Pub Date : 2026-01-21DOI: 10.1016/j.tcs.2026.115768
Wing-Kai Hon, Yu-Jia Huang, Wang-Yang Li
Let be a graph with a vertex set V, an edge set E, and a distinguished subset of vertices B⊆V, called beer stores. A beer path from u to v is a path that starts at u, ends at v, and visits at least one beer store. The notion of a beer path was recently introduced and studied, with a focus on finding the shortest beer path in a graph. In this work, we explore the natural extension of this notion into what we call a beer flow.
We show that the maximum beer flow problem can be formulated by linear programming so that it can be solved in polynomial time. However, when the flow on each edge is restricted to be integral, the problem suddenly becomes -hard, even in the very simple settings. This result is in stark contrast with the traditional maximum flow problem, where integrality of flow will not complicate the problem.
{"title":"The maximum beer flow problem","authors":"Wing-Kai Hon, Yu-Jia Huang, Wang-Yang Li","doi":"10.1016/j.tcs.2026.115768","DOIUrl":"10.1016/j.tcs.2026.115768","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> be a graph with a vertex set <em>V</em>, an edge set <em>E</em>, and a distinguished subset of vertices <em>B</em>⊆<em>V</em>, called <strong>beer stores</strong>. A <strong>beer path</strong> from <em>u</em> to <em>v</em> is a path that starts at <em>u</em>, ends at <em>v</em>, and visits at least one beer store. The notion of a beer path was recently introduced and studied, with a focus on finding the shortest beer path in a graph. In this work, we explore the natural extension of this notion into what we call a <strong>beer flow</strong>.</div><div>We show that the <strong>maximum beer flow</strong> problem can be formulated by linear programming so that it can be solved in polynomial time. However, when the flow on each edge is restricted to be <em>integral</em>, the problem suddenly becomes <span><math><mi>NP</mi></math></span>-hard, even in the very simple settings. This result is in stark contrast with the traditional maximum flow problem, where integrality of flow will not complicate the problem.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115768"},"PeriodicalIF":1.0,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039655","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}
Pub Date : 2026-01-21DOI: 10.1016/j.tcs.2026.115772
Jia-Jie Liu
This paper investigates the structural symmetry of the spined cube SQn, a notable hypercube variant recognized for its low diameter which speeds up communication. We quantify this symmetry using the concept of orbits, where two vertices are in the same orbit if they can be mapped to one another by a graph automorphism. The orbit number Orb(G), which represents the number of distinct orbits, is a measure of a graph’s structural equivalence; a graph is vertex-transitive if its orbit number is 1. While previous research has shown that SQn is vertex-transitive for n ≤ 3, the orbit number for higher dimensions has remained an open question. In this work, we definitively answer this question, proving that and that for all n ≥ 6, the orbit number is 4. Our findings collectively show that the Spined Cube achieves a compelling balance between a low diameter and a low orbit number, which ensures structural simplicity, making it a strong candidate for future large-scale interconnection networks.
