Line digraphs and coreflexive vertex sets

Vertex-monochromatic connectivity of strong digraphs

A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ of vertices in $D$ there exists an $(u,v)$-path having all its internal vertices of the same color. Let $smc_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $smc_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $smc_v(T)$.

On the vertex-degree based invariants of digraphs

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $φ$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{φ(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randić index, the Zagreb index, the…

On non-Hamiltonian circulant digraphs of outdegree three

We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself.

Notes on the connectivity of Cayley coset digraphs

Hamidoune's connectivity results for hierarchical Cayley digraphs are extended to Cayley coset digraphs and thus to arbitrary vertex transitive digraphs. It is shown that if a Cayley coset digraph can be hierarchically decomposed in a certain way, then it is optimally vertex connected. The results are obtained by extending the methods used by Hamidoune. They are used to show that cycle-prefix graphs are optimally vertex connected. This implies that cycle-prefix graphs have good fault tolerance properties.

Highly connected infinite digraphs without edge-disjoint back and forth paths between a certain vertex pair

We construct for all $ k\in \mathbb{N} $ a $ k $-edge-connected digraph $ D $ with $ s,t\in V(D) $ such that there are no edge-disjoint $ s \rightarrow t $ and $t\rightarrow s $ paths. We use in our construction "self-similar" graphs which technique could be useful in other problems as well.

Rainbow vertex connection of digraphs

An edge-coloured path is \emph{rainbow} if its edges have distinct colours. An edge-coloured connected graph is said to be \emph{rainbow connected} if any two vertices are connected by a rainbow path, and \emph{strongly rainbow connected} if any two vertices are connected by a rainbow geodesic. The (\emph{strong}) \emph{rainbow connection number} of a connected graph is the minimum number of colours needed to make the graph (strongly) rainbow connected. These two graph parameters were introduced by Chartrand, Johns, McKeon and Zhang in 2008. As an extension, Krivelevich and Yuster proposed the…

准素性和双准素性高度弧传递有向图与有限单群

Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups

本文研究了准素性和双准素性高度弧传递有向图与有限单群的关系，提出新的构造方法并揭示了相关开放问题。

Vertex-Pancyclism in the Generalized Sum of Digraphs

A digraph $D=(V(D)$, $A(D))$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in \{3,\ldots,n\}$; and $D$ is vertex-pancyclic iff, for each vertex $v\in V(D)$ and each $k\in \{3,\ldots,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. Let $D_1, D_2, \ldots, D_k$ be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of $D_1, D_2, \ldots, D_k$, denoted by $\oplus_{i=1}^k D_i$ or $D_1\oplus D_2 \oplus \cdots \oplus D_k$, is the set of all digraphs $D$ satisfying: (i) $V(D)=\bigcup_{i=1}^k V(D_i)$, (ii…

Vertex coloring acyclic digraphs and their corresponding hypergraphs

We consider vertex coloring of an acyclic digraph $\Gdag$ in such a way that two vertices which have a common ancestor in $\Gdag$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding {\em down-chromatic number} and derive an upper bound as a function of $D(\Gdag)$, the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices …

Classification of vertex-transitive digraphs via automorphism group

In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.

Critical digraphs with few vertices

We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical graphs.

Vertices with the Second Neighborhood Property in Eulerian Digraphs

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but …

Disjoint cycles with length constraints in digraphs of large connectivity or minimum degree

A conjecture by Lichiardopol states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles …

顶点传递有向图中的长有向圈

Long Directed Cycles in Vertex-Transitive Digraphs

在图论与群论交叉领域，寻找顶点传递图和有向图中的哈密顿圈是经典问题。本文构造无限多个最长有向圈遗漏至少\(n/12\)个顶点的连通顶点传递有向图，证实相关猜想，还将有向圈长度下限从\(\Omega(n^{1/3})\)提高到\(\Omega(\sqrt n)\) 。

交错群与对称群的顶点本原 s-弧传递有向图中 s 的界

Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

本文针对 Praeger 提出的有限顶点本原 s-弧传递有向图的上界问题，在群为交错群或对称群的情形下证明了 s ≤ 2。

Arc-connectivity and super arc-connectivity of mixed Cayley digraph

A digraph X=(V, E) is max-λ, if λ(X)=δ(X). A digraph X is super-λif every minimum cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that for all but a few exceptions, the strongly connected mixed Cayley digraphs are max-λand super-λ.

