Discrete Applied Mathematics最新文献

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Computational complexity of the recoverable robust shortest path problem with discrete recourse

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-19 DOI: 10.1016/j.dam.2025.03.004

Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński

In this paper, the recoverable robust shortest path problem is investigated. Discrete budgeted interval uncertainty representation is used to model uncertain second-stage arc costs. The known complexity results for this problem are strengthened. Namely, it is shown that the recoverable robust shortest path problem is

Σ_{3}^{p}

-hard for the arc exclusion and arc symmetric difference neighborhoods. Furthermore, it is also proven that the inner adversarial problem for these neighborhoods is

Π_{2}^{p}

-hard.

引用次数: 0

K4-free planar minimal bricks with the maximum number of edges

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-18 DOI: 10.1016/j.dam.2025.02.044

Jinqiu Zhou, Xing Feng, Weigen Yan

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick

G

is minimal if

G - e

is not a brick, for every edge

e

G

. Lovász (1983) showed that every brick is

K_{4}

-based or

{\bar{C}}_{6}

-based. A brick is

K_{4}

-free (respectively,

{\bar{C}}_{6}

-free) if it is not

K_{4}

-based (respectively,

{\bar{C}}_{6}

-based). Kothari and Murty (2016) proved that a planar brick is

K_{4}

-free if and only if it has precisely two odd faces and determined the list of all

{\bar{C}}_{6}

-free planar bricks. In this paper, we show that the

K_{4}

-free planar (minimal) bricks

G

have at most

2 | V (G) | - 3

edges. Furthermore, we characterize all the extremal graphs that meet this upper bound.

引用次数: 0

Generating subgraphs in chordal graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-18 DOI: 10.1016/j.dam.2025.02.042

Vadim E. Levit , David Tankus

<div><div>A graph <math><mi>G</mi></math> is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function <math><mi>w</mi></math> is defined on the vertex set of <math><mi>G</mi></math>. Then <math><mi>G</mi></math> is <math><mi>w</mi></math>-well-covered if all maximal independent sets are of the same weight. For every graph <math><mi>G</mi></math>, the set of weight functions <math><mi>w</mi></math> such that <math><mi>G</mi></math> is <math><mi>w</mi></math>-well-covered is a vector space, denoted <math><mrow><mi>W</mi><mi>C</mi><mi>W</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>.</div><div>Let <math><mi>B</mi></math> be a complete bipartite induced subgraph of <math><mi>G</mi></math> on vertex sets of bipartition <math><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub></math> and <math><msub><mrow><mi>B</mi></mrow><mrow><mi>Y</mi></mrow></msub></math>. Then <math><mi>B</mi></math> is generating if there exists an independent set <math><mi>S</mi></math> such that <math><mrow><mi>S</mi><mo>∪</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>X</mi></mrow></msub></mrow></math> and <math><mrow><mi>S</mi><mo>∪</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>Y</mi></mrow></msub></mrow></math> are both maximal independent sets of <math><mi>G</mi></math>. In the restricted case that a generating subgraph <math><mi>B</mi></math> is isomorphic to <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math>, the unique edge in <math><mi>B</mi></math> is called a relating edge. Generating subgraphs play an important role in finding <math><mrow><mi>W</mi><mi>C</mi><mi>W</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>.</div><div>Deciding whether an input graph <math><mi>G</mi></math> is well-covered is co-NP-complete. Hence, finding <math><mrow><mi>W</mi><mi>C</mi><mi>W</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well.</div><div>A graph is chordal if every induced cycle is a triangle. It is known that finding <math><mrow><mi>W</mi><mi>C</mi><mi>W</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> can be done polynomially in the restricted case that <math><mi>G</mi></math> is chordal. Thus, recognizing well-covered chordal graphs is a polynomial problem. We present a polynomial algorithm for recognizing relating edges and gen

引用次数: 0

Variants of the Erdős distinct sums problem and variance method

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-18 DOI: 10.1016/j.dam.2025.03.003

Simone Costa , Stefano Della Fiore , Andrea Ferraguti

<div><div>Let <math><mrow><mi>Σ</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></math> be a set of positive integers with <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mo>…</mo><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> such that all <math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math> subset sums are pairwise distinct. A famous conjecture of Erdős states that <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>></mo><mi>C</mi><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> for some constant <math><mi>C</mi></math>, while the best result known to date is of the form <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>></mo><mi>C</mi><mi>⋅</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></mrow></math>. In this paper, we propose a generalization of the Erdős distinct sum problem that is in the same spirit as those of the Davenport and the Erdős–Ginzburg–Ziv constants recently introduced in Caro et al. (2022) and in Caro and Schmitt (2022). More precisely, we require that the non-zero evaluations of the <math><mi>m</mi></math>th degree symmetric polynomial are all distinct over the subsequences of <math><mi>Σ</mi></math> whose size is at most <math><mrow><mi>λ</mi><mi>n</mi></mrow></math>, for a given <math><mrow><mi>λ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math>, considering <math><mi>Σ</mi></math> as a sequence in <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>k</mi></mrow></msup></math> with each coordinate of each <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math> in <math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>M</mi><mo>]</mo></mrow></math>. If <math><msub><mrow><mi>F</mi></mrow><mrow><mi>λ</mi><mo>,</mo><mi>n</mi></mrow></msub></math> denotes the family of subsets of <math><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></mrow></math> whose size is at most <math><mrow><mi>λ</mi><mi>n</mi></mrow></math>, our main result is that, for each <math><mrow><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo></mrow></math> and <math><mi>λ</mi></math>, there exists an explicit constant <math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow></msub></math> such that <math><mrow><mi>M</mi><mo>≥</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>λ</mi></mrow>

