{"title":"On disjunction convex hulls by big-M lifting","authors":"Yushan Qu, Jon Lee","doi":"10.1016/j.dam.2025.03.013","DOIUrl":"10.1016/j.dam.2025.03.013","url":null,"abstract":"<div><div>We study the natural extended-variable formulation for the disjunction of <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> polytopes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We demonstrate that the convex hull <span><math><mi>D</mi></math></span> in the natural extended-variable space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span> is given by full optimal big-M lifting (i) when <span><math><mrow><mi>d</mi><mo>≤</mo><mn>2</mn></mrow></math></span> (and that it is not generally true for <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span>), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary <span><math><mrow><mi>d</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. We give a broad family of examples with <span><math><mrow><mi>d</mi><mo>≥</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow></math></span>, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of <span><math><mi>D</mi></math></span>, and we demonstrate that all facets of <span><math><mi>D</mi></math></span> can be enumerated in polynomial time when <span><math><mi>d</mi></math></span> is fixed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 31-45"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683253","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}
{"title":"Grundy packing coloring of graphs","authors":"Didem Gözüpek , Iztok Peterin","doi":"10.1016/j.dam.2025.03.024","DOIUrl":"10.1016/j.dam.2025.03.024","url":null,"abstract":"<div><div>A map <span><math><mrow><mi>c</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><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> is a packing <span><math><mi>k</mi></math></span>-coloring if every two different vertices of the same color <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> are at distance more than <span><math><mi>i</mi></math></span>. The packing chromatic number <span><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></span> of <span><math><mi>G</mi></math></span> is the smallest integer <span><math><mi>k</mi></math></span> such that there exists a packing <span><math><mi>k</mi></math></span>-coloring. In this paper we introduce the notion of <em>Grundy packing chromatic number</em>, analogous to the Grundy chromatic number of a graph. We first present a polynomial-time algorithm that is based on a greedy approach and gives a packing coloring of any graph <span><math><mi>G</mi></math></span>. We then define the Grundy packing chromatic number <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span> as the maximum value that this algorithm yields in <span><math><mi>G</mi></math></span>. We present several properties of <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, provide results on the complexity of the problem as well as bounds and some exact results for <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 17-30"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143683268","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}
{"title":"Pareto-scheduling of two competing agents with total weighted tardiness being one criterion","authors":"Jinwen Sun , Rubing Chen , Qiulan Zhao","doi":"10.1016/j.dam.2025.03.026","DOIUrl":"10.1016/j.dam.2025.03.026","url":null,"abstract":"<div><div>We study the Pareto-scheduling of two competing agents on a single machine, in which the jobs of at least one agent have their own equal processing times. When the criterion of one agent is the total weighted tardiness and the criterion of the other agent is the total completion time, the total tardiness or the total weighted completion time, the exact complexities of these problems remain open as posed by Chen et al. (2022). In this paper, we design a unified algorithm for solving these problems. As consequences, we show that these problems are solvable either in polynomial time or in pseudo-polynomial time. Combining the known results in the literature, we determine the complexity classification of nine problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 137-148"},"PeriodicalIF":1.0,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681869","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}
{"title":"The planar Turán number of double stars S2,l","authors":"Xin Xu , Jiawei Shao , Qiang Zhou","doi":"10.1016/j.dam.2025.03.016","DOIUrl":"10.1016/j.dam.2025.03.016","url":null,"abstract":"<div><div>The planar Turán number <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>H</mi></math></span> is the maximum number of edges in an <span><math><mi>H</mi></math></span>-free planar graph on <span><math><mi>n</mi></math></span> vertices. The double star <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>l</mi></mrow></msub></math></span> is obtained from joining the centers of two stars each having <span><math><mi>k</mi></math></span> leaves and <span><math><mi>l</mi></math></span> leaves, respectively. In this paper, we give the exact value of <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, which determines the planar Turán number for all double stars <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>l</mi></mrow></msub></math></span> when <span><math><mrow><mi>l</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 131-136"},"PeriodicalIF":1.0,"publicationDate":"2025-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681914","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}
Bo Zhu , Shumin Zhang , Jou-Ming Chang , Jinyu Zou
