Theoretical Computer Science最新文献

{"title":"Computing subset vertex covers in H-free graphs","authors":"Nick Brettell ,&nbsp;Jelle J. Oostveen ,&nbsp;Sukanya Pandey ,&nbsp;Daniël Paulusma ,&nbsp;Johannes Rauch ,&nbsp;Erik Jan van Leeuwen","doi":"10.1016/j.tcs.2025.115088","DOIUrl":"10.1016/j.tcs.2025.115088","url":null,"abstract":"<div><div>We consider a natural generalization of <span>Vertex Cover</span>: the <span>Subset Vertex Cover</span> problem, which is to decide for a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>, a subset <span><math><mi>T</mi><mo>⊆</mo><mi>V</mi></math></span> and integer <em>k</em>, if <em>V</em> has a subset <em>S</em> of size at most <em>k</em>, such that <em>S</em> contains at least one end-vertex of every edge incident to a vertex of <em>T</em>. A graph is <em>H</em>-free if it does not contain <em>H</em> as an induced subgraph. We solve two open problems from the literature by proving that <span>Subset Vertex Cover</span> is <span>NP</span>-complete on subcubic (claw, diamond)-free planar graphs and on 2-unipolar graphs, a subclass of <span><math><mn>2</mn><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-free weakly chordal graphs. Our results show for the first time that <span>Subset Vertex Cover</span> is computationally harder than <span>Vertex Cover</span> (under <span><math><mi>P</mi><mo>≠</mo><mrow><mi>NP</mi></mrow></math></span>). We also prove new polynomial time results, some of which follow from a reduction to <span>Vertex Cover</span> restricted to classes of probe graphs. We first give a dichotomy on graphs where <span><math><mi>G</mi><mo>[</mo><mi>T</mi><mo>]</mo></math></span> is <em>H</em>-free. Namely, we show that <span>Subset Vertex Cover</span> is polynomial-time solvable on graphs <em>G</em>, for which <span><math><mi>G</mi><mo>[</mo><mi>T</mi><mo>]</mo></math></span> is <em>H</em>-free, if <span><math><mi>H</mi><mo>=</mo><mi>s</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>t</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span>NP</span>-complete otherwise. Moreover, we prove that <span>Subset Vertex Cover</span> is polynomial-time solvable for <span><math><mo>(</mo><mi>s</mi><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><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for <span>Subset Vertex Cover</span> on <em>H</em>-free graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1032 ","pages":"Article 115088"},"PeriodicalIF":0.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143302052","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computable classifications of continuous, transducer, and regular functions","authors":"Johanna N.Y. Franklin ,&nbsp;Rupert Hölzl ,&nbsp;Alexander Melnikov ,&nbsp;Keng Meng Ng ,&nbsp;Daniel Turetsky","doi":"10.1016/j.tcs.2025.115086","DOIUrl":"10.1016/j.tcs.2025.115086","url":null,"abstract":"<div><div>We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwise linear-time Lipschitz functions is <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-complete. (Every regular function is pointwise linear-time Lipschitz.) We show that a function <span><math><mi>f</mi><mo>:</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo><mo>→</mo><mi>R</mi></math></span> is (binary) transducer if and only if it is continuous regular. As one of many consequences, our <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-completeness result covers the class of transducer functions as well. Finally, we show that the Banach space <span><math><mi>C</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> of real-valued continuous functions admits an arithmetical classification among separable Banach spaces. Our proofs combine methods of abstract computability theory, automata theory, and functional analysis.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1032 ","pages":"Article 115086"},"PeriodicalIF":0.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143302606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Securing data in the cloud using pairing-free inner product functional encryption with unbounded vector size","authors":"Subhranil Dutta,&nbsp;Ratna Dutta,&nbsp;Sourav Mukhopadhyay","doi":"10.1016/j.tcs.2025.115085","DOIUrl":"10.1016/j.tcs.2025.115085","url":null,"abstract":"<div><div>Cloud computing has emerged as a paradigm shift from traditional computing, enabling data processing, storage, and sharing in potentially insecure environments with applications across sectors such as healthcare, online data storage, media platforms, big data analytics, and e-learning. An advanced cryptographic primitive called <em>inner product functional encryption</em> (<span>IPFE</span>) for the linear function class offers fine-grained access control for sensitive cloud storage, with significant applications in the Internet of Things (IoT), network privacy, mobile device security, cloud security, healthcare access control, and more. Most <span>IPFE</span> schemes compute inner products for fixed-length vectors, requiring predefined bounds on message and key vector sizes at the parameter generation stage. To address this, Tomida et al. (ASIACRYPT 2018) and Dufour-Sans et al. (ACNS 2019) introduced <em>unbounded <span>IPFE</span></em> (<span>UIPFE</span>), which allows for flexible vector lengths in key and ciphertext generation. This adaptability extends <span>IPFE</span>'s applicability to contexts where vector lengths may vary or be unknown in advance. However, existing <span>UIPFE</span> schemes rely on pairing operations, incurring substantial computational costs. This work addresses these limitations by introducing <span>UIPFE</span> schemes that eliminate the need for pairing operations. Our main contributions are as follows:<ul><li><span>–</span><span><div>We propose the <em>first</em> post-quantum secure <em>public-key</em> <span>UIPFE</span> (<span>pubUIPFE</span>) scheme in the random oracle model with adaptive security, leveraging the Learning With Errors (<span>LWE</span>) assumption to reduce computational overhead.</div></span></li><li><span>–</span><span><div>We construct a <em>private-key</em> <span>UIPFE</span> (<span>privUIPFE</span>) scheme in the standard model that offers selective security based on the Decisional Diffie-Hellman (<span>DDH</span>) assumption, featuring linear-size ciphertexts and constant-size keys.</div></span></li></ul></div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115085"},"PeriodicalIF":0.9,"publicationDate":"2025-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A categorical model for organic chemistry","authors":"Ella Gale ,&nbsp;Leo Lobski ,&nbsp;Fabio Zanasi","doi":"10.1016/j.tcs.2025.115084","DOIUrl":"10.1016/j.tcs.2025.115084","url":null,"abstract":"<div><div>We introduce a mathematical framework for organic chemistry, with three interrelated perspectives on chemical processes: reaction schemes, formal reactions and disconnection rules. We apply the framework to retrosynthetic analysis, an important research method in synthetic chemistry. Our approach represents molecules as labelled graphs, while the interactions between them are represented either as double pushout graph rewriting, partial bijections or local edge rewrite rules. In particular, we show that the formal reactions are generated by reaction schemes using double pushout rewriting, and that the disconnection rules are sound, complete and universal with respect to chemical reactions. The mathematical formulation of retrosynthesis is based on layered props – a recently introduced categorical model for partial explanations in scientific reasoning.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1032 ","pages":"Article 115084"},"PeriodicalIF":0.9,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143301910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Visibility extension via reflection","authors":"Arash Vaezi ,&nbsp;Bodhayan Roy ,&nbsp;Mohammad Ghodsi","doi":"10.1016/j.tcs.2025.115083","DOIUrl":"10.1016/j.tcs.2025.115083","url":null,"abstract":"<div><div>This paper studies a variant of the Art Gallery problem in which the “walls” can be replaced by <em>reflecting edges</em>, which allows the guards to see further and thereby see a larger portion of the gallery. Given a simple polygon <span><math><mtext>P</mtext></math></span>, first, we consider one guard as a point viewer, and we intend to use reflection to add a certain amount of area to the visibility polygon of the guard. We study visibility with specular and diffuse reflections where the specular type of reflection is the mirror-like reflection, and in the diffuse type of reflection, the angle between the incident and reflected ray may assume all possible values between 0 and <em>π</em>. Lee and Aggarwal already proved that several versions of the general Art Gallery problem are <span><math><mtext>NP</mtext></math></span>-hard. We show that several cases of adding an area to the visible area of a given point guard are <span><math><mtext>NP</mtext></math></span>-hard, too.</div><div>Second,<span><span>¹</span></span> we assume that all edges are reflectors, and we intend to decrease the minimum number of guards required to cover the whole gallery.</div><div>Chao Xu proved that even considering <em>r</em> specular reflections, one may need <span><math><mo>⌊</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌋</mo></math></span> guards to cover the polygon, let <em>r</em> be the maximum number of reflections of a guard's visibility ray.