Discrete Applied Mathematics最新文献

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Neighbor sum distinguishing total choosability of planar graphs without intersecting 4-cycles 无相交 4 循环平面图的邻域和区分总可选性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-13 DOI: 10.1016/j.dam.2024.11.005

Yuan-yuan Duan, Liang-ji Sun, Wen-yao Song

<div><div>Given a simple graph <math><mi>G</mi></math>, a proper total-<math><mi>k</mi></math>-coloring <math><mrow><mi>c</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> <math><mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math> is called neighbor sum distinguishing if <math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> for any two adjacent vertices <math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, where <math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> denote the sum of the color of <math><mi>v</mi></math> and the colors of edges incident with <math><mi>v</mi></math>. The least number <math><mi>k</mi></math> needed for such a coloring of <math><mi>G</mi></math> is the neighbor sum distinguishing total chromatic number, denoted by <math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. Pilśniak and Woźniak conjected that <math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>Σ</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>Δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn></mrow></math> for any simple graph <math><mi>G</mi></math>. Let <math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math> be a set of lists of real numbers and each of size <math><mi>k</mi></math>. The least number <math><mi>k</mi></math> for which for any specified collection of such lists, there exists a neighbor sum distinguish total coloring of <math><mi>G</mi></math> with colors from <math><msub><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow></msub></math> for each <math><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> is called the neighbor sum distinguishing total choosability of <math><mi>G</mi></math>, denoted by <math><mrow><mi>c</mi><msubsup><mrow><mi>h</mi></mrow><mrow><mi>Σ</mi></mrow><mrow><mo

给定一个简单图 G，对于任意两个相邻顶点 u,v∈V(G)，如果∑c(u)≠∑c(v)，则适当的总 k 着色 c:V(G)∪E(G) →{1,2,...,k} 称为邻和区分，其中∑c(v) 表示 v 的颜色与 v 所带边的颜色之和。对 G 进行着色所需的最小数 k 是邻域和区分总色度数，用 χΣ′′(G) 表示。让 Lx（x∈V(G)∪E(G)）是一组实数列表，每个列表的大小为 k。对于任意指定的此类列表集合，存在对每个 x∈V(G)∪E(G)用来自 Lx 的颜色对 G 进行邻域和区分总着色的最小数 k，称为 G 的邻域和区分总可选性，用 chΣ′′(G) 表示。本文证明，对于任何没有相交 4 循环的平面图 G，chΣ′′(G)≤max{Δ(G)+3,10}。

引用次数: 0

The α-index of graphs without intersecting triangles/quadrangles as a minor 无相交三角形/四边形的图形的 α 指数为次要指数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-13 DOI: 10.1016/j.dam.2024.10.027

Yanting Zhang, Ligong Wang

The

A_{α}

-matrix of a graph

G

is the convex linear combination of the adjacency matrix

A (G)

and the diagonal matrix of vertex degrees

D (G)

, i.e.,

A_{α} (G) = α D (G) + (1 - α) A (G)

, where

0 \leq α \leq 1

. The

α

-index of

G

is the largest eigenvalue of

A_{α} (G)

. In this paper, we characterize the extremal graphs with the maximum

α

-index among all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor for any

0 < α < 1

, respectively. As by-products, we determine the extremal graphs with the maximum signless Laplacian spectral radius over all graphs of sufficiently large order without intersecting triangles and quadrangles as a minor, respectively.

图 G 的 Aα 矩阵是邻接矩阵 A(G) 和顶点度对角矩阵 D(G) 的凸线性组合，即 Aα(G)=αD(G)+(1-α)A(G)，其中 0≤α≤1.G 的 α 指数是 Aα(G) 的最大特征值。在本文中，我们描述了在任意 0<α<1 的情况下，在所有阶数足够大且不相交三角形和四边形的图形中，具有最大 α-index 的极值图形。作为副产品，我们分别确定了在所有阶数足够大且不以三角形和四边形相交为次要特征的图形中具有最大无符号拉普拉斯谱半径的极值图形。

引用次数: 0

Rigidity of symmetric linearly constrained frameworks in the plane 平面内对称线性约束框架的刚度

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-12 DOI: 10.1016/j.dam.2024.10.024

Anthony Nixon, Bernd Schulze, Joseph Wall

A bar-joint framework

(G, p)

is the combination of a finite simple graph

G = (V, E)

and a placement

p : V \to R^{d}

. The framework is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of the space. Motivated by applications where boundary conditions play a significant role, one may generalise and consider linearly constrained frameworks where some vertices are constrained to move on fixed affine subspaces. Streinu and Theran characterised exactly which linearly constrained frameworks are generically rigid in 2-dimensional space. In this article we extend their characterisation to symmetric frameworks. In particular necessary combinatorial conditions are given for a symmetric linearly constrained framework in the plane to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In the case of rotation symmetry groups whose order is either 2 or odd, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.

