Discrete Applied Mathematics最新文献

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A note on degree conditions for Ramsey goodness of paths 关于路径拉姆齐良度条件的一个注记

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.012

Chunlin You

<div><div>For graphs <math><mi>G</mi></math>, <math><mi>F</mi></math> and <math><mi>H</mi></math>, we write <math><mrow><mi>G</mi><mo>→</mo><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> if every red/blue edge coloring of <math><mi>G</mi></math> contains a red copy of <math><mi>F</mi></math> or a blue copy of <math><mi>H</mi></math>. The Ramsey number <math><mrow><mi>r</mi><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> is the smallest number <math><mi>N</mi></math> such that the complete graph <math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>F</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math>. A classical result of Chvátal implies that if <math><mrow><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math>, then every red/blue edge-coloring of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math> contains a red <math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math> or a blue <math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math>. We study a natural generalization of this result, determining the exact minimum degree condition for a graph <math><mi>G</mi></math> with at least <math><mrow><mi>r</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math> vertices to guarantee <math><mrow><mi>G</mi><mo>→</mo><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math>. Recently, Aragão et al. (2025) proposed the following conjecture: for all integers <math><mrow><mi>r</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi></mrow></math> with <math><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></math>, let <math><mi>G</mi></math> be a graph with <math><mi>n</mi></math> vertices such that <math><mrow><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>k</mi><mo><</mo><mi>n</mi><mo>≤</mo><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math> and <math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>n</mi><mo>−</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfra

对于图G， F和H，如果G的每一个红/蓝边着色都包含F的一个红副本或H的一个蓝副本，则我们写出G→（F，H）。拉姆齐数r（F，H）是使完全图KN→（F，H）的最小数N。Chvátal的一个经典结果表明，如果n≥(r−1)(t−1)+1，则Kn的每一个红/蓝边染色包含一个红色Kr或一个蓝色Pt。我们研究了这个结果的一个自然推广，确定了至少有r（Kr,Pt）顶点的图G的精确最小度条件，以保证G→（Kr,Pt）。最近,阿拉冈et al。(2025)提出了以下猜想:所有整数r, k, t和r≥2,让G是一个有n个顶点的图,这样(r−1)(t−1)k< n≤(r−1)(t−1)(k + 1)和δ(G)≥n−⌈kk + 1⌈nr−1⌉⌉,那么G→(Kr, Pt)。在本文中，我们对所有k≥3r−4证明了这个猜想。

引用次数: 0

On near optimal colorable graphs 关于近似最优可着色图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.004

C.U. Angeliya , Arnab Char , T. Karthick

<div><div>A class of graphs <math><mi>G</mi></math> is said to be near optimal colorable if there exists a constant <math><mrow><mi>c</mi><mo>∈</mo><mi>N</mi></mrow></math> such that every graph <math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math> satisfies <math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mo>max</mo><mrow><mo>{</mo><mi>c</mi><mo>,</mo><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math>, where <math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> respectively denote the chromatic number and clique number of <math><mi>G</mi></math>. The class of near optimal colorable graphs is an important subclass of the class of <math><mi>χ</mi></math>-bounded graphs which is well-studied in the literature. In this paper, we show that the class of (<math><mrow><mi>F</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mi>e</mi></mrow></math>)-free graphs is near optimal colorable, where <math><mrow><mi>F</mi><mo>∈</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></mrow></mrow></math> and <math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mi>e</mi></mrow></math>, also known as the diamond, is the graph obtained from the complete graph on four vertices by deleting one edge. This partially answers a question of Ju and Huang (2024) and is related to a question of Schiermeyer (unpublished). Furthermore, using these results with some earlier known results, we also provide an alternate proof to the fact that the Chromatic Number problem for the class of (<math><mrow><mi>F</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>−</mo><mi>e</mi></mrow></math>)-free graphs is solvable in polynomial time, where <math><mrow><mi>F</mi><mo>∈</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></mrow></mrow>

