Discrete Applied Mathematics最新文献

英文中文

Approximation algorithm of maximizing non-submodular functions under non-submodular constraint 非次模化约束条件下的非次模化函数最大化近似算法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-26 DOI: 10.1016/j.dam.2024.09.022

Nowadays, maximizing the non-negative and non-submodular objective functions under Knapsack constraint or Cardinality constraint is deeply researched. Nevertheless, few studies study the objective functions with non-submodularity under the non-submodular constraint. And there are many practical applications of the situations, such as Epidemic transmission, and Sensor Placement and Feature Selection problem. In this paper, we study the maximization of the non-submodular objective functions under the non-submodular constraint. Based on the non-submodular constraint, we discuss the maximization of the objective functions with some specific properties, which includes the property of negative, and then, we obtain the corresponding approximate ratios by the greedy algorithm. What is more, these approximate ratios could be improved when the constraint becomes tight.

如今，在Knapsack约束或Cardinality约束下最大化非负和非次模态目标函数的研究已经非常深入。然而，很少有人研究非次模化约束下的非次模化目标函数。而这种情况在实际应用中很多，如流行病传播、传感器安置和特征选择问题等。本文研究了非次模化约束下的非次模化目标函数最大化问题。基于非次模化约束，我们讨论了目标函数最大化的一些特定属性，其中包括负属性，然后通过贪婪算法得到了相应的近似比率。更重要的是，当约束条件变得严格时，这些近似比率可以得到改善。

引用次数: 0

Unraveling the enigmatic irregular coloring of Honeycomb Networks 揭开蜂巢网络神秘的不规则着色面纱

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.dam.2024.09.013

The honeycomb mesh, based on the hexagonal tessellation, is considered a multiprocessor interconnection network. Cellular phone station placement, computer graphics and image processing are some applications where hexagonal tessellations are used. They are also used for the representation of benzenoid hydrocarbons. In this paper, we study the irregular chromatic number of Honeycomb Network, Honeycomb Cup Network and Honeycomb Torus and establish an exact value for the same. Also, we find an upper bound for the Honeycomb Rhombic and Rectangular Torus networks.

基于六边形细分的蜂巢网格被认为是一种多处理器互连网络。蜂窝电话站布局、计算机制图和图像处理都是六边形网格的应用领域。它们还用于表示苯碳氢化合物。本文研究了蜂巢网络、蜂巢杯状网络和蜂巢环状网络的不规则色度数，并确定了其精确值。此外，我们还找到了蜂巢菱形网和矩形环网的上限。

引用次数: 0

Maximum energy bicyclic graphs containing two odd cycles with one common vertex 包含两个奇数循环和一个共同顶点的最大能量双环图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.dam.2024.09.014

The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let

P_{n}^{6, 6}

be the graph obtained from two copies of

C_{6}

joined by a path

P_{n - 10}

. In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is

P_{n}^{6, 6}

. This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.

图形的能量是其邻接矩阵所有特征值的绝对值之和。假设 Pn6,6 是由 C6 的两个副本通过路径 Pn-10 连接而成的图。2001 年，Gutman 和 Vidović（2001 年）猜想能量最大的双环图是 Pn6,6。这一猜想适用于双方位双环图。对于非双方形双环图，Ji 和 Li（2012 年）证明了双环图的猜想，这些双环图恰好有两个边缘相交的循环，其中一个循环是偶数循环，另一个循环是奇数循环。本文将证明包含两个奇数循环且有一个共同顶点的双环图的猜想。

引用次数: 0

Toughness and distance spectral radius in graphs involving minimum degree 涉及最小度的图中的韧性和距离谱半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.dam.2024.09.019

The toughness

τ (G) = \min {\frac{| S |}{c (G - S)} : S is a cut set of vertices in G}

for

G ≇ K_{n} .

The concept of toughness initially proposed by Chvátal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph

G

is called

t

-tough if

τ (G) \geq t .

It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree

δ

and

t

-tough with

t \geq 1

being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present a sufficient condition based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree

δ .

Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be

t

-tough, where

t

\frac{1}{t}

is a positive integer.

