maximum degree $Δ\ge k$, then $F_k(G) \le \frac{(Δ-k+1)n}{Δ- k + 1 +\min{\{δ,k\}}}$. This simplifies to, for the zero forcing number case of $k=1$, $Z(G)=F_1(G) \le \frac{Δn}{Δ+1}$. Moreover, when $Δ\ge 2$ and the graph is $k$-connected, we prove that $F_k(G) \leq \frac{(Δ-2)n+2}{Δ+k-2}$, which is an improvement when $k\leq 2$, and specializes to, for the zero forcing number case, $Z(G)= F_1(G) \le \frac{(Δ-2)n+2}{Δ-1}$. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the $k$-forcing number and the connected $k$-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.

研究动机与目标

• 建立以最大度Δ和图阶n为参数的k-强迫数Fₖ(G)的紧致上界。
• 解决关于正则二分循环图零强迫数的开放问题。
• 探讨k-强迫数与连通k-支配数之间的关系。
• 推导特殊图族（包括k-连通图和K₁,r-自由图）中k-强迫数的界。
• 刻画所推导界中等号成立的情形，特别是零强迫数的情形。

提出的方法

• 通过基于度数的极值论证和极值图构造，推导Fₖ(G)的上界。
• 应用k-强迫过程：一个有至多k个未着色邻居的着色顶点会强迫其所有未着色邻居变为着色。
• 在K₁,r-自由图中，利用k-独立数αₖ(G)来界定Fₖ(G)，证明Fₖ₍ᵣ₋₁₎(G) ≤ n − αₖ(G)。
• 施加连通性约束（k-连通性）以改进界，尤其在k=1时效果显著。
• 应用极值图论和独立数界，推导出紧致不等式。
• 通过归纳法和对树与完全图的结构分析，验证界值的紧致性。

实验结果

研究问题

• RQ1 对于一般图，以最大度Δ和阶n为参数，k-强迫数Fₖ(G)的最紧可能上界是什么？
• RQ2 对于Δ ≥ 2的k-连通图，零强迫数Z(G)能否得到更紧的上界？
• RQ3 在连通图中，k-强迫数与连通k-支配数之间有何关系？
• RQ4 在什么极值图中，F₁(G) ≤ (Δ−2)n+2 / (Δ−1) 的等号成立？
• RQ5 能否利用k-独立数在K₁,r-自由图中界定k-强迫数？

主要发现

• 对于任意满足n ≥ 2且Δ ≥ k的图G，k-强迫数满足Fₖ(G) ≤ (Δ−k+1)n / (Δ−k+1 + min{δ,k})。
• 当k=1（即零强迫数）时，该式简化为Z(G) ≤ Δn / (Δ+1)，优于已知界。
• 对于Δ ≥ 2的k-连通图，有Fₖ(G) ≤ ((Δ−2)n + 2) / (Δ + k − 2)，当k ≤ 2时优于先前结果。
• 对于零强迫数，该式推出Z(G) ≤ ((Δ−2)n + 2) / (Δ−1)，且在K_{Δ+1}与K_{Δ,Δ}中等号成立。
• 在满足δ ≥ 1的K₁,r-自由图中，有Fₖ₍ᵣ₋₁₎(G) ≤ n − αₖ(G)，将k-强迫数与k-独立数联系起来。
• 作为主要结果的推论，连通图中零强迫数与连通支配数之和至多为n。