如果存在一个常数c∈N，使得每一个图G∈G满足χ(G)≤max{c,ω(G)}，其中χ(G)和ω(G)分别表示G的色数和团数，则称一类图G是近最优可着色的。近最优可着色图是文献中研究较多的χ-有界图的一个重要子类。在本文中，我们证明了一类（F,K4−e）自由图是接近最优可着色的，其中F∈{P1+2P2,2P1+P3,3P1+P2}和K4−e，也称为菱形图，是由四个顶点上的完全图通过删除一条边得到的图。这部分回答了Ju和Huang（2024）的一个问题，并与Schiermeyer（未发表）的一个问题有关。此外，利用这些结果和一些先前已知的结果，我们还提供了另一种证明，证明（F,K4−e）自由图的色数问题在多项式时间内可解，其中F∈{P1+2P2,2P1+P3,3P1+P2}。

引用次数: 0

Hamiltonicity and bipancyclicity of balanced bipartite digraphs 平衡二部有向图的哈密顿性和双环性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.022

Ruixia Wang, Lie Ma, Qiaoping Guo

In 2017, Adamus et al. studied the degree-sum condition of dominating and dominated pair of vertices for the existence of hamiltonian cycles in balanced bipartite digraphs, yielding the following result: Let

D

be a balanced bipartite digraph of order

2 a

with

a \geq 3

. If

d (x) + d (y) \geq 3 a

for every dominating or dominated pair of vertices

{x, y}

, then

D

is hamiltonian. In this paper, we introduce a new semi-degree condition for a balanced bipartite digraph to be hamiltonian. We replace the degree condition

d (x) + d (y) \geq 3 a

with

min {d^{+} (x) + d^{+} (y), d^{-} (x) + d^{-} (y)} \geq a + 2

and prove that

D

is hamiltonian; moreover it is also bipancyclic. Using two examples, we show that these two degree conditions are independent. In addition, we also show that the lower bound

a + 2

is indeed sharp.

2017年，Adamus等研究了平衡二部有向图中哈密顿环存在的支配顶点对和支配顶点对的度和条件，得到如下结果：设D为a≥3的2a阶平衡二部有向图。如果d(x)+d(y)≥3a对于每一个支配或支配顶点对{x，y}，则d是哈密顿的。本文给出了平衡二部有向图是哈密顿的一个新的半次条件。用min{d+(x)+d+(y),d−(x)+d−(y)}≥a+2代替d(x)+d(y)≥3a，证明d是哈密顿量；而且它也是双环的。通过两个例子，我们证明了这两个度条件是独立的。此外，我们还证明了a+2的下界确实是尖锐的。

引用次数: 0

Charging station placement for limited energy robots 有限能量机器人的充电站安置

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.020

Arun Kumar Das

We address the minimum charging station (Min-Station) placement problem for unlabeled robots. The input consists of a graph

G = (V, E)

, with given starting and target positions

S \subset V

and

T \subset V

, respectively, for a set of

m

robots. A robot moves one step by transitioning from one vertex to an adjacent vertex in

G

and has a movement capacity of

r

steps, meaning it can move up to

r

steps without recharging from either its starting position or a charging station. Each target position must be occupied by exactly one robot to complete the motion, hence

| S | = | T |

. The objective is to determine the minimum number of charging stations needed to ensure all target positions are reached. We prove that this problem is NP-hard, even for bounded degree graphs with a maximum degree of 6. However, on the positive side, we present linear and quadratic time algorithms for the Min-Station problem on paths and cycles.

我们解决了无标签机器人的最小充电站（Min-Station）安置问题。输入由一个图G=（V，E）组成，对于一组m个机器人，起始位置和目标位置分别为S∧V和T∧V。机器人通过从G中的一个顶点过渡到相邻的顶点来移动一步，并且具有r步的移动能力，这意味着它可以移动r步而无需从其起始位置或充电站充电。每个目标位置必须恰好有一个机器人占据才能完成运动，因此|S|=|T|。目标是确定充电站的最小数量，以确保达到所有目标位置。我们证明了这个问题是np困难的，即使对于最大度为6的有界度图也是如此。然而，在积极的方面，我们提出了线性和二次时间算法的最小站问题的路径和循环。

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引用次数: 0

The ϵ-spectral radii of k-uniform hypertrees k-一致超树的ϵ-spectral半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.015

Weimin Li , Jianping Li , Weihua He , Jianbin Zhang

The

ϵ

-spectral radius of a connected hypergraph is the largest eigenvalue of its eccentricity matrix. Let

T_{n, d}

be the set of

k

-uniform hypertrees with order

n

and diameter

d

. In this paper, we characterize the unique hypertree with the minimum

ϵ

-spectral radius among

⋃_{d \neq 3} T_{n, d}

, which solves the problem prosed by Zhou and Zhu (Zhou and Zhu, 2024).