条形连接框架（G,p）是有限简单图 G=（V,E）与位置 p:V→Rd 的组合。如果顶点的唯一保持边长的连续运动来自于空间的等距，那么这个框架就是刚性的。在边界条件起重要作用的应用中，我们可以推广并考虑线性约束框架，其中一些顶点受限于在固定的仿射子空间上移动。Streinu 和 Theran 准确地描述了哪些线性约束框架在二维空间中是一般刚性的。在本文中，我们将他们的描述扩展到对称框架。特别是给出了在平面上的对称线性约束框架在任何有限点群对称下等静态（即最小无限刚度）的必要组合条件。在阶数为 2 或奇数的旋转对称群的情况下，这些条件在适当的通性假设下被证明是充分的，从而给出了在这些情况下对称等静止图形的精确组合描述。

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引用次数: 0

On enumerating short projected models 关于枚举简短预测模型

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-12 DOI: 10.1016/j.dam.2024.10.021

Sibylle Möhle , Roberto Sebastiani , Armin Biere

Propositional model enumeration, or All-SAT, is the task to record all models of a propositional formula. It is a key task in software and hardware verification, system engineering, and predicate abstraction, to mention a few. It also provides a means to convert a CNF formula into DNF, which is relevant in circuit design. While in some applications enumerating models multiple times causes no harm, in others avoiding repetitions is crucial. We therefore present two model enumeration algorithms which adopt dual reasoning in order to shorten the found models. The first method enumerates pairwise contradicting models. Repetitions are avoided by the use of so-called blocking clauses for which we provide a dual encoding. In our second approach we relax the uniqueness constraint. We present an adaptation of the standard conflict-driven clause learning procedure to support model enumeration without blocking clauses. Our procedures are expressed by means of a calculus and proofs of correctness are provided.

命题模型枚举或 All-SAT 是记录命题式所有模型的任务。它是软件和硬件验证、系统工程和谓词抽象等领域的一项关键任务。它还提供了一种将 CNF 公式转换为 DNF 的方法，这与电路设计相关。在某些应用中，多次枚举模型并无坏处，而在其他应用中，避免重复则至关重要。因此，我们提出了两种采用双重推理的模型枚举算法，以缩短找到的模型。第一种方法枚举成对的矛盾模型。通过使用我们提供了对偶编码的所谓阻塞条款来避免重复。在第二种方法中，我们放宽了唯一性约束。我们对标准的冲突驱动条款学习程序进行了调整，以支持无阻塞条款的模型枚举。我们的程序通过微积分来表达，并提供了正确性证明。

引用次数: 0

Recoloring some hereditary graph classes 对一些遗传图类进行重新染色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-11 DOI: 10.1016/j.dam.2024.10.026

Manoj Belavadi , Kathie Cameron

The reconfiguration graph of the

k

-colorings, denoted

R_{k} (G)

, is the graph whose vertices are the

k

-colorings of

G

and two colorings are adjacent in

R_{k} (G)

if they differ in color on exactly one vertex. A graph

G

is said to be recolorable if

R_{ℓ} (G)

is connected for all

ℓ \geq χ (G) + 1

. In this paper, we study the recolorability of several graph classes restricted by forbidden induced subgraphs. We prove some properties of a vertex-minimal graph

G

which is not recolorable. We show that every (triangle,

H

)-free graph is recolorable if and only if every (paw,

H

)-free graph is recolorable. Every graph in the class of

(2 K_{2}, H)

-free graphs, where

H

is a 4-vertex graph except

P_{4}

P_{3} + P_{1}

, is recolorable if

H

is either a triangle, paw, claw, or diamond. Furthermore, we prove that every (

P_{5}

C_{5}

, house, co-banner)-free graph is recolorable.

如果两个颜色在 Rk(G) 中正好有一个顶点的颜色不同，那么这两个颜色就是相邻的。如果 Rℓ(G)对所有 ℓ≥χ(G)+1 都是连通的，则称图 G 为可再着色图。在本文中，我们研究了几个由禁止诱导子图限制的图类的可重色性。我们证明了不可再色的顶点最小图 G 的一些性质。我们证明，当且仅当每个不含（三角形，H）的图都是可再色的时候，每个不含（爪形，H）的图都是可再色的。如果 H 是三角形、爪形、爪形或菱形，则 (2K2,H)-free graph（其中 H 是除 P4 或 P3+P1 以外的 4 顶点图）类中的每个图都是可再色的。此外，我们还证明了每个（P5, C5, house, co-banner）无边图都是可再色的。

引用次数: 0

New theoretical results on the Monotone Boolean Duality and the Monotone Boolean Dualization problems 关于单调布尔对偶性和单调布尔对偶化问题的新理论成果

