Discrete Applied Mathematics最新文献

英文中文

Nut digraphs 螺母有向图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.dam.2025.12.037

Nino Bašić , Patrick W. Fowler , Maxine M. McCarthy , Primož Potočnik

A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is dextro-nut (resp. laevo-nut); a bi-nut digraph is both laevo- and dextro-nut; an ambi-nut digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is inter-nut if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.

坚果图是一个简单的图，它的核是由一个完整向量张成的（即邻接矩阵有一个零特征值，所有非零核特征向量没有零条目）。我们将坚果图的泛化分类为坚果有向图：一个有向图的核(相对于。协核（Co-kernel）是由一个完整的向量（右旋螺母）张成的。laevo-nut);双螺母有向图既是左螺母也是右螺母；双坚果有向图是双坚果有向图，其中核和协核由同一个向量张成；如果一个有向图的核和协核的交集是由一个完整的向量张成的，那么这个有向图就是内核。已知坚果图是连通的、无叶的、非二部的。本文证明了一个双核有向图是强连通的，非二部的（即，有一个非二部的底图），并且具有至少2的最小入度和最小出度。改进的核心点和禁止核心点的概念适用于奇异有向图。介绍了坚果有向图的无穷族以及系统的聚并、交叉和乘子构造。讨论了螺母有向图与拓扑物理的相关性。

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引用次数: 0

Factors of bipartite graphs with degree conditions imposed on each partite set 对每个部集施加程度条件的二部图的因子

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.dam.2025.12.027

Michitaka Furuya , Mikio Kano

Let

G

be a bipartite graph with bipartition

(X, Y)

, and let

a, b : X \to Z_{\geq 0}

and

c : Y \to Z_{\geq 0}

be functions. In this paper, we give a sufficient condition for

G

to have a factor

F

satisfying

{deg}_{F} (x) \in {a (x), b (x)}

for all

x \in X

and

{deg}_{F} (y) \in {c (y), c (y) + 1}

for all

y \in Y

. Our theorem modifies a result in Addario-Berry et al. (2008).

设G为二分图（X，Y），设a，b:X→Z≥0，c:Y→Z≥0为函数。本文给出了G有一个因子F满足对所有x∈x degF(x)∈{a(x),b(x)}，对所有y∈y degF(y)∈{c(y),c(y)+1}的充分条件。我们的定理修正了adario - berry et al.（2008）的一个结果。

引用次数: 0

When can Cluster Deletion with bounded weights be solved efficiently? 如何有效地解决有界权的聚类删除问题？

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.dam.2025.12.028

Jaroslav Garvardt , Christian Komusiewicz, Nils Morawietz

In the NP-hard Weighted Cluster Deletion problem, the input is an undirected graph

G = (V, E)

and an edge-weight function

ω : E \to N

, and the task is to partition the vertex set

V

into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that Weighted Cluster Deletion is NP-hard on some graph classes where Cluster Deletion, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value

t

of the largest edge weight assigned by

ω

on the problem complexity for such graph classes. Our main results are that Weighted Cluster Deletion is fixed-parameter tractable with respect to

t

on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for Weighted Cluster Deletion. For example, we show that Weighted Cluster Deletion is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.

在NP-hard加权聚类删除问题中，输入是无向图G=（V，E）和边权函数ω:E→N，任务是将顶点集V划分为团，使团中边的总权值最大化。最近的研究表明，在某些图类上，加权聚类删除是np困难的，在这些图类中，每条边都有单位权值的特殊情况下，聚类删除可以在多项式时间内解决。我们研究了ω赋值的最大边权值t对这类图的问题复杂度的影响。我们的主要结果是加权聚类删除是相对于t的固定参数可处理的图类，其图由稀疏外围连接的分离良好的聚类组成。这类的具体例子是分割图和接近聚类图的图。我们通过加强先前加权聚类删除的硬度结果来补充我们的结果。例如，我们证明了加权聚类删除在图的受限子类上是np困难的，即使每个边的权值为1或2。