连通超图的ϵ-spectral半径是其偏心矩阵的最大特征值。设Tn，d为k个阶为n、直径为d的一致超树的集合。在本文中，我们刻画了在∈d≠3Tn，d中具有最小ϵ-spectral半径的唯一超树，解决了Zhou和Zhu （Zhou and Zhu, 2024）提出的问题。

引用次数: 0

On budget-constrained coverage in Multi-Interface networks: Branchwidth and treewidth perspectives 多接口网络的预算约束覆盖：分支宽度和树宽度视角

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.005

Alessandro Aloisio , Alfredo Navarra

Multi-Interface networks concern to achieve efficient communications while managing heterogeneous devices. The requirement to establish connections may occur in various contexts, including emergency situations where conventional connections are unavailable.

We investigate the so-called Coverage problem in budget-constrained Multi-Interface networks. The budget is the total amount of energy in the network, while the coverage means the model aims to activate every required communication among the available devices.

The network is modeled by an undirected graph, where vertices represent devices, and edges represent the desired communication links. Each device is endowed with a set of interfaces it can use to communicate with other devices. Activating an interface at the end of an edge yields a profit for the edge but also incurs an energy cost for both the related devices. The profits incentivize solutions that enhance the network’s quality in terms of bandwidth. The energy cost plays a crucial role in extending the network’s lifespan by optimizing battery usage. We propose a generalization of this model by introducing a function that assigns an upper bound on the number of interfaces that can be used on each device. Additionally, we use a

p

-norm function to represent the cost across the network. Finally, we analyze different cases in which the profits and costs depend either solely on the interfaces or also on the vertices and edges.

Since the original problem has been proven to be NP-hard, we study our generalization from the perspective of parameterized complexity. We present several results for the modified problem, where the parameter is the well-known branchwidth, combined in various ways with the number of available interfaces, the maximum energy,

p

, and an upper bound on the total profit. We then extend these results to treewidth, another classical parameter. Finally, we note that when the graph is complete, the problem is para-NP-hard if the profits also depend on the links and the parameter is the number of available interfaces.

多接口网络关注的是在管理异构设备的同时实现高效通信。建立连接的要求可能发生在各种情况下，包括无法使用常规连接的紧急情况。我们研究了预算受限的多接口网络中所谓的覆盖问题。预算是网络中的总能量，而覆盖意味着模型旨在激活可用设备之间所需的所有通信。该网络由无向图建模，其中顶点表示设备，边表示所需的通信链路。每个设备都被赋予了一组接口，可以用来与其他设备通信。在边缘的末端激活接口会为边缘带来利润，但也会为相关设备带来能源成本。利润会激励那些在带宽方面提高网络质量的解决方案。通过优化电池使用，能源成本在延长网络寿命方面起着至关重要的作用。我们通过引入一个函数来推广这个模型，该函数指定了每个设备上可以使用的接口数量的上界。此外，我们使用p-范数函数来表示整个网络的成本。最后，我们分析了利润和成本仅取决于接口或也取决于顶点和边的不同情况。由于原来的问题已经被证明是np困难的，我们从参数化复杂性的角度来研究我们的泛化。我们给出了修正问题的几个结果，其中参数是众所周知的分支宽度，以各种方式与可用界面的数量、最大能量、p和总利润的上界相结合。然后我们将这些结果扩展到树宽，另一个经典参数。最后，我们注意到，当图完全时，如果利润也依赖于链路，且参数是可用接口的数量，则问题是类np困难的。

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引用次数: 0

On graceful colorings in subcubic trees 论次立方树的优美着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.027

Paola T. Pantoja , Simone Dantas , Atílio G. Luiz

A graceful

k

-coloring of a graph

G

is a proper vertex coloring

π : V (G) \to {1, 2, \dots, k}

k \geq 1

, that induces a proper edge coloring

π^{'} : E (G) \to {1, 2, \dots, k - 1}

defined by

π^{'} (u v) = | π (u) - π (v) |

, where

u v \in E (G)