G≇Kn 的韧性 τ(G)=min{|S|c(G-S):Sis a cut set of vertices inG} 。韧性的概念最初是由 Chvátal 于 1973 年提出的，它是一种简单的方法来衡量图中各个部分的紧密程度。如果 τ(G)≥t ，则图 G 称为 t-韧性图。研究图的韧性和特征值之间的关系非常有趣。Fan、Lin 和 Lu [European J. Combin.通过使用一些典型的距离谱技术和结构分析，我们在本文中提出了一个基于距离谱半径的充分条件，以保证图是最小度为 δ 的 1-韧图。此外，我们还证明了关于距离谱半径的充分条件，以保证图是 t-韧图，其中 t 或 1t 是正整数。

{"title":"Toughness and distance spectral radius in graphs involving minimum degree","authors":"","doi":"10.1016/j.dam.2024.09.019","DOIUrl":"10.1016/j.dam.2024.09.019","url":null,"abstract":"<div><div>The toughness <math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>min</mi><mrow><mo>{</mo><mfrac><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mrow></mfrac><mo>:</mo><mi>S</mi><mspace></mspace><mtext>is a cut set of vertices in</mtext><mspace></mspace><mi>G</mi><mo>}</mo></mrow></mrow></math> for <math><mrow><mi>G</mi><mo>≇</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>.</mo></mrow></math> The concept of toughness initially proposed by Chvátal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph <math><mi>G</mi></math> is called <math><mi>t</mi></math>-tough if <math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>t</mi><mo>.</mo></mrow></math> It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree <math><mi>δ</mi></math> and <math><mi>t</mi></math>-tough with <math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math> being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present a sufficient condition based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree <math><mrow><mi>δ</mi><mo>.</mo></mrow></math> Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be <math><mi>t</mi></math>-tough, where <math><mi>t</mi></math> or <math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></mfrac></math> is a positive integer.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Independence, matching and packing coloring of the iterated Mycielskian of graphs 图的迭代密西尔斯基的独立性、匹配和包装着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.dam.2024.09.015

<div><div>Let <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> denote respectively the independence, matching, 2-matching and packing chromatic numbers of a graph <math><mi>G</mi></math>. A well-known construction on graphs, called the Mycielskian of a graph, transforms any <math><mi>k</mi></math>-chromatic graph <math><mi>G</mi></math> into a <math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math>-chromatic graph <math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> having an equal clique number to <math><mi>G</mi></math>. The <math><mi>t</mi></math>th iterated Mycielskian of a graph <math><mi>G</mi></math>, denoted <math><mrow><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, is obtained by iteratively repeating the Mycielskian transformation <math><mi>t</mi></math> times. In this paper, we give <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math> in terms of <math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. Then we show that for all <math><mrow><mi>t</mi><mo>≥</mo><mn>2</mn></mrow></math>, <math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mo>max</mo><mrow><mo>{</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math>. We characterize for all <math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math>, the connected graphs having <math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math> and those having <math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>

让 α(G)、ν(G)、ν2(G) 和 χρ(G) 分别表示图 G 的独立色度数、匹配色度数、2-匹配色度数和打包色度数。图的第 t 次迭代 Mycielskian（记为 Mt(G)）是通过重复进行 t 次 Mycielskian 变换得到的。在本文中，我们用 ν2(G)给出了 α(M(G))。然后我们证明，对于所有 t≥2，α(Mt(G))=max{|Mt-1(G)|,2t-1α(M(G))}。对于所有 t≥1，我们描述了具有 α(Mt(G))=|Mt-1(G)|的连通图和具有 α(Mt(G))=2tα(G)的连通图的特征。然后，我们给出所有 t≥1 时的ν(Mt(G))和ν2(Mt(G))。然后我们证明，对于所有 t≥1，当且仅当 G 没有完美的 2 匹配时，Mt(G) 是一个柯尼希-埃格瓦里图。随后，我们将研究 Mt(G) 的包装色度数。我们为 χρ(Mt(G))提出了几个尖锐的上界和下界，其中一些是以迭代次数 t、G 的阶、k∈{1,2,3} 的 k-independence 数和 χρ(G)来表示的。我们证明，如果 G 的直径最大为 2，χρ(Mt(G)) 可以在多项式时间内计算。最近，在 Bidine 等人 (2023) 的文章中，作者研究了 t≥1 时具有 χρ(Mt(G))=2tχρ(G)的直径为 2 的图。他们还提出了一个关于以 χρ(G) 表示的 χρ(Mt(G))增长的问题。我们证明，对于 t≥1，χρ(Mt(G)) 不可能仅由χρ(G) 的函数上界。此外，我们还讨论了 χρ(Mt(G))的可实现值，并描述了具有最小可能 χρ(Mt(G))的图的特征。