{"title":"Fault-tolerability analysis of hypercubes based on 3-component path-structure connectivity","authors":"Bo Zhu , Shumin Zhang , Jou-Ming Chang , Jinyu Zou","doi":"10.1016/j.dam.2025.03.021","DOIUrl":"10.1016/j.dam.2025.03.021","url":null,"abstract":"<div><div>Interconnection networks are essential in parallel computing and network science nowadays. Network failures are inevitable during operation and result in inestimable losses. Hence, designing an interconnection network with excellent performance is necessary. Reliability is a key indicator of the performance of interconnection networks, and its research originated from the first telecommunication switching network system. The failure of elements in a network system reduces overall communication capacity, leading to network congestion and system failure, typically measured by connectivity. In this paper, we introduce a new type of conditional connectivity of a graph <span><math><mi>G</mi></math></span>, termed <span><math><mi>r</mi></math></span>-component <span><math><mi>H</mi></math></span>-structure connectivity and denoted as <span><math><mrow><mi>c</mi><msub><mrow><mi>κ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>;</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>. Then, we investigate 3-component <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>-structure connectivity for hypercube networks <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and acquire the result <span><span><span><math><mrow><mi>c</mi><msub><mrow><mi>κ</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mtable><mtr><mtd><mn>2</mn><mi>n</mi><mo>−</mo><mn>4</mn><mspace></mspace></mtd><mtd><mtext>if </mtext><mi>k</mi><mo>=</mo><mn>2</mn><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mo>⌈</mo><mrow><mfrac><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow><mrow><mi>k</mi></mrow></mfrac></mrow><mo>⌉</mo></mrow><mspace></mspace></mtd><mtd><mtext>for </mtext><mi>k</mi><mo>≥</mo><mn>4</mn><mtext> even</mtext><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mo>⌈</mo><mrow><mfrac><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>5</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>⌉</mo></mrow><mspace></mspace></mtd><mtd><mtext>for </mtext><mi>k</mi><mo>≥</mo><mn>3</mn><mtext> odd</mtext><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></mrow></math></span></span></span></div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 111-123"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143680942","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}
Kedson Alves Silva , Tanilson Dias dos Santos , Uéverton dos Santos Souza
{"title":"Clique Cover on L-EPG representations of graphs","authors":"Kedson Alves Silva , Tanilson Dias dos Santos , Uéverton dos Santos Souza","doi":"10.1016/j.dam.2025.03.027","DOIUrl":"10.1016/j.dam.2025.03.027","url":null,"abstract":"<div><div><span>Clique Cover</span> is a classical graph theory problem where we are given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, an integer <span><math><mi>k</mi></math></span>, and asked if <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> can be partitioned into at most <span><math><mi>k</mi></math></span> cliques. Edge intersection graphs of paths in grids (EPG graphs) are graphs whose vertices can be represented as nontrivial paths in a grid such that two vertices are adjacent if and only if the corresponding paths share at least one edge of the grid. When the paths have at most one change of direction (bend), these graphs are called <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-EPG graphs; in addition, they are called <span><math><mi>⌞</mi></math></span>-EPG graphs (or <span><math><mi>⌞</mi></math></span>-shaped graphs) if all the paths are represented with one of the following shapes: “<span><math><mi>⌞</mi></math></span>”, “–”, or “<span><math><mo>∣</mo></math></span>”. Such a representation is called <span><math><mi>⌞</mi></math></span>-EPG representation. The class of <span><math><mi>⌞</mi></math></span>-EPG graphs is a natural superclass of interval graphs. Since all maximal cliques of a <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-EPG graph <span><math><mi>G</mi></math></span> can be computed in polynomial time, and <span>Clique Cover</span> on interval graphs is polynomial-time solvable; in this paper, we study the complexity of <span>Clique Cover</span> on <span><math><mi>⌞</mi></math></span>-EPG graphs. We show that given an <span><math><mi>⌞</mi></math></span>-EPG representation of a graph <span><math><mi>G</mi></math></span>, it is NP-complete to determine whether <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> can be partitioned into at most <span><math><mi>k</mi></math></span> cliques, but it can be solved in FPT time when parameterized by <span><math><mi>k</mi></math></span>. Furthermore, we adapt our algorithm to the case where the input is a graph without its <span><math><mi>⌞</mi></math></span>-EPG representation. Finally, we also show that <span>Clique Cover</span> on <span><math><mi>∂</mi></math></span>EPG<span><math><msub><mrow></mrow><mrow><mn>2</mn></mrow></msub></math></span> representations (a particular class of <span><math><mi>⌞</mi></math></span>-EPG representations) is polynomial-time solvable.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 145-156"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681990","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}
{"title":"PSPACE-completeness of k-Atropos","authors":"Chao Yang , Zhujun Zhang","doi":"10.1016/j.dam.2025.03.010","DOIUrl":"10.1016/j.dam.2025.03.010","url":null,"abstract":"<div><div>Burke and Teng introduced a two-player combinatorial game Atropos based on Sperner’s lemma, and showed that deciding whether one has a winning strategy for Atropos is <span>PSPACE</span>-complete. In the original Atropos game, the players must color a node adjacent to the last colored node. Burke and Teng also mentioned a variant <span><math><mi>k</mi></math></span>-Atropos in which each move is at most of distance <span><math><mi>k</mi></math></span> of the previous move, and asked a question on determining the computational complexity of this variant. In this paper, we answer this question by showing that for any fixed integer <span><math><mrow><mi>k</mi><mspace></mspace><mrow><mo>(</mo><mi>k</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>, the associated decision problem <span><math><mi>k</mi></math></span>-<span>Atropos</span> is <span>PSPACE</span>-complete by reduction from True Quantified Boolean Formula (<span>TQBF</span>).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 190-198"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686565","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}
