Zero forcing in iterated line digraphs

Iterated Arc Graphs

The arc graph $δ(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $δ(G)$ join consecutive arcs of $G$. In 1981, Poljak and Rödl characterised the chromatic number of $δ(G)$ in terms of the chromatic number of $G$ when $G$ is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph $δ^k(G)$ still only depends on the chromatic number of $G$ wh…

Outdegree conditions forcing short cycles in digraphs

Given a positive integer $m\ge 3$, let $ch(m)$ be the smallest positive constant with the following property: \emph{ Every simple directed graph on $n\ge 3$ vertices all whose outdegrees are at least $ch(m)\cdot n$ contains a directed cycle of length at most $m$.} Caccetta and Häggkvist conjectured that $ch(m)=1/m$, which if true, would be the best possible. In this paper, we prove the following result: \emph{ For every integer $m\ge 3$, let $α(m)$ be the unique real root in $(0,1)$ of the equation} \begin{equation*} (1-x)^{m-2}=\frac{3x}{2-x}. \end{equation*} Then $ch(m)\le α(m)$. This genera…

Identifying codes in line digraphs

Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$. In this paper, we prove that every $k$-iterated line digraph of minimum in-degree at least 2 and $k\geq2$, or minimum in-degree at least 3 and $k\geq1$, admits a $(1,\le \ell)$-identifying code with $\ell\leq2$, and in any case it does not admit a $(1,\le \ell)$-identifying code for $\ell\geq3$. Moreover, we find that the identifying number of a line digraph …

迭代图模型上的零强迫

Zero Forcing on Iterated Graph Models

本文研究 ILT 与 ILAT 迭代图模型上的零强迫信息扩散问题，给出零强迫数和失败零强迫数的界，并证明 ILAT 图的失败零强迫数仅可能取四个值之一，同时简要讨论更一般迭代图模型上的图燃烧问题。

The Iterated Local Directed Transitivity Model for Social Networks

We introduce a new directed graph model for social networks, based on the transitivity of triads. In the Iterated Local Directed Transitivity (ILDT) model, new nodes are born over discrete time-steps, and inherit the link structure of their parent nodes. The ILDT model may be viewed as a directed analogue of the ILT model for undirected graphs introduced in \cite{ilt}. We investigate network science and graph theoretical properties of ILDT digraphs. We prove that the ILDT model exhibits a densification power law, so that the digraphs generated by the models densify over time. The number of dir…

On $d$-Fibonacci digraphs

The $d$-Fibonacci digraphs $F(d,k)$, introduced here, have the number of vertices following generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their nice properties. For instance, $F(2,k)$ has diameter $d+k-2$ and is semi-pancyclic, that is, it has a cycle of every length between 1 and $\ell$, with $\ell\in\{2k-2,2k-1\}$. Moreover, it turns out that several other numbers of $F(d,k)$ (of closed $l$-walks, classes of vertices, etc.) also follow the same linear recurrences as the numbers of vertices of the $…

On Cyclic Kautz Digraphs

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs $MCK(d,\ell)$ and it is derived from the Kautz digraphs $K(d,\ell)$. %It is not common to find non-regular digraphs with minimal diameter given their number of vertices and out-degree. It is well-known that the Kautz digraphs $K(d,\ell)$ h…

有向图上标准零强迫的节流

Throttling for standard zero forcing on directed graphs

本文研究有向图上的节流问题，刻画节流数至多为t的简单有向图，分析翻转弧与删点的影响，并引入无向图的方向节流区间，给出一般界及路径定向的精确结果。

有向图中哈密顿圈平方的精确最小总度数阈值

The exact total degree threshold for the square of a Hamilton cycle in digraphs

本文解决了德比亚西奥等人关于有向图中哈密顿圈平方最小总度数阈值的猜想，证明了每个足够大的\(n\)顶点有向图，当最小总度数至少为\(8n/5 - c\)（\(n\)取不同值时\(c\)取值不同）时，包含哈密顿圈的平方。

在多项式时间内计算外平面图的forcing谱

Computing the forcing spectrum of outerplanar graphs in polynomial time

针对平面图最小forcing数计算复杂度这一开放问题，本文提出首个多项式时间算法，可计算外平面图的所有可能forcing数构成的forcing谱，为该问题的求解迈出重要一步。