</div><div>In this work, we prove that considering <em>r diffuse</em> reflections, the minimum number of <em>vertex or boundary</em> guards required to cover a given simple polygon <span><math><mi>P</mi></math></span> decreases to <span><math><mo>⌈</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>+</mo><mo>⌊</mo><mfrac><mrow><mi>r</mi></mrow><mrow><mn>8</mn></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌉</mo></math></span>, where <em>α</em> indicates the minimum number of guards required to cover the polygon without reflection. We also generalize the <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span>-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115083"},"PeriodicalIF":0.9,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Powers of low rank sparse matrices","authors":"Keren Cohen","doi":"10.1016/j.tcs.2025.115082","DOIUrl":"10.1016/j.tcs.2025.115082","url":null,"abstract":"<div><div>Let <em>A</em> be a matrix of order <em>n</em> over an arbitrary field <em>F</em>, and let <em>k</em> be a positive integer. We present an algorithm that computes <span><math><msup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span> which is faster than the known methods whenever <em>A</em> is sufficiently sparse and its rank is sufficiently small. In fact, already when <span><math><mi>r</mi><mi>a</mi><mi>n</mi><mi>k</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> for a fixed <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span> our method yields improved running times.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1032 ","pages":"Article 115082"},"PeriodicalIF":0.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143302051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Recursive and iterative approaches to generate rotation Gray codes for stamp foldings and semi-meanders","authors":"Bowie Liu,&nbsp;Dennis Wong,&nbsp;Chan-Tong Lam,&nbsp;Marcus Im","doi":"10.1016/j.tcs.2024.115053","DOIUrl":"10.1016/j.tcs.2024.115053","url":null,"abstract":"<div><div>We first present a simple recursive algorithm that generates cyclic rotation Gray codes for stamp foldings and semi-meanders, where consecutive strings differ by a stamp rotation. These are the first known Gray codes for stamp foldings and semi-meanders, and we thus solve an open problem posted by Sawada and Li (2012) <span><span>[17]</span></span>. We then introduce an iterative algorithm that generates the same rotation Gray codes for stamp foldings and semi-meanders. Both the recursive and iterative algorithms generate stamp foldings and semi-meanders in constant amortized time and <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>-amortized time per string respectively, using a linear amount of memory.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115053"},"PeriodicalIF":0.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Conflict-Free Spanning Tree: Mapping tractable and hard instances through the lenses of graph classes","authors":"Bruno José S. Barros ,&nbsp;Luiz Satoru Ochi ,&nbsp;Rian Gabriel S. Pinheiro ,&nbsp;Uéverton S. Souza","doi":"10.1016/j.tcs.2025.115081","DOIUrl":"10.1016/j.tcs.2025.115081","url":null,"abstract":"&lt;div&gt;&lt;div&gt;A natural constraint in real-world applications is avoiding conflicting elements in problem solutions. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a graph where each edge &lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; has a positive integer weight &lt;span&gt;&lt;math&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be a conflict graph such that &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and each edge &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; represents a conflict between two edges &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In the &lt;span&gt;Minimum Conflict-Free Spanning Tree (MCFST)&lt;/span&gt; problem, we are asked to find a spanning tree avoiding pairs of conflicting edges (if such a tree exists) with minimum cost. In contrast to the polynomial-time solvability of &lt;span&gt;Minimum Spanning Tree&lt;/span&gt;, to determine whether an instance &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;MCFST&lt;/span&gt; admits a feasible solution is an &lt;span&gt;&lt;math&gt;&lt;mi&gt;NP&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-complete problem. In this paper, we present a multivariate complexity analysis of &lt;span&gt;MCFST&lt;/span&gt; by considering particular classes of graphs &lt;em&gt;G&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;. We show that the problem of determining whether an instance &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;MCFST&lt;/span&gt; has a feasible solution is &lt;span&gt;&lt;math&gt;&lt;mi&gt;NP&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-complete even if &lt;em&gt;G&lt;/em&gt; is a bipartite planar subcubic graph, and &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is a disjoint union of paths with three vertices. Contrastingly, we