. The smallest positive integer

k

for which a graph

G

has a graceful

k

-coloring is called the graceful chromatic number of

G

and is denoted by

χ_{g} (G)

. It is well known that any tree

T

of maximum degree

Δ

has graceful chromatic number

χ_{g} (T) \leq ⌈ \frac{5}{3} Δ ⌉

. In this paper, we refine this bound by showing that, under certain restrictions, a tree with maximum degree

Δ

has a graceful chromatic number of either

Δ + 1

or at most

Δ + 2

. Additionally, we identify two families of lobsters with maximum degree 3: one with a graceful chromatic number of 4 and another with a graceful chromatic number of 5, showing that both lower and upper bounds for this parameter are attained within this class of graphs. Furthermore, we develop a polynomial-time algorithm to determine whether a given tree

T

with maximum degree three has a graceful chromatic number of 4. Finally, we present a proof that the Graceful 4-Coloring Problem is NP-complete.

图G的优美k-着色是一个适当的顶点着色π:V(G)→{1,2，…，k}, k≥1，它引出一个适当的边着色π ':E(G)→{1,2，…，k−1}，定义为π ' (uv)=|π(u)−π(V)|，其中uv∈E(G)。图G具有优美k着色的最小正整数k称为G的优美色数，用χg(G)表示。众所周知，任何最大次为Δ的树T都具有优美的色数χg(T)≤≤53Δ²。在本文中，我们通过证明在一定的限制条件下，最大度为Δ的树有一个优美的色数Δ+1或最多Δ+2来改进这个界。此外，我们确定了两个最大度为3的龙虾族：一个具有优美色数4，另一个具有优美色数5，表明该参数的下界和上界都是在这类图中获得的。此外，我们开发了一个多项式时间算法来确定给定的最大次为3的树T是否具有优美的色数4。最后，我们证明了优美4-着色问题是np完全的。

{"title":"On graceful colorings in subcubic trees","authors":"Paola T. Pantoja , Simone Dantas , Atílio G. Luiz","doi":"10.1016/j.dam.2025.11.027","DOIUrl":"10.1016/j.dam.2025.11.027","url":null,"abstract":"<div><div>A graceful <math><mi>k</mi></math>-coloring of a graph <math><mi>G</mi></math> is a proper vertex coloring <math><mrow><mi>π</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></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>, <math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math>, that induces a proper edge coloring <math><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></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><mn>1</mn><mo>}</mo></mrow></mrow></math> defined by <math><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>π</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>−</mo><mi>π</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math>, where <math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. The smallest positive integer <math><mi>k</mi></math> for which a graph <math><mi>G</mi></math> has a graceful <math><mi>k</mi></math>-coloring is called the graceful chromatic number of <math><mi>G</mi></math> and is denoted by <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. It is well known that any tree <math><mi>T</mi></math> of maximum degree <math><mi>Δ</mi></math> has graceful chromatic number <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>g</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌈</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mi>Δ</mi><mo>⌉</mo></mrow></mrow></math>. In this paper, we refine this bound by showing that, under certain restrictions, a tree with maximum degree <math><mi>Δ</mi></math> has a graceful chromatic number of either <math><mrow><mi>Δ</mi><mo>+</mo><mn>1</mn></mrow></math> or at most <math><mrow><mi>Δ</mi><mo>+</mo><mn>2</mn></mrow></math>. Additionally, we identify two families of lobsters with maximum degree 3: one with a graceful chromatic number of 4 and another with a graceful chromatic number of 5, showing that both lower and upper bounds for this parameter are attained within this class of graphs. Furthermore, we develop a polynomial-time algorithm to determine whether a given tree <math><mi>T</mi></math> with maximum degree three has a graceful chromatic number of 4. Finally, we present a proof that the Graceful 4-Coloring Problem is NP-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 98-111"},"PeriodicalIF":1.0,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580214","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

Structural parameterization of minus domination 负支配的结构参数化

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.030

Sangam Balchandar Reddy, Anjeneya Swami Kare

Given a graph

G = (V, E)