{"title":"The orbits of spined cubes","authors":"Jia-Jie Liu","doi":"10.1016/j.tcs.2026.115772","DOIUrl":"10.1016/j.tcs.2026.115772","url":null,"abstract":"<div><div>This paper investigates the structural symmetry of the spined cube <em>SQ<sub>n</sub></em>, a notable hypercube variant recognized for its low diameter which speeds up communication. We quantify this symmetry using the concept of orbits, where two vertices are in the same orbit if they can be mapped to one another by a graph automorphism. The orbit number <em>Orb</em>(<em>G</em>), which represents the number of distinct orbits, is a measure of a graph’s structural equivalence; a graph is vertex-transitive if its orbit number is 1. While previous research has shown that <em>SQ<sub>n</sub></em> is vertex-transitive for <em>n</em> ≤ 3, the orbit number for higher dimensions has remained an open question. In this work, we definitively answer this question, proving that <span><math><mrow><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>S</mi><msub><mi>Q</mi><mn>4</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mi>r</mi><mi>b</mi><mrow><mo>(</mo><mi>S</mi><msub><mi>Q</mi><mn>5</mn></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span> and that for all <em>n</em> ≥ 6, the orbit number is 4. Our findings collectively show that the Spined Cube achieves a compelling balance between a low diameter and a low orbit number, which ensures structural simplicity, making it a strong candidate for future large-scale interconnection networks.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115772"},"PeriodicalIF":1.0,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039651","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}
Pub Date : 2026-01-21DOI: 10.1016/j.tcs.2026.115765
Changan Liu , Haoxin Sun , Ahad N. Zehmakan , Zhongzhi Zhang
We study the notion of unfairness in social networks, where a group such as females in a male-dominated industry are disadvantaged in access to important information, e.g. job posts, due to their less favorable positions in the network. We investigate a well-established network-based formulation of fairness called PageRank fairness, which refers to a fair allocation of the PageRank weights among distinct groups. Our goal is to enhance the PageRank fairness by modifying the underlying network structure. More precisely, we study the problem of maximizing PageRank fairness with respect to a disadvantaged group, when we are permitted to rewire a fixed number of edges in the network. Building on a greedy approach, we leverage techniques from fast sampling of rooted spanning forests to devise an effective linear-time algorithm for this problem. To evaluate the accuracy and performance of our proposed algorithm, we conduct a large set of experiments on various real-world network data. Our experiments demonstrate that the proposed algorithm significantly outperforms the existing ones. Our algorithm is capable of generating accurate solutions for networks of million nodes in just a few minutes.
{"title":"Efficient edge rewiring strategies for enhancing PageRank fairness","authors":"Changan Liu , Haoxin Sun , Ahad N. Zehmakan , Zhongzhi Zhang","doi":"10.1016/j.tcs.2026.115765","DOIUrl":"10.1016/j.tcs.2026.115765","url":null,"abstract":"<div><div>We study the notion of unfairness in social networks, where a group such as females in a male-dominated industry are disadvantaged in access to important information, e.g. job posts, due to their less favorable positions in the network. We investigate a well-established network-based formulation of fairness called PageRank fairness, which refers to a fair allocation of the PageRank weights among distinct groups. Our goal is to enhance the PageRank fairness by modifying the underlying network structure. More precisely, we study the problem of maximizing PageRank fairness with respect to a disadvantaged group, when we are permitted to rewire a fixed number of edges in the network. Building on a greedy approach, we leverage techniques from fast sampling of rooted spanning forests to devise an effective linear-time algorithm for this problem. To evaluate the accuracy and performance of our proposed algorithm, we conduct a large set of experiments on various real-world network data. Our experiments demonstrate that the proposed algorithm significantly outperforms the existing ones. Our algorithm is capable of generating accurate solutions for networks of million nodes in just a few minutes.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115765"},"PeriodicalIF":1.0,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039656","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}
Pub Date : 2026-01-20DOI: 10.1016/j.tcs.2026.115773
Julien Bensmail , Romain Bourneuf , Paul Colinot , Samuel Humeau , Timothée Martinod
The 1-2-3 Conjecture, introduced by Karoński, Łuczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph G different from K2, we can turn G into a locally irregular multigraph M(G), i.e., in which no two adjacent vertices have the same degree, by replacing some of its edges with at most three parallel edges. In this work, we introduce and study a restriction of this problem under the additional constraint that edges added to G to reach M(G) must form a walk (i.e., a path with possibly repeated edges and vertices) of G. We investigate the general consequences of having this additional constraint, and provide several results of different natures (structural, combinatorial, algorithmic) on the length of the shortest irregularising walks, for general graphs and more restricted classes.