show that when &lt;em&gt;G&lt;/em&gt; is complete and &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is bipartite, then a solution for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˆ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; can be found in linear time, while the problem of finding an optimal solution is &lt;span&gt;&lt;math&gt;&lt;mi&gt;NP&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-h","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115081"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Conflict-free coloring on subclasses of perfect graphs and bipartite graphs","authors":"Sriram Bhyravarapu ,&nbsp;Subrahmanyam Kalyanasundaram ,&nbsp;Rogers Mathew","doi":"10.1016/j.tcs.2025.115080","DOIUrl":"10.1016/j.tcs.2025.115080","url":null,"abstract":"<div><div>A <em>Conflict-Free Open Neighborhood coloring</em>, abbreviated CFON^⁎ coloring, of a graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> using <em>k</em> colors is an assignment of colors from a set of <em>k</em> colors to a subset of vertices of <em>V</em> such that every vertex sees some color exactly once in its open neighborhood. The minimum <em>k</em> for which <em>G</em> has a CFON^⁎ coloring using <em>k</em> colors is called the <em>CFON</em>^⁎ <em>chromatic number</em> of <em>G</em>, denoted by <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>O</mi><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The analogous notion for closed neighborhood is called CFCN^⁎ coloring and the analogous parameter is denoted by <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The problem of deciding whether a given graph admits a CFON^⁎ (or CFCN^⁎) coloring that uses <em>k</em> colors is <span>NP</span>-complete. Below, we describe briefly the main results of this paper.<ul><li><span>•</span><span><div>We show that it is <span>NP</span>-hard to determine the CFCN^⁎ chromatic number of chordal graphs. We also show the existence of a family of chordal graphs <em>G</em> that requires <span><math><mi>Ω</mi><mo>(</mo><msqrt><mrow><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msqrt><mo>)</mo></math></span> colors to CFCN^⁎ color <em>G</em>, where <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> represents the size of a maximum clique in <em>G</em>.</div></span></li><li><span>•</span><span><div>We give a polynomial time algorithm to compute <span><math><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>O</mi><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for interval graphs <em>G</em>. This answers in positive the open question posed by Reddy [Theoretical Comp. Science, 2018] to determine whether CFON^⁎ chromatic number can be computed in polynomial time for interval graphs.</div></span></li><li><span>•</span><span><div>We explore biconvex graphs, a subclass of bipartite graphs, and give a polynomial time algorithm to compute their CFON^⁎ chromatic number.</div></span></li></ul></div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115080"},"PeriodicalIF":0.9,"publicationDate":"2025-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Monitoring arc-geodetic sets of oriented graphs","authors":"Tapas Das ,&nbsp;Florent Foucaud ,&nbsp;Clara Marcille ,&nbsp;P.D. Pavan ,&nbsp;Sagnik Sen","doi":"10.1016/j.tcs.2025.115079","DOIUrl":"10.1016/j.tcs.2025.115079","url":null,"abstract":"<div><div>Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna and Ramasubramony Sulochana with relation to several prior notions in the area of network monitoring like distance edge-monitoring.</div><div>In this work, we explore the extension of those notions unto oriented graphs, modelling oriented networks, and call these objects monitoring arc-geodetic sets. We also define the lower and upper monitoring arc-geodetic number of an undirected graph as the minimum and maximum of the monitoring arc-geodetic number of all orientations of the graph. We determine the monitoring arc-geodetic number of fundamental graph classes such as bipartite graphs, trees, cycles, etc. Then, we characterize the graphs for which every monitoring arc-geodetic set is the entire set of vertices, and also characterize the solutions for tournaments. We also cover some complexity aspects by studying two algorithmic problems. We show that the problem of determining if an undirected graph has an orientation with the minimal monitoring arc-geodetic set being the entire set of vertices, is NP-hard. We also show that the problem of finding a monitoring arc-geodetic set of size at most <em>k</em> is NP-complete when restricted to oriented graphs with maximum degree 4.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1031 ","pages":"Article 115079"},"PeriodicalIF":0.9,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143265848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}