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-11 DOI: 10.1016/j.dam.2024.10.019

Alice Raffaele , Romeo Rizzi

This work presents a new decomposition for the Monotone Boolean Duality problem, which consists of checking whether two monotone Boolean functions

f

(described by its unique irredundant DNF) and

g

(described by its unique irredundant CNF) are equivalent. This coNP problem has several applications and relevance in many research areas (e.g., data mining and knowledge discovery, artificial intelligence, matroid theory, computational biology, and, last but not least, mathematical programming). Best-known algorithms run in quasi-polynomial time; no polynomial-time algorithm has been discovered yet, even if for many special classes these are known. The exact complexity of the general problem is still an open question. Based on both classical results by Berge (1989) and Fredman and Khachiyan (1996), and also on the concept of full covers used by Boros and Makino (2009) and Elbassioni (2008), we propose a new approach to decompose the problem and obtain new theoretical results. Our scheme offers a strong bound which, in the worst case only, has the same time complexity as Fredman and Khachiyan (1996). Anyway, it better highlights the inherent symmetry of the problem, lets us present another polynomial-space algorithm for the Monotone Boolean Dualization problem, and motivates further study on full covers.

该问题包括检查两个单调布尔函数 f（由其唯一的非冗余 DNF 描述）和 g（由其唯一的非冗余 CNF 描述）是否等价。这个 coNP 问题在许多研究领域（如数据挖掘和知识发现、人工智能、矩阵理论、计算生物学，最后但并非最不重要的是数学编程）都有一些应用和相关性。最著名的算法都是在准多项式时间内运行的；目前还没有发现任何多项式时间算法，即使对许多特殊类别来说，这些算法是已知的。一般问题的确切复杂性仍是一个未决问题。基于 Berge (1989) 和 Fredman 和 Khachiyan (1996) 的经典结果，以及 Boros 和 Makino (2009) 和 Elbassioni (2008) 使用的全覆盖概念，我们提出了一种分解问题的新方法，并获得了新的理论结果。我们的方案提供了一个强约束，仅在最坏情况下，其时间复杂度与 Fredman 和 Khachiyan（1996）相同。总之，它更好地突出了问题的内在对称性，让我们为单调布尔二化问题提出了另一种多项式空间算法，并激发了对全覆盖的进一步研究。

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引用次数: 0

Star-factors with large components, fractional k-extendability and spectral radius in graphs 图中的大分量星形因子、分数 k 扩展性和谱半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-11 DOI: 10.1016/j.dam.2024.10.018

Sizhong Zhou , Hongxia Liu

Let

G

be a graph, and let

m

and

k

be two integers with

m \geq 2

and

k \geq 1

. A

{K_{1, j} : m \leq j \leq 2 m}

-factor of

G

is a spanning subgraph of

G

, in which every component is isomorphic to a member in

{K_{1, j} : m \leq j \leq 2 m}

. A graph

G

is fractional

k

-extendable if every

k

-matching in

G

can be extended to a fractional perfect matching of

G

. In this paper, we first establish a lower bound on the

A_{a}

-spectral radius of

G

to guarantee that

G

has a

{K_{1, j} : m \leq j \leq 2 m}

-factor, where

a \in {0, 1}

. Then we determine a lower bound on the

A_{α}

-spectral radius of

G

to ensure that

G

is fractional

k

-extendable, where

α \in [0, 1)

让 G 是一个图，让 m 和 k 是两个整数，其中 m≥2 和 k≥1。G 的一个 {K1,j:m≤j≤2m} 因子是 G 的一个跨子图，其中每个分量都与 {K1,j:m≤j≤2m} 中的一个成员同构。本文首先建立了 G 的 Aa 谱半径下限，以保证 G 具有 {K1,j:m≤j≤2m} 因子，其中 a∈{0,1}。然后，我们确定 G 的 Aα 谱半径下限，以确保 G 是分数 k 可扩展的，其中 α∈[0,1)。

引用次数: 0

On the pebbling numbers of Flower, Blanuša and Watkins snarks 关于《花》、《布拉努沙》和《沃特金斯》的鹅卵石数量

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-08 DOI: 10.1016/j.dam.2024.10.020

Matheus Adauto , Celina de Figueiredo , Glenn Hurlbert , Diana Sasaki

Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number

π (G)

is the smallest

t

so that from any initial configuration of

t

pebbles it is possible, after a sequence of pebbling moves, to place a pebble on any given target vertex. In this paper, we provide the first results on the pebbling numbers of snarks. Until now, only the Petersen graph had its pebbling number correctly established, although attempts had been made for the Flower and Watkins snarks.