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引用次数: 0

On polynomial kernelization for Stable Cutset 稳定割集的多项式核化

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-22 DOI: 10.1016/j.dam.2025.12.026

Stefan Kratsch , Van Bang Le

A stable cutset in a graph

G

is a set

S \subseteq V (G)

such that vertices of

S

are pairwise non-adjacent and such that

G - S

is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is

NP

-complete to determine whether a given graph

G

has any stable cutset. Recently, Rauch et al. [FCT 2023 & JCSS 2025] gave a number of fixed-parameter tractable (FPT) algorithms, running in time

f (k) \cdot {| V (G) |}^{c}

, for Stable Cutset under a variety of parameters

k

such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to

P_{5}

-free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size.

We complement these findings by giving the first results on the existence of polynomial kernelizations for Stable Cutset, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that

NP ⊈ coNP/poly

, we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given

(G, k)

answering whether there is a stable cutset of size at most

k

. On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.

图G中的稳定切集是S的顶点不相邻且G−S不连通的集S⊥V(G)，即它既是稳定（或独立）集，又是切集（或分隔符）。与一般切集不同，确定给定图G是否存在稳定切集是np完全的。最近，Rauch等人[FCT 2023 & JCSS 2025]给出了一些固定参数可处理（FPT）算法，运行时间为f(k)⋅|V(G)|c，用于稳定割集在各种参数k下，如（给定）支配集的大小，奇环截线的大小或P5-free图的删除距离。早期的工作暗示FPT算法相对于派系宽度和相对于解决方案大小。我们通过给出稳定割集多项式核化存在的第一个结果来补充这些发现，即有效的预处理算法返回参数值中大小多项式的等效实例。在NP - coNP/poly的标准假设下，我们证明了不可能对单个路径的删除距离进行多项式核化，将删除距离推广到各种图类，也不可能对（给定）支配集的大小进行多项式核化。我们还表明，在相同的假设下，多项式核化不可能相对于解的大小，即给定（G,k）回答是否存在最大数为k的稳定割集。在积极的一面，我们展示了通过调制器对单个团，对簇或共簇图，以及通过双覆盖进行参数化的多项式核化。

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引用次数: 0

Maximal non-empty cross s-union families 最大非空交叉组合族

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.038

Yongjiang Wu , Yongtao Li , Zhiyi Liu , Lihua Feng , Tingzeng Wu

Two families of sets

F

and

G

are said to be cross

s

-union if for any

F \in F

and

G \in G

| F \cup G | \leq s

. In 2021, Frankl and Wong proved that if

F, G \subseteq 2^{[n]}

are non-empty cross

s

-union, then

| F | + | G | \leq \sum_{i = 0}^{s} (\frac{n}{i}) + 1 .

Moreover, for

s < n - 1

, equality holds if and only if

(F, G) = ({0̸}, {G \subseteq [n] : | G | \leq s})

. In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona

s

-union theorem due to Frankl and Tokushige, and characterize the extremal families for the case

s = n - 1

如果对于任意F∈F且G∈G， |F∪G|≤s，则称集合F和G的两个族是交叉并集。2021年，Frankl和Wong证明了若F、G≥≥n≤∑i=0sni+1，则|F≥≥|+|G≤∑i=0sni+1。并且，对于s<；n−1，当且仅当F，G={0},{G [n]:|G b1≤s}，等式成立。本文给出了一种新的方法来证明这一结果。我们的方法也允许我们建立一个向量空间版本和一个遗传的家族延伸。作为一个副产品，我们重新审视了Frankl和Tokushige的Katona s-并定理的向量空间版本，并描述了s=n−1情况下的极值族。

引用次数: 0

On the Kőnig–Egerváry index of a graph 在Kőnig-Egerváry图的索引上

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.039

Daniel A. Jaume , Vadim E. Levit , Eugen Mandrescu , Gonzalo Molina , Kevin Pereyra

A graph is said to be Kőnig–Egerváry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a Kőnig–Egerváry graph. Several properties of this difference, called the Kőnig–Egerváry index or Kőnig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving Kőnig–Egerváry graphs are, in fact, general statements about graphs that can be expressed in terms of their Kőnig–Egerváry indices.