引用次数: 0

The time complexity of oriented chromatic number for acyclic oriented connected subcubic subgraphs of grids

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-18 DOI: 10.1016/j.dam.2025.03.001

E.M.M. Coelho , H. Coelho , L. Faria , M.P. Ferreira , S. Klein

An oriented

k

-coloring of an oriented graph

\vec{G} = (V, A)

is a partition of

V

into

k

color classes, such that there is no pair of adjacent vertices belonging to the same class and all the arcs between a pair of classes have the same orientation. The smallest

k

such that

\vec{G}

admits an oriented

k

-coloring is the oriented chromatic number

χ_{o} (\vec{G})

\vec{G}

. The

k

-oriented chromatic number decision problem asks whether an oriented graph

\vec{G}

has oriented chromatic number at most

k

k

-oriented chromatic number is a polynomial problem, when

k \leq 3

and

N P

-complete, when

k \geq 4

even if input

\vec{G}

is acyclic and the underlying graph

G

is bipartite, cubic and planar. In 2003, Fertin, Raspaud and Roychowdhury established exact values and bounds on the oriented chromatic number for several grid subgraph classes. But, the time complexity of

k

-oriented chromatic number was unknown for grid subgraphs. In this work we prove that the

k

-oriented chromatic number problem is NP-complete even if

\vec{G}

is acyclic oriented,

k \geq 4

, and the underlying graph

G

is a connected and subcubic subgraph of a grid.

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引用次数: 0

The interval coloring impropriety of planar graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-15 DOI: 10.1016/j.dam.2025.03.005

Seunghun Lee

For a graph

G

, we call an edge coloring of

G

an improper interval edge coloring if for every

v \in V (G)

the colors, which are integers, of the edges incident with

v

form an integral interval. The interval coloring impropriety of

G

, denoted by

μint (G)

, is the smallest value

k

such that

G

has an improper interval edge coloring where at most

k

edges of

G

with a common endpoint have the same color.

The purpose of this note is to communicate solutions to two previous questions on interval coloring impropriety, mainly regarding planar graphs. First, we prove

μint (G) \leq 2

for every outerplanar graph

G

. This confirms a conjecture by Casselgren and Petrosyan in the affirmative. Secondly, we prove that for each

k \geq 2

, the interval coloring impropriety of

k

-trees is unbounded. This refutes a conjecture by Carr, Cho, Crawford, Iršič, Pai and Robinson.

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引用次数: 0

The chromatic number of odd-hole-free graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-14 DOI: 10.1016/j.dam.2025.02.043

Rong Chen , Kaiyang Lan , Xinheng Lin , Yidong Zhou

In this paper, we prove that for each connected odd-hole-free graph

G

χ (G) \leq max {Δ (G) - 1, ω (G)}

if and only if

G

is not isomorphic to the complement of

C_{7}

引用次数: 0

Alternating L-functions of finite digraphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-14 DOI: 10.1016/j.dam.2025.03.009

Yongjiang Wu, Lihua Feng, Weijun Liu

In this paper, we provide a new proof for the Ihara expression of the alternating

L

-function of a digraph. Furthermore, we present a relation between the weighted alternating

L

-function of a digraph and that of the quotient of its regular covering. As an application, we give a decomposition formula for the weighted alternating zeta function of the quotient of a regular covering of a digraph by its weighted alternating

L

-functions.

引用次数: 0

Coherent domains and improved lower bounds for the maximum size of Condorcet domains

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-14 DOI: 10.1016/j.dam.2025.03.007

Alexander Karpov , Klas Markström , Søren Riis , Bei Zhou

In this paper, we study Condorcet domains, sets of linear orders from which majority ranking produces a linear order. We introduce a new class of Condorcet domains, called coherent domains, which is natural from both a voting theoretic and combinatorial perspective. After studying the properties of these domains we introduce set-alternating schemes. This is a method for constructing well-behaved coherent domains. Using this we show that, for sufficiently large numbers of alternatives

n

, there are coherent domains of size more than

2.197 3^{n}

. This improves the best existing asymptotic lower bounds for the size of the largest general Condorcet domains.

引用次数: 0

Injective edge chromatic index of sparse graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-03-14 DOI: 10.1016/j.dam.2025.03.006

Xiaolan Hu, Guixian Zhang

An injective

k

-edge coloring of a graph

G

is a

k

-edge coloring

ϕ

G

such that

ϕ (e_{1}) \neq ϕ (e_{3})

for any three consecutive edges

e_{1}, e_{2}

and

e_{3}

of a path or a triangle. The injective edge chromatic index

χ_{i}^{'} (G)

G

is the smallest value of

k

such that

G

has an injective

k

-edge coloring. Let

m a d (G)

and

Δ (G)

denote the maximum average degree and the maximum degree of a graph

G

, respectively. In this paper we show that if

Δ (G) \geq 3

and

m a d (G) < \frac{7}{3}

, then

χ_{i}^{'} (G) \leq Δ (G) + 1

; and if

Δ (G) \geq 3

and

m a d (G) < \frac{5}{2}

, then

χ_{i}^{'} (G) \leq Δ (G) + 2

引用次数: 0

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