引用次数: 0

New results on edge-coloring and total-coloring of split graphs 关于分裂图的边着色和总着色的新结果

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-21 DOI: 10.1016/j.dam.2024.09.008

A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph $G$ is said to be $t$ -admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most $t$ . Given a graph $G$ , determining the smallest $t$ for which $G$ is $t$ -admissible, i.e., the stretch index of $G$ , denoted by $σ (G)$ , is the goal of the $t$ -admissibility problem. Split graphs are 3-admissible and can be partitioned into three subclasses: split graphs with $σ = 1, 2$ or 3. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with $Δ$ or $Δ + 1$ colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, it is conjectured that any graph can be colored with $Δ + 1$ or $Δ + 2$ colors, and thus can be classified as Type 1 or Type 2. Both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with $σ = 2$ . For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.

分裂图是指顶点集可以划分为一个群集和一个独立集的图。给定一个图 G，确定 G 是 t-admissible 的最小 t，即 G 的伸展指数（用 σ(G)表示），是 t-admissibility 问题的目标。分裂图是 3-admissible 的，可以划分为三个子类：σ=1、2 或 3 的分裂图。在这项工作中，我们在处理分裂图着色问题时考虑了这种划分。Vizing 证明，任何图的边都可以用 Δ 或 Δ+1 种颜色着色，因此可以分别归为第 1 类或第 2 类。当边和顶点同时着色时，可以推测任何图形都可以用 Δ+1 或 Δ+2 种颜色着色，因此可以分为第 1 类或第 2 类。对于分裂图，这两种变体都还没有定论。在本文中，我们利用上面介绍的分裂图分区，考虑了 σ=2 的分裂图的边着色问题和总着色问题。对于这一类图，我们描述了第 2 类图和第 2 类图的特征，并提供了对任何第 1 类图或第 1 类图着色的多项式时间算法。

{"title":"New results on edge-coloring and total-coloring of split graphs","authors":"","doi":"10.1016/j.dam.2024.09.008","DOIUrl":"10.1016/j.dam.2024.09.008","url":null,"abstract":"<div>A split graph is a graph whose vertex set can be partitioned into a clique and an independent set. A connected graph <math><mi>G</mi></math> is said to be <math><mi>t</mi></math>-admissible if admits a special spanning tree in which the distance between any two adjacent vertices is at most <math><mi>t</mi></math>. Given a graph <math><mi>G</mi></math>, determining the smallest <math><mi>t</mi></math> for which <math><mi>G</mi></math> is <math><mi>t</mi></math>-admissible, i.e., the stretch index of <math><mi>G</mi></math>, denoted by <math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, is the goal of the <math><mi>t</mi></math>-admissibility problem. Split graphs are 3-admissible and can be partitioned into three subclasses: split graphs with <math><mrow><mi>σ</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math> or 3. In this work we consider such a partition while dealing with the problem of coloring a split graph. Vizing proved that any graph can have its edges colored with <math><mi>Δ</mi></math> or <math><mrow><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></math> colors, and thus can be classified as Class 1 or Class 2, respectively. When both, edges and vertices, are simultaneously colored, it is conjectured that any graph can be colored with <math><mrow><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></math> or <math><mrow><mi>Δ</mi><mo>+</mo><mn>2</mn></mrow></math> colors, and thus can be classified as Type 1 or Type 2. Both variants are still open for split graphs. In this paper, using the partition of split graphs presented above, we consider the edge coloring problem and the total coloring problem for split graphs with <math><mrow><mi>σ</mi><mo>=</mo><mn>2</mn></mrow></math>. For this class, we characterize Class 2 and Type 2 graphs and we provide polynomial-time algorithms to color any Class 1 or Type 1 graph.</div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Super graphs on groups, II 群上的超级图形，II