2度顶点传递有向图中的生成因子分解

Spanning Factorizations in Vertex-Transitive Digraphs of Degree 2

针对出度为d的顶点传递有向图中生成1-因子分解的存在性问题，本文聚焦d=2的情况，利用交替环和块系结构提出块/阶段框架，得到同时包含两条出边的充分条件，证明块障碍的影响及强传递集的存在性，并给出可行块大小约束等结果。

关于分裂有向图的2-连通性问题

On the 2-Linkage Problem for Split Digraphs

本文证明了6强分裂有向图和5强半完全分裂有向图均为2-连通，解决了相关问题。

Vertex-flames in countable rooted digraphs preserving an Erdős-Menger separation for each vertex

It follows from a theorem of Lovász that if $ D $ is a finite digraph with $ r\in V(D) $ then there is a spanning subdigraph $ E $ of $ D $ such that for every vertex $ v\neq r $ the following quantities are equal: the local connectivity from $ r $ to $ v $ in $ D $, the local connectivity from $ r $ to $ v $ in $ E $ and the indegree of $ v $ in $ E $. In infinite combinatorics cardinality is often an overly rough measure to obtain deep results and it is more fruitful to capture structural properties instead of just equalities between certain quantities. The best known example for such a resu…

On the number of vertex-disjoint cycles in digraphs

Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, Springer-Verlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979-992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284-302] that …

Two disjoint cycles in digraphs

Bermond and Thomassen conjectured that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex disjoint cycles. So far the conjecture was verified for $k\le 3$. Here we generalise the question asking for all outdegree sequences which force $k$ vertex disjoint cycles and give the full answer for $k\le 2$.

Vertex-disjoint directed and undirected cycles in general digraphs

The dicycle transversal number t(D) of a digraph D is the minimum size of a dicycle transversal of D, i. e. a set T of vertices of D such that D-T is acyclic. We study the following problem: Given a digraph D, decide if there is a dicycle B in D and a cycle C in the underlying undirected graph of D such such that B,C are disjoint. It is known that there is a polynomial time algorithm for this problem when restricted to strongly connected graphs, which actually finds B,C if they exist. We generalize this to any class of digraphs D with either t(D) not equal to 1 or t(D)=1 and a bounded number o…

On energy ordering of vertex-disjoint bicyclic sidigraphs

The energy and iota energy of signed digraphs are respectively defined as the sum of absolute values of real parts and sum of absolute values of imaginary parts of its eigenvalues. Recently, Yang and Wang (2018) find the energy and iota energy ordering of digraphs in D(n) and compute the maximal energy and iota energy, where D(n) denotes the set of vertex-disjoint bicyclic digraphs of a fixed order n. In this paper, we investigate the energy ordering of signed digraphs in D(n,s) and finds the maximal energy, where D(n,s) denotes the set of vertex-disjoint bicyclic sidigraphs of a fixed order n…

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose unde…

Majority Colourings of Digraphs

We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.

Extremal digraphs avoiding distinct walks of length 3 with the same endpoints

In this paper, we determine the maximum size of digraphs on $n$ vertices in which there are no two distinct walks of length $3$ with the same initial vertex and the same terminal vertex. The digraphs attaining this maximum size are also characterized. Combining this with previous results, we obtain a full solution to a problem proposed by X. Zhan in 2007.

Small feedback vertex sets in planar digraphs

Let $G$ be a directed planar graph on $n$ vertices, with no directed cycle of length less than $g\ge 4$. We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\le \tfrac{5n-5}9$ if $g=4$, $|X|\le \tfrac{2n-5}4$ if $g=5$, and $|X|\le \tfrac{2n-6}{g}$ if $g\ge 6$. This improves recent results of Golowich and Rolnick.

Vertex-primitive digraphs having vertices with almost equal neighbourhoods

We consider vertex-primitive digraphs having two vertices with almost equal neighbourhoods (that is, the set of vertices that are neighbours of one but not the other is small). We prove a structural result about such digraphs and then apply it to answer a question of Araújo and Cameron about synchronising groups.

有向图Yama Nim

Digraph Yama Nim

本文研究两堆Nim变体Yama Nim的获胜策略与SG值，提出其推广形式Digraph Yama Nim，证明该推广游戏的胜者判定问题即使在二分有向无环图下仍是PSPACE完全的，同时给出了部分可解情形。

The Number of Seymour Vertices in Random Tournaments and Digraphs

Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements along lines suggested by Seymour's conjecture, proving that almost surely there are a "large" number of Seymour vertices in random tournaments and "even more" in general random digraphs.