如果一个图的匹配数等于它的顶点覆盖数，我们就说它是Kőnig-Egerváry。这两个图参数的差值，顶点覆盖数减去匹配数，在某种意义上，衡量了一个图离Kőnig-Egerváry图有多远。介绍了这种差异的几个性质，称为Kőnig-Egerváry指数或Kőnig缺陷，包括一些重要的结构特征。此外，还证明了涉及Kőnig-Egerváry图的各种语句实际上是关于图的一般语句，这些语句可以用它们的Kőnig-Egerváry索引来表示。

引用次数: 0

Planar graphs without cycles of length 4 or 5 are (7:2)-colorable 没有周期长度为4或5的平面图是（7:2）可着色的

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.033

Yingqian Wang

Given positive integers

a

and

b

with

a \geq b

, a graph

G

is said to be

(a : b)

-colorable, if we can assign

b

colors from color set

{1, 2, \dots, a}

to each vertex of

G

so that adjacent vertices receive no common colors. Let

S

denote the family of planar graphs without cycles of length 4 or 5. It is known that there are graphs in

S

which are not

(3 : 1)

-colorable (Cohen-Addad et al., 2017), and every graph in

S

is list

(11 : 3)

-colorable (Dvořák and Hu, 2019). This paper shows that every graph in

S

(7 : 2)

-colorable. This gives a new relaxation for refuted Steinberg’s conjecture in the setting of

(a : b)

-coloring.

给定a≥b的正整数a和b，图G是（a:b）可着色的，如果我们可以从颜色集{1,2，…，a}中为G的每个顶点分配b种颜色，使得相邻的顶点没有共同的颜色。设S表示无环长为4或5的平面图族。已知S中存在非（3:1）可着色的图（Cohen-Addad et al., 2017）， S中的每个图都是list（11:3）可着色的（Dvořák and Hu, 2019）。本文证明了S中的每一个图都是（7:2）可着色的。这给在(a:b)-着色集合中被驳倒的Steinberg猜想提供了一个新的松弛。

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引用次数: 0

Graph problems and monotone classes 图问题与单调类

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.029

Vadim Lozin

We study properties of graph classes that are closed under taking subclasses, such as boundedness of graph parameters or polynomial-time solvability of algorithmic problems. In the universe of minor-closed classes of graphs, any such property can be described by a set of minimal classes that do not possess the property, because the minor relation is a well-quasi-order. This, however, is not the case for the subgraph relation, implying that in the universe of monotone classes, which extends the family of minor-closed classes, the existence of minimal classes is not guaranteed. To overcome this difficulty, we employ the notion of boundary classes. Together with minimal classes they play a critical role for classes defined by finitely many forbidden subgraphs. In the present paper, we identify several levels in the hierarchy of monotone classes and describe respective critical classes. In particular, we show that a finitely-defined monotone class

X

has bounded chromatic number, degeneracy, functionality and admits an implicit representation if and only if

X

excludes a forest. We also show that

X

has bounded tree-, clique- and twin-width and admits polynomial-time solutions for a variety of algorithmic problems if and only if

X

excludes a tripod, i.e. a subcubic forest every connected component of which has at most one cubic vertex. The last result, however, does not apply to the Hamiltonian cycle problem. Towards identifying critical classes for this problem we determine complexity of the Hamiltonian cycle problem in some monotone classes.

我们研究了图类在取子类时闭合的性质，如图参数的有界性或算法问题的多项式时间可解性。在图的小闭类的宇宙中，任何这样的性质都可以用一组不具有该性质的最小类来描述，因为小关系是一个良拟序。然而，对于子图关系却不是这样，这意味着在单调类的宇宙中，它扩展了小闭类族，最小类的存在性是不能保证的。为了克服这个困难，我们采用了边界类的概念。它们与最小类一起，对于由有限多个禁止子图定义的类起着关键作用。在本文中，我们在单调类的层次中识别了几个层次，并描述了各自的临界类。特别地，我们证明了有限定义单调类X具有有界色数、简并性、泛函性，并且当且仅当X排除森林时允许隐式表示。我们还证明了X具有有界的树宽度、团宽度和双宽度，并且当且仅当X不包含三脚架，即每个连通成分最多有一个三次顶点的次三次森林时，各种算法问题都有多项式时间解。然而，最后的结果并不适用于哈密顿循环问题。为了确定这个问题的临界类，我们确定了一些单调类中哈密顿循环问题的复杂度。