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.dam.2024.09.012

In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz. the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same order; for each choice of a graph type $A$ and an equivalence relation $B$ , there is a graph, the $B$ super $A$ graph defined on $G$ . The resulting nine graphs (of which eight were shown to be in general distinct) form a two-dimensional hierarchy. In the present paper, we consider these graphs further. We prove universality properties for the conjugacy supergraphs of various types, adding the nilpotent, solvable and enhanced power graphs to the commuting graphs considered in the rest of the paper, and also examine their relation to the invariably generating graph of the group. We also show that supergraphs can be expressed as graph compositions, in the sense of Schwenk, and use this representation to calculate their Wiener index. We illustrate these by computing Wiener index of equality supercommuting and conjugacy supercommuting graphs for dihedral and dicyclic groups.

在早先的一篇论文中，作者考虑了定义在一个群上的三类图和三种等价关系，即幂图，增强幂图和交换图，以及相等、共轭和同阶关系；每选择一种图类型 A 和一种等价关系 B，就有一个图，即定义在 G 上的 B superA 图。在本文中，我们将进一步研究这些图。我们证明了各种类型共轭超图的普遍性，在本文其余部分所考虑的换元图的基础上增加了零势图、可解图和增强幂图，还研究了它们与群的不变生成图的关系。我们还证明，超图可以用施文克意义上的图合成来表示，并用这种表示法计算它们的维纳指数。我们通过计算二面群和二环群的相等超容图和共轭超容图的维纳指数来说明这一点。

{"title":"Super graphs on groups, II","authors":"","doi":"10.1016/j.dam.2024.09.012","DOIUrl":"10.1016/j.dam.2024.09.012","url":null,"abstract":"<div>In an earlier paper, the authors considered three types of graphs, and three equivalence relations, defined on a group, viz. the power graph, enhanced power graph, and commuting graph, and the relations of equality, conjugacy, and same order; for each choice of a graph type <math><mi>A</mi></math> and an equivalence relation <math><mi>B</mi></math>, there is a graph, the <math><mi>B</mi></math> super<math><mi>A</mi></math> graph defined on <math><mi>G</mi></math>. The resulting nine graphs (of which eight were shown to be in general distinct) form a two-dimensional hierarchy. In the present paper, we consider these graphs further. We prove universality properties for the conjugacy supergraphs of various types, adding the nilpotent, solvable and enhanced power graphs to the commuting graphs considered in the rest of the paper, and also examine their relation to the invariably generating graph of the group. We also show that supergraphs can be expressed as graph compositions, in the sense of Schwenk, and use this representation to calculate their Wiener index. We illustrate these by computing Wiener index of equality supercommuting and conjugacy supercommuting graphs for dihedral and dicyclic groups.</div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Approximation ratio of the min-degree greedy algorithm for Maximum Independent Set on interval and chordal graphs 区间图和和弦图上最大独立集最小度贪婪算法的近似率

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.dam.2024.09.009

In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a $(2 / 3)$ -approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a $(1 / 2)$ -approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of $\frac{3}{Δ + 2}$ of the greedy algorithm for general graphs of maximum degree $Δ$ .

在这篇文章中，我们证明了最小度贪婪算法与对抗性平局打破是区间图上最大独立集问题的 (2/3)- 近似。我们证明，即使在最大度数为 3 的单位区间图上，这一算法也是严密的。我们还证明，在和弦图上，贪婪算法是 (1/2)- 近似算法，而且这一算法也是严密的。这些结果与已知的贪婪算法对最大度数为 Δ 的一般图的 3Δ+2 的（紧密）近似率形成了鲜明对比。

引用次数: 0

Recognizing unit multiple interval graphs is hard 认识单位多重区间图很难

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.dam.2024.09.011

Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A $d$ -interval is the union of $d$ disjoint intervals on the real line, and a graph is a $d$ -interval graph if it is the intersection graph of $d$ -intervals. In particular, it is a unit $d$ -interval graph if it admits a $d$ -interval representation where every interval has unit length.

Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is $NP$ -complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also $NP$ -complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit $d$ -interval graphs for any $d \geq 2$ , which does not follow directly in graph recognition problems — as an example, it took almost 20 years to close the gap between $d = 2$ and $d > 2$ for the recognition of $d$ -track interval graphs. Our result has several implications, including that for every $d \geq 2$ , recognizing $(x, \dots, x)$ $d$ -interval graphs and depth $r$ unit $d$ -interval graphs is $NP$ -complete for every $x \geq 11$ and every $r \geq 4$ .

多区间图是 20 世纪 70 年代引入的对区间图的著名概括，用于处理调度和分配中自然出现的情况。d 个区间是实线上 d 个互不相交的区间的联合，如果一个图是 d 个区间的交集图，那么它就是一个 d 个区间图。特别是，如果一个图可以用 d 个区间表示，其中每个区间的长度都是单位，那么它就是一个单位 d 个区间图。长期以来，人们都知道识别 2 个区间图和其他相关类别（如 2 轨区间图）是 NP-完全的，但识别单位 2 个区间图的复杂性仍然是个未知数。在这里，我们通过证明单位 2 间隔图的识别也是 NP-完全来解决这个问题。我们的证明技术采用了一种完全不同于其他识别相关类的硬度结果的方法。此外，我们还扩展了对任意 d≥2 的单位 d 间隔图的结果，这在图识别问题中并不直接适用--例如，在识别 d 轨道间隔图时，我们花了近 20 年时间才缩小了 d=2 和 d>2 之间的差距。我们的结果有几个意义，包括对于每一个 d≥2，对于每一个 x≥11 和每一个 r≥4，识别（x,...,x）d-区间图和深度 r 单位 d-区间图都是 NP-完全的。

{"title":"Recognizing unit multiple interval graphs is hard","authors":"","doi":"10.1016/j.dam.2024.09.011","DOIUrl":"10.1016/j.dam.2024.09.011","url":null,"abstract":"<div>Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A <math><mi>d</mi></math>-interval is the union of <math><mi>d</mi></math> disjoint intervals on the real line, and a graph is a <math><mi>d</mi></math>-interval graph if it is the intersection graph of <math><mi>d</mi></math>-intervals. In particular, it is a unit <math><mi>d</mi></math>-interval graph if it admits a <math><mi>d</mi></math>-interval representation where every interval has unit length.Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is <math><mi>NP</mi></math>-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also <math><mi>NP</mi></math>-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit <math><mi>d</mi></math>-interval graphs for any <math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math>, which does not follow directly in graph recognition problems — as an example, it took almost 20 years to close the gap between <math><mrow><mi>d</mi><mo>=</mo><mn>2</mn></mrow></math> and <math><mrow><mi>d</mi><mo>></mo><mn>2</mn></mrow></math> for the recognition of <math><mi>d</mi></math>-track interval graphs. Our result has several implications, including that for every <math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math>, recognizing <math><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mo>…</mo><mo>,</mo><mi>x</mi><mo>)</mo></mrow></math>\u0000<math><mi>d</mi></math>-interval graphs and depth <math><mi>r</mi></math> unit <math><mi>d</mi></math>-interval graphs is <math><mi>NP</mi></math>-complete for every <math><mrow><mi>x</mi><mo>≥</mo><mn>11</mn></mrow></math> and every <math><mrow><mi>r</mi><mo>≥</mo><mn>4</mn></mrow></math>.</div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166218X24004013/pdfft?md5=435fad9c65782c974becdf669baa5302&pid=1-s2.0-S0166218X24004013-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Characterizations of graph classes via convex geometries: A survey 通过凸几何学描述图类的特征：调查

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.dam.2024.09.010

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a convex geometry. In this work, we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.

图凸性一直被用作更好地理解图类结构的重要工具。许多研究都致力于确定具有凸性的图是否是凸几何。在这项工作中，我们将概述通过凸几何对知名图类进行表征的结果。我们还对这一主题做出了一些贡献。

引用次数: 0

下一页尾页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

Discrete Applied Mathematics

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