{"title":"Making graphs irregular through irregularising walks","authors":"Julien Bensmail , Romain Bourneuf , Paul Colinot , Samuel Humeau , Timothée Martinod","doi":"10.1016/j.tcs.2026.115773","DOIUrl":"10.1016/j.tcs.2026.115773","url":null,"abstract":"<div><div>The 1-2-3 Conjecture, introduced by Karoński, Łuczak, and Thomason in 2004, was recently solved by Keusch. This implies that, for any connected graph <em>G</em> different from <em>K</em><sub>2</sub>, we can turn <em>G</em> into a locally irregular multigraph <em>M</em>(<em>G</em>), <em>i.e.</em>, in which no two adjacent vertices have the same degree, by replacing some of its edges with at most three parallel edges. In this work, we introduce and study a restriction of this problem under the additional constraint that edges added to <em>G</em> to reach <em>M</em>(<em>G</em>) must form a walk (<em>i.e.</em>, a path with possibly repeated edges and vertices) of <em>G</em>. We investigate the general consequences of having this additional constraint, and provide several results of different natures (structural, combinatorial, algorithmic) on the length of the shortest irregularising walks, for general graphs and more restricted classes.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115773"},"PeriodicalIF":1.0,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080052","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}
Pub Date : 2026-01-19DOI: 10.1016/j.tcs.2026.115770
Riccardo Dondi , Alexandru Popa
In this contribution we introduce two combinatorial problems related to graph string matching, motivated by recent approaches in computational genomics. Given a DAG where each node is labeled by a symbol, the problems aim to find a path in the DAG whose nodes contain all (or the maximum number of) symbols of the alphabet. We introduce a decision problem, Σ-Representing Path, that asks whether there exists a path that contains all the symbols of the alphabet, and an optimization problem, called Maximum Representing Path, that asks for a path that contains the maximum number of symbols. We analyze the complexity of the problems, showing the NP-completeness of Σ-Representing Path when each symbol labels at most three nodes in the DAG, and showing the APX-hardness of Maximum Representing Path when each symbol labels at most two nodes in the DAG. We complement the first result by giving a polynomial-time algorithm for Σ-Representing Path when each symbol labels at most two nodes in the DAG. Then we investigate the parameterized complexity of the two problems for two parameters: (1) the number of symbols in a solution and (2) the distance from a set of disjoint paths. We show that both problems are FPT when parameterized by the former parameter, and W[1]-hard for the latter. We consider the approximation of Maximum Representing Path, and we give an approximation algorithm of factor the maximum number of occurrences of a symbol and an approximation algorithm of factor , where OPT is the number of distinct symbols in an optimal solution. We also show that Maximum Representing Path cannot be approximated within factor , for any constant α > 0, unless NP⊆DTIME(|V|O(log log |V|)) (V is the set of nodes of the DAG).
{"title":"Complexity results and algorithms for representing paths in digraphs","authors":"Riccardo Dondi , Alexandru Popa","doi":"10.1016/j.tcs.2026.115770","DOIUrl":"10.1016/j.tcs.2026.115770","url":null,"abstract":"<div><div>In this contribution we introduce two combinatorial problems related to graph string matching, motivated by recent approaches in computational genomics. Given a DAG where each node is labeled by a symbol, the problems aim to find a path in the DAG whose nodes contain all (or the maximum number of) symbols of the alphabet. We introduce a decision problem, <span>Σ-Representing Path</span>, that asks whether there exists a path that contains all the symbols of the alphabet, and an optimization problem, called <span>Maximum Representing Path</span>, that asks for a path that contains the maximum number of symbols. We analyze the complexity of the problems, showing the NP-completeness of <span>Σ-Representing