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引用次数: 0

Semi-total domination in unit disk graphs and general graphs 单位盘图和一般图的半全支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.016

Sasmita Rout , Gautam Kumar Das

Let

G = (V, E)

be a simple undirected graph with no isolated vertex. A set

D \subseteq V

is a dominating set if each vertex

u \in V

is either in

D

or is adjacent to a vertex

v \in D

. A set

D_{t 2} \subseteq V

is called a semi-total dominating set if

(i)

D_{t 2}

is a dominating set, and

(i i)

for every vertex

u \in D_{t 2}

, there exists another vertex

v \in D_{t 2}

such that the distance between

u

and

v

G

is at most 2. Given a graph

G

, the semi-total domination problem finds a semi-total dominating set of minimum size. This problem is known to be NP-complete for general graphs and remains NP-complete for some special graph classes, such as planar, split, and chordal bipartite graphs. In this paper, we demonstrate that the problem is also NP-complete for unit disk graphs and propose a

6 -

factor approximation algorithm. The algorithm’s running time is

O (n log n)

, where

n

is the number of vertices in the given unit disk graph. In addition, we show that the minimum semi-total domination problem in a graph with maximum degree

Δ

admits a

2 + ln (Δ + 1) -

factor approximation algorithm, which is an improvement over the best-known result

2 + 3 ln (Δ + 1)

设G=（V，E）是一个没有孤立顶点的简单无向图。如果每个顶点u∈V在D中或与顶点V∈D相邻，则集合D是支配集。若(i) Dt2是一个控制集，且（ii）对于每一个顶点u∈Dt2，存在另一个顶点V∈Dt2，且在G中u与V的距离不大于2，则称集合Dt2为半全控制集。给定一个图G，半全支配问题求一个最小大小的半全支配集。对于一般图，这个问题是np完全的，对于一些特殊的图类，如平面图、分割图和弦二部图，这个问题仍然是np完全的。在本文中，我们证明了这个问题对于单位磁盘图也是np完全的，并提出了一个六因子逼近算法。该算法的运行时间为O(nlogn)，其中n为给定单位磁盘图中的顶点数。此外，我们还证明了最大度为Δ的图的最小半全控制问题允许使用2+ln（Δ+1）因子逼近算法，这是对最著名的结果2+3ln（Δ+1）的改进。

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引用次数: 0

An upper bound on path cover number of regular graphs and its application to Hamiltonian cycle in tough graphs 正则图的路径覆盖数上界及其在难图哈密顿循环中的应用

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.036

Xiaodan Chen, Xiaoning Yang

The path cover number of a graph

G

is the minimum integer

β

such that

G

contains

β

vertex-disjoint paths that cover all of its vertices. In this paper, we first establish an upper bound on the path cover number for regular graphs. Then we apply this bound to help to derive sufficient conditions for a

t

-tough graph to be Hamiltonian with integer

t \geq 1

, in terms of the edge number of the graph, which improve some known results in the literature. Another key tool we used to derive these sufficient conditions is the (complete) toughness closure lemma due to Hoàng and Robin (2024) and Shan and Tanyel (2025).

图G的路径覆盖数是最小整数β，使得G包含覆盖其所有顶点的不相交路径β。本文首先建立了正则图的路径覆盖数的上界。然后，我们利用这个界，从图的边数出发，得到了t-tough图是整数t≥1的哈密顿算子的充分条件，改进了文献中一些已知的结果。我们用来推导这些充分条件的另一个关键工具是（完全）韧性闭合引理，这是由Hoàng和Robin（2024）以及Shan和Tanyel（2025）得出的。

引用次数: 0

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