, a minus dominating function

f : V \to {- 1, 0, 1}

is a labeling of vertices such that

\sum_{v \in N [u]} f (v) \geq 1

, for each vertex

u \in V

. The weight of

f

is the sum of

f (u)

over all the vertices

u \in V

. The objective of Minus Domination problem is to compute a minus dominating function of minimum weight. The problem is known to be NP-complete even on split graphs and bipartite graphs. The parameterized complexity of the problem for the parameter treewidth is a long standing open question. In this paper, we answer this by proving that the problem is W[1]-hard for the parameter distance to disjoint paths, which is larger than treewidth and feedback vertex set number. Later, we show that no polynomial kernel exists for the parameter vertex cover number. For the parameter weight, we prove that the problem is W[2]-hard on bipartite graphs and W[1]-hard on circle graphs, respectively. On the positive front, we provide an FPT algorithm for the parameter feedback edge set number. In addition, we show that the problem is APX-hard on graphs with maximum degree 5.

给定一个图G=(V，E)，一个负支配函数f:V→{- 1,0,1}是顶点的标记，使得∑V∈N[u]f(V)≥1，对于每个顶点u∈V。f的权值是f(u)对所有u∈V的和。负支配问题的目标是计算一个权值最小的负支配函数。已知该问题在分裂图和二部图上是np完全的。参数树宽问题的参数化复杂性是一个长期悬而未决的问题。在本文中，我们通过证明不相交路径的参数距离大于树宽和反馈顶点集数的问题是W[1]-hard来回答这个问题。随后，我们证明了参数顶点覆盖数不存在多项式核。对于参数权值，我们分别证明了问题在二部图上是W[2]-hard，在圆图上是W[1]-hard。在正面，我们提供了一种参数反馈边集数的FPT算法。此外，我们还证明了问题在最大度为5的图上是APX-hard的。

{"title":"Structural parameterization of minus domination","authors":"Sangam Balchandar Reddy, Anjeneya Swami Kare","doi":"10.1016/j.dam.2025.11.030","DOIUrl":"10.1016/j.dam.2025.11.030","url":null,"abstract":"<div><div>Given a graph <math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math>, a minus dominating function <math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></math> is a labeling of vertices such that <math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>N</mi><mrow><mo>[</mo><mi>u</mi><mo>]</mo></mrow></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math>, for each vertex <math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi></mrow></math>. The weight of <math><mi>f</mi></math> is the sum of <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math> over all the vertices <math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi></mrow></math>. The objective of Minus Domination problem is to compute a minus dominating function of minimum weight. The problem is known to be NP-complete even on split graphs and bipartite graphs. The parameterized complexity of the problem for the parameter treewidth is a long standing open question. In this paper, we answer this by proving that the problem is W[1]-hard for the parameter distance to disjoint paths, which is larger than treewidth and feedback vertex set number. Later, we show that no polynomial kernel exists for the parameter vertex cover number. For the parameter weight, we prove that the problem is W[2]-hard on bipartite graphs and W[1]-hard on circle graphs, respectively. On the positive front, we provide an FPT algorithm for the parameter feedback edge set number. In addition, we show that the problem is APX-hard on graphs with maximum degree 5.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 137-150"},"PeriodicalIF":1.0,"publicationDate":"2025-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580195","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

On Bollobás-type theorems of d-tuples d元组的Bollobás-type定理

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.025

Erfei Yue

<div><div>In 1965, Bollobás (1965) proved that for a Bollobás set-pair system <math><mrow><mo>{</mo><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>∣</mo><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>m</mi><mo>]</mo></mrow><mo>}</mo></mrow></math>, the maximum value of <math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mrow><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow></mfrac></mrow></mfenced></mrow></math>−1 is 1. Hegedüs and Frankl (2024) recently extended the concept of Bollobás systems to <math><mi>d</mi></math>-tuples, conjecturing that for a Bollobás system of <math><mi>d</mi></math>-tuples, <math><mrow><mo>{</mo><mrow><mo>(</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow><mo>∣</mo><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>m</mi><mo>]</mo></mrow><mo>}</mo></mrow></math>, the maximum value of <math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mrow><mo>|</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mo>|</mo></mrow><mo>+</mo><mo>⋯</mo><mo>+</mo><mrow><mo>|</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></msubsup><mo>|</mo></mrow></mrow><mrow><mrow><mo>|</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msubsup><mo>|</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mrow><mo>|</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow></msubsup><mo>|</mo></mrow></mrow></mfrac></mrow></mfenced></mrow></math>−1 is also 1. This paper refutes this conjecture and establishes an upper bound for the sum. In the case <math><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow></math>, the derived upper bound is asymptotically tight. Furthermore, we sharpen an inequality for skew Bollobás systems of <math><mi>d</mi></math>-tuples in Hegedüs and Frankl (2024), Finally, we determine the maximum size of a uniform skew Bollobás system of <math><mi>d</mi></math>-tuples on both sets an