有向图反馈顶点集问题中的最大度有界情况

Feedback vertex sets of digraphs with bounded maximum degree

该研究在最大度有界的情况下，给出了有向图反馈顶点集问题的最优上界，针对不同图结构给出了不同的界值。

A dichotomy for the kernel by $H$-walks problem in digraphs

Let $H = (V_H, A_H)$ be a digraph which may contain loops, and let $D = (V_D, A_D)$ be a loopless digraph with a coloring of its arcs $c: A_D \to V_H$. An $H$-walk of $D$ is a walk $(v_0, \dots, v_n)$ of $D$ such that $(c(v_{i-1}, v_i), c(v_i, v_{i+1}))$ is an arc of $H$, for every $1 \le i \le n-1$. For $u, v \in V_D$, we say that $u$ reaches $v$ by $H$-walks if there exists an $H$-walk from $u$ to $v$ in $D$. A subset $S \subseteq V_D$ is a kernel by $H$-walks of $D$ if every vertex in $V_D \setminus S$ reaches by $H$-walks some vertex in $S$, and no vertex in $S$ can reach another vertex in…

Point- and arc-reaching sets of vertices in a digraph

In a digraph $D = (X, \mathcal{U})$, not necessarily finite, an arc $(x, y) \in \mathcal{U}$ is reachable from a vertex $u$ if there exists a directed walk $W$ that originates from $u$ and contains $(x, y)$. A subset $S \subseteq X$ is an arc-reaching set of $D$ if for every arc $(x, y)$ there exists a diwalk $W$ originating at a vertex $u \in S$ and containing $(x, y)$. A minimal arc-reaching set is an arc-basis. $S$ is a point-reaching set if for every vertex $v$ there exists a diwalk $W$ to $v$ originating at a vertex $u \in S$. A minimal point-reaching set is a point-basis. We extend the r…

关于有向图的 dib-色数

On the dib-chromatic number of a digraph

本文针对有向图的dib-色数展开研究，明确了其存在性，并对二部有向图的dib-色数进行了分析，该色数是满足特定顶点指向条件的无环着色的最大颜色数。

Double domination and total $2$-domination in digraphs and their dual problems

A subset $S$ of vertices of a digraph $D$ is a double dominating set (total $2$-dominating set) if every vertex not in $S$ is adjacent from at least two vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double domination number (total $2$-domination number) of a digraph $D$ is the minimum cardinality of a double dominating set (total $2$-dominating set) in $D$. In this work, we investigate these concepts which can be considered as two extensions of double domination in graphs to digraphs, along wit…

Maximally connected and super arc-connected Bi-Cayley digraphs

Let X=(V, E) be a digraph. X is maximally connected, if κ(X)=δ(X). X is maximally arc-connected, if λ(X)=δ(X). And X is super arc-connected, if every minimum arc-cut of X is either the set of inarcs of some vertex or the set of outarcs of some vertex. In this paper, we will prove that the strongly connected Bi-Cayley digraphs are maximally connected and maximally arc-connected, and the most of strongly connected Bi-Cayley digraphs are super arc-connected.

顶点传递有向图中的长环

Long cycles in vertex transitive digraphs

研究了顶点传递有向图中长环的存在性，证明了在n个顶点的图中可找到长度至少为Ω(n^{1/3})的有向环。

Whitney's Theorem for 2-Regular Planar Digraphs

A digraph is 2-regular if every vertex has both indegree and outdegree two. We define an embedding of a 2-regular digraph to be a 2-cell embedding of the underlying graph in a closed surface with the added property that for every vertex~$v$, the two edges directed away from $v$ are not consecutive in the local rotation around $v$. In other words, at each vertex the incident edges are oriented in-out-in-out. The goal of this article is to provide an analogue of Whitney's theorem on planar embeddings in the setting of 2-regular digraphs. In the course of doing so, we note that Tutte's Theorem on…

阿贝尔群上2价Cayley有向图中的完美码

Perfect codes in 2-valent Cayley digraphs on abelian groups

研究阿贝尔群上强连通2价Cayley有向图的完美码问题，完成图的分类并完全确定其所有完美码。

Vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups

A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.

具有大固定性的顶点本原有向图

Vertex-primitive digraphs with large fixity

本文定义了有向图的相对固定性，刻画了相对固定性至少为 $1/3$ 的顶点本原有向图，并证明了有界出度下仅有有限多个此类图。

有向图中的距离、阶、大小与连通性约束

Remoteness, order, size and connectivity constraints in digraphs

本文针对强连通有向图，给出了在给定阶、大小和顶点连通性条件下remoteness的尖锐上界，刻画了极值图，并将相关上界推广至欧拉有向图。

关于有向图着色与顶点树荫度的复杂性

On the Complexity of Digraph Colourings and Vertex Arboricity

本文研究有向图与图的分数量着色相关决策问题的复杂性，证明星色数、分数量色数及循环顶点树荫度的判定问题在特定条件下均为NP完全，并引入循环同态概念以证明关键结果。

On the total Rainbow domination of digraphs

For a positive integer $k$, a $k$-rainbow dominating function ($k$RDF) on a digraph $D$ is a function $f$ from the vertex set $V(D)$ to the set of all subsets of $\{1,2,\ldots,k\}$ such that for any vertex $v$ with $f(v)=\emptyset$, $\bigcup_{u\in N^-(v)}f(u)=\{1,2,\ldots,k\}$, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a $k$RDF $f$ is defined as $\sum_{v\in V(D)}|f(v)|$. A $k$RDF $f$ on $D$ with no isolated vertex is called a total $k$-rainbow dominating function if the subdigraph of $D$ induced by the set $\{v\in V(D):f(v)\ne\emptyset\}$ has no isolated vertex. The total…