图卵石游戏是一种在顶点上有卵石的图上进行的游戏。每次移动鹅卵石都会从一个顶点移走两颗鹅卵石，并在相邻顶点放置一颗鹅卵石。卵石数 π(G)是最小的 t，即从任何 t 个卵石的初始配置出发，经过一连串的卵石移动后，有可能在任何给定的目标顶点上放置一个卵石。在本文中，我们首次提出了关于 "咆哮图 "卵石数的结果。到目前为止，只有彼得森图的鹅卵石数被正确确定，尽管弗劳尔图和沃特金斯图的鹅卵石数也曾被尝试确定。

引用次数: 0

Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size 给定阶数和大小的图的基于度的加权邻接矩阵的极谱半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-07 DOI: 10.1016/j.dam.2024.10.025

Chenghao Shen, Haiying Shan

The

f

-adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge

i j

f (d_{i}, d_{j})

, where

f

is a real symmetric function and

d_{i}, d_{j}

are the degrees of vertex

i

and vertex

j

. The

f

-spectral radius of a graph is the spectral radius of its

f

-adjacency matrix. In this paper, the effect of subdividing an edge on

f

-spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an application of these results, we obtain the bicyclic graph(s) with the smallest

f

-spectral radius for fixed order

n \geq 8

by applying generalized Lu–Man method.

f-adjacency 矩阵是一种边缘加权邻接矩阵，其边缘 ij 的权重为 f(di,dj)，其中 f 是实对称函数，di,dj 是顶点 i 和顶点 j 的度数。本文讨论了细分边对 f 谱半径的影响。本文推导了具有给定阶数和大小的极值图的一些必要条件。作为这些结果的应用，我们应用广义鲁曼法得到了固定阶数 n≥8 时 f 谱半径最小的双环图。

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引用次数: 0

The phylogeny number of a graph in the aspect of its triangles and diamonds 三角形和菱形图的系统发生数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-07 DOI: 10.1016/j.dam.2024.10.028

Soogang Eoh, Suh-Ryung Kim, Hojun Lee

<div><div>The phylogeny graph of a digraph <math><mi>D</mi></math>, denoted by <math><mrow><mi>P</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math>, has the vertex set <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math> and an edge <math><mrow><mi>u</mi><mi>v</mi></mrow></math> if and only if <math><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></math> or <math><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow></math> is an arc of <math><mi>D</mi></math> or <math><mi>u</mi></math> and <math><mi>v</mi></math> have a common out-neighbor in <math><mi>D</mi></math>. The notion of phylogeny graphs was introduced by Roberts and Sheng (1997) as a variant of competition graph. Moral graphs having arisen from studying Bayesian networks are the same as phylogeny graphs. Any acyclic digraph <math><mi>D</mi></math> for which <math><mi>G</mi></math> is an induced subgraph of <math><mrow><mi>P</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math> and such that <math><mi>D</mi></math> has no arcs from vertices outside of <math><mi>G</mi></math> to vertices in <math><mi>G</mi></math> is called a phylogeny digraph for <math><mi>G</mi></math>.</div><div>The phylogeny number <math><mrow><mi>p</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> of <math><mi>G</mi></math> is the smallest <math><mi>r</mi></math> so that <math><mi>G</mi></math> has a phylogeny digraph <math><mi>D</mi></math> with <math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>∖</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><mi>r</mi></mrow></math>. In this paper, we integrate the existing theorems computing phylogeny numbers of connected graph with a small number of triangles into one proposition: for a graph <math><mi>G</mi></math> containing at most two triangle, <math><mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>2</mn><mi>t</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>≤</mo><mi>p</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mi>t</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math> where <math><mrow><mi>t</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo><

当且仅当(u,v)或(v,u)是 D 的弧或 u 和 v 在 D 中有共同的外邻时，数字图 D 的系统发育图才有顶点集 V(D)和边 uv，用 P(D) 表示。道德图产生于贝叶斯网络研究，与系统发育图相同。任何非循环数图 D，如果 G 是 P(D) 的诱导子图，且 D 没有从 G 外顶点到 G 内顶点的弧，则称为 G 的系统发育数图。G 的系统发育数 p(G) 是使 G 具有|V(D)∖V(G)|=r 的系统发育数图 D 的最小 r。本文将现有的计算具有少量三角形的连通图的系统发生数的定理整合为一个命题：对于最多包含两个三角形的图 G，|E(G)|-|V(G)|-2t(G)+d(G)+1≤p(G)≤|E(G)|-|V(G)|-t(G)+1，其中 t(G) 和 d(G) 分别表示 G 中的三角形数和菱形数。然后，我们将证明这些不等式对于有许多三角形的图形是成立的。在证明过程中，我们推导出一个有用的定理，该定理在推导各种有意义的结果中发挥了关键作用，其中包括一个回答 Wu 等人（2019）所提问题的定理。

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