{"title":"Spectra of the Mycielskian of a signed graph and related products","authors":"Mir Riyaz Ul Rashid , S. Pirzada , Zoran Stanić","doi":"10.1016/j.dam.2025.03.017","DOIUrl":"10.1016/j.dam.2025.03.017","url":null,"abstract":"<div><div>In a search for graphs with arbitrarily large chromatic number, Mycielski has offered a construction that transforms a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> into a new graph <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which is called the Mycielskian of <span><math><mi>G</mi></math></span>. Recently, the same concept is transferred to the framework of signed graphs. Accordingly, if <span><math><mrow><mi>V</mi><mo>=</mo><mrow><mo>{</mo><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><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><mi>E</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow></mrow></math></span>, then the vertex set of <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the disjoint union of <span><math><mi>V</mi></math></span>, <span><math><mrow><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mrow><mo>{</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>}</mo></mrow></mrow></math></span> and an isolated vertex <span><math><mi>w</mi></math></span>, and its edge set is <span><math><mrow><mi>E</mi><mo>∪</mo><mrow><mo>{</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>:</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mo>}</mo></mrow><mo>∪</mo><mrow><mo>{</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mi>w</mi><mo>:</mo><msubsup><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>∈</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>}</mo></mrow></mrow></math></span>. The Mycielskian of a signed graph <span><math><mrow><mi>Σ</mi><mo>=</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></mrow></mrow></math></span> is the signed graph <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>Σ</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, w","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 124-144"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681012","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}
{"title":"Some spectral conditions for star-factors in bipartite graphs","authors":"Sizhong Zhou","doi":"10.1016/j.dam.2025.03.014","DOIUrl":"10.1016/j.dam.2025.03.014","url":null,"abstract":"<div><div>A spanning subgraph <span><math><mi>F</mi></math></span> of <span><math><mi>G</mi></math></span> is called an <span><math><mi>F</mi></math></span>-factor if every component of <span><math><mi>F</mi></math></span> is isomorphic to some member of <span><math><mi>F</mi></math></span>, where <span><math><mi>F</mi></math></span> is a set of connected graphs. We denote by <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> the adjacency matrix of <span><math><mi>G</mi></math></span>, and by <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> the distance matrix of <span><math><mi>G</mi></math></span>. The largest eigenvalue of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, denoted by <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is called the adjacency spectral radius of <span><math><mi>G</mi></math></span>. The largest eigenvalue of <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, denoted by <span><math><mrow><mi>μ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is called the distance spectral radius of <span><math><mi>G</mi></math></span>. In this paper, we aim to provide two spectral conditions to ensure the existence of star-factors with given properties. Let <span><math><mi>G</mi></math></span> be a <span><math><mi>k</mi></math></span>-edge-connected bipartite graph with bipartition <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> and <span><math><mrow><mrow><mo>|</mo><mi>B</mi><mo>|</mo></mrow><mo>=</mo><mi>k</mi><mrow><mo>|</mo><mi>A</mi><mo>|</mo></mrow><mo>=</mo><mi>k</mi><mi>n</mi></mrow></math></span>, where <span><math><mi>n</mi></math></span> is a sufficiently large positive integer. Then the following two results are true.</div><div>(i) If <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>ρ</mi><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>∇</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> contains a star-factor <span><math><mi>F</mi></math></span> with <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span> for any <span><math><mrow><mi>u</mi><mo>∈</mo><mi>A</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span> for any <span><math><mrow><mi>v</mi><mo>∈</mo><mi>B</mi></mrow></math></span>, unless <span><math><mrow><mi>G</","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 124-130"},"PeriodicalIF":1.0,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143681870","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}
Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński
{"title":"Computational complexity of the recoverable robust shortest path problem with discrete recourse","authors":"Marcel Jackiewicz, Adam Kasperski, Paweł Zieliński","doi":"10.1016/j.dam.2025.03.004","DOIUrl":"10.1016/j.dam.2025.03.004","url":null,"abstract":"<div><div>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 <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>-hard for the arc exclusion and arc symmetric difference neighborhoods. Furthermore, it is also proven that the inner adversarial problem for these neighborhoods is <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>p</mi></mrow></msubsup></math></span>-hard.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 103-110"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644992","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}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