具有简单对称有向图的矩阵及其群逆

Matrices with simple symmetric digraphs and their group inverses

本文定义了简单对称有向图类 $\mathcal{D}$，给出了其群逆元素的图论描述，并对实方阵 $A$ 及其群逆 $A^{\\#}$ 均属于该类的矩阵进行了分类。

Generating infinite digraphs by derangements

A set $\mathcal{S}$ of derangements (fixed-point-free permutations) of a set $V$ generates a digraph with vertex set $V$ and arcs $(x,x^σ)$ for $x\in V$ and $σ\in\mathcal{S}$. We address the problem of characterising those infinite (simple loopless) digraphs which are generated by finite sets of derangements. The case of finite digraphs was addressed in earlier work by the second and third authors. A criterion is given for derangement generation which resembles the criterion given by De Bruijn and Erdős for vertex colourings of graphs in that the property for an infinite digraph is determined …

A note on heavy cycles in weighted digraphs

A weighted digraph is a digraph such that every arc is assigned a nonnegative number, called the weight of the arc. The weighted outdegree of a vertex $v$ in a weighted digraph $D$ is the sum of the weights of the arcs with $v$ as their tail, and the weight of a directed cycle $C$ in $D$ is the sum of the weights of the arcs of $C$. In this note we prove that if every vertex of a weighted digraph $D$ with order $n$ has weighted outdegree at least 1, then there exists a directed cycle in $D$ with weight at least $1/\log_2 n$. This proves a conjecture of Bollobás and Scott up to a constant facto…

A Remark on the Second Neighborhood Problem

Seymour's second neighborhood conjecture states that every simple digraph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. Such a vertex is said to have the second neighborhood property (SNP). We define "good" digraphs and prove a statement that implies that every feed vertex of a tournament has the SNP. In the case of digraphs missing a matching, we exhibit a feed vertex with the SNP by refining a proof due to Fidler and Yuster and using good digraphs. Moreover, in some cases we exhibit two vertices with SNP.

Transpose on vertex symmetric digraphs

We discuss transpose (sometimes called universal exchange or all-to-all) on vertex symmetric networks. We provide a method to compare the efficiency of transpose schemes on two different networks with a cost function based on the number processors and wires needed to complete a given algorithm in a given time.

有向图的细分乘积的谱

Spectra of Subdivision Products of Digraphs

本文介绍了四种扩展自无向情况的有向图细分乘积类型，包括细分顶点连接、细分弧连接、细分顶点冠和细分弧冠。分析了这些构造的结构和谱性质，重点研究邻接谱、拉普拉斯谱和无符号拉普拉斯谱。

Disjoint cycles of different lengths in graphs and digraphs

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct lengths in a given graph (resp. digraph). In the context of undirected graphs, we prove that, for every $k \geq 1$, every graph with minimum degree at least $\frac{k^2+5k-2}{2}$ has $k$ vertex-disjoint cycles of different lengths, where the degree bound is best possible. We also consider stronger situations, and exhibit degree bounds (some of which are best…

有向图在删除一个顶点或一条弧后的 $α$ 迹范数变化及其应用

Variation in $α$ trace norm of a digraph by deletion of a vertex or an arc and its applications

本文研究了有向图在删除顶点或弧后其 $α$ 迹范数的变化规律，并据此刻画了具有最大 $α$ 迹范数的定向树与单圈有向图。

On existence of noncritical vertices in digraphs

Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then there exists a noncritical vertex in $D$, and if sum of the degrees of any two adjacent vertices of $D$ is at least $n+2$, then there exist two noncritical vertices in $D$. A series of examples confirm that these bounds are tight.

Pancyclism in the Generalized Sum of Digraphs

A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a directed cycle of length $k$ passing through $v$. Let $D_1$, $D_2$,..., $D_k$ be a collection of pairwise vertex disjoint digraphs. The generalized sum (g.s.) of $D_1$, $D_2$,..., $D_k$, denoted by $\oplus_{i=1}^k D_i$ or $D_1\oplus D_2 \oplus \cdots \oplus D_k$, is the set of all digraphs D satisfying: (i) $V(D)=\bigcup_{i=1}^k V(D_i)$, (ii) $D\langle V(D_i) \ran…