1965年Bollobás（1965）证明了对于Bollobás集对系统{(Ai,Bi)∣i∈[m]},∑i=1m|Ai|+|Bi||Ai|−1的最大值为1。heged s和Frankl（2024）最近将Bollobás系统的概念扩展到d元组，推测对于d元组的Bollobás系统，{(Ai(1)，…，Ai(d))∣i∈[m]},∑i=1m|Ai(1)|+⋯+|Ai(d)||Ai(1)|，…，|Ai(d)|−1的最大值也是1。本文驳斥了这一猜想，并建立了和的上界。在d=3的情况下，导出的上界是渐近紧的。此外，我们在heged和Frankl（2024）中尖锐了d元组的倾斜Bollobás系统的不等式，最后，我们确定了集合和空间上d元组的均匀倾斜Bollobás系统的最大尺寸。

引用次数: 0

Characterizations of graphs with equal vertex and coalition number 等顶点和等联合体数图的刻画

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-11-18 DOI: 10.1016/j.dam.2025.11.002

Fuyuan Yang , Qiang Sun , Chao Zhang

<div><div>Let <math><mrow><mi>Φ</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow></math> be a vertex partition of vertex set <math><mi>V</mi></math> of a graph <math><mi>G</mi></math>. A set <math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math> is a dominating set of a graph <math><mi>G</mi></math> if every vertex in <math><mrow><mi>V</mi><mo>−</mo><mi>S</mi></mrow></math> is adjacent to a vertex in <math><mi>S</mi></math>. Furthermore, two disjoint sets <math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mi>V</mi></mrow></math> form a coalition in <math><mi>G</mi></math> if neither of them is a dominating set of <math><mi>G</mi></math> but their union <math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∪</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> is a dominating set. Denote <math><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></math> by the cardinality of the vertex set <math><mi>V</mi></math>. A vertex partition <math><mi>Φ</mi></math> of <math><mi>V</mi></math> is a coalition partition of <math><mi>G</mi></math> if every set <math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>Φ</mi></mrow></math> either is a dominating set consisting of a single vertex of degree <math><mrow><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math>, or is not a dominating set, but for some <math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>Φ</mi></mrow></math>, <math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and <math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub></math> form a coalition. The coalition number <math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> is the maximum cardinality of a coalition partition of <math><mi>G</mi></math>. In this paper, we characterize all graphs <math><mi>G</mi></math> with <math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow></math> and solve one of the open problems posed by Haynes et al. Moreover, we give shorter proofs of some theorems on the characterization of all graphs <math><mi>G</mi></math> with <math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn></mrow></math> and <math><mrow><mi>C</mi><mrow><mo>(</mo><mi>G</m

让Φ= {V1,…,Vk}是一个顶点的顶点集合V分区图G S⊆V是一组控制的一组一个图G如果每个顶点V−S是相邻的顶点S .此外,两个不相交的集V1、V2⊆V组建联合政府在G如果他们两人主导的G但工会V1∪V2控制集。用顶点集V的基数表示|V|，如果每个集合Vi∈Φ是由单个度为|V|−1的顶点组成的控制集，或者不是控制集，但对于某些Vj∈Φ， Vi和Vj形成一个联盟，则V的顶点划分Φ是G的一个联合划分。联盟数C(G)是G的一个联盟划分的最大基数。本文用C(G)=|V|刻画了所有图G，并解决了Haynes等人提出的一个开放问题。此外，对于所有图G δ(G)≤1且C(G)=n，以及所有树T C(G)=n−1的刻画，我们给出了一些定理的简短证明。

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