Path</span> when each symbol labels at most three nodes in the DAG, and showing the APX-hardness of <span>Maximum Representing Path</span> when each symbol labels at most two nodes in the DAG. We complement the first result by giving a polynomial-time algorithm for <span>Σ-Representing Path</span> when each symbol labels at most two nodes in the DAG. Then we investigate the parameterized complexity of the two problems for two parameters: (1) the number of symbols in a solution and (2) the distance from a set of disjoint paths. We show that both problems are FPT when parameterized by the former parameter, and W[1]-hard for the latter. We consider the approximation of <span>Maximum Representing Path</span>, and we give an approximation algorithm of factor the maximum number of occurrences of a symbol and an approximation algorithm of factor <span><math><msqrt><mrow><mi>O</mi><mi>P</mi><mi>T</mi></mrow></msqrt></math></span>, where <em>OPT</em> is the number of distinct symbols in an optimal solution. We also show that <span>Maximum Representing Path</span> cannot be approximated within factor <span><math><mrow><mfrac><mi>e</mi><mrow><mi>e</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mi>α</mi></mrow></math></span>, for any constant <em>α</em> > 0, unless <em>NP</em>⊆<em>DTIME</em>(|<em>V</em>|<sup><em>O</em>(log log |<em>V</em>|)</sup>) (<em>V</em> is the set of nodes of the DAG).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115770"},"PeriodicalIF":1.0,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080050","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}
Pub Date : 2026-01-18DOI: 10.1016/j.tcs.2026.115769
Ching-Lueh Chang
We show how to find a minimum-diameter spanning tree (resp., the center of a minimum-diameter spanning star) of an n-point ultrametric space in deterministic O(n) (resp., O(1)) time.
{"title":"Finding ultrametric minimum-diameter spanning trees","authors":"Ching-Lueh Chang","doi":"10.1016/j.tcs.2026.115769","DOIUrl":"10.1016/j.tcs.2026.115769","url":null,"abstract":"<div><div>We show how to find a minimum-diameter spanning tree (resp., the center of a minimum-diameter spanning star) of an <em>n</em>-point ultrametric space in deterministic <em>O</em>(<em>n</em>) (resp., <em>O</em>(1)) time.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115769"},"PeriodicalIF":1.0,"publicationDate":"2026-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039652","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}
Pub Date : 2026-01-17DOI: 10.1016/j.tcs.2026.115766
Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci
Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph G of n vertices asks to find the largest set of vertices X⊆V(G), also called μ-set, such that for any two vertices u, v ∈ X, there is a shortest u, v-path P where all internal vertices of P are not in X. This means that u and v are visible w.r.t. X. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside X. The mutual-visibility problem and all its variants are known to be -complete on graphs of diameter 4.
We design a polynomial-time algorithm that finds a μ-set of size , where D is the average distance in G, we show inapproximability results for all visibility problems on graphs of diameter 2, and we strengthen the inapproximability ratios for graphs of diameter 3 or larger. More precisely, assuming , the mutual-visibility and dual mutual-visibility problems are not approximable within a factor of on graphs of diameter at least 3, while the outer and total mutual-visibility problems are not approximable within a factor of , for any constant ε > 0. Finally, we study the relationship between the mutual-visibility number and the general position number, in which no three distinct vertices u, v, w of X belong to any shortest path of G.
由于可见性问题是具有挑战性的组合问题，并且与机器人导航问题密切相关，因此长期以来人们在不同的假设下对其进行了研究。有n个顶点的图G中的互可见性问题要求求出最大的顶点集X V(G)，也称为μ-set，使得对于任意两个顶点u， V ∈ X，存在一个最短的u， V路径P，且P的所有内部顶点都不在X中。这意味着u和V在w.r.t.x中是可见的。这个问题的变体被称为总、外和对偶互可见性问题。依赖于x内外顶点的可见性，互可见性问题及其所有变体在直径为4的图上已知是np完全的。我们设计了一个多项式时间算法，它找到一个大小为Ω（n/D）的μ集，其中D是G中的平均距离，我们在直径为2的图上展示了所有可见性问题的不近似结果，并且我们加强了直径为3或更大的图的不近似比率。更准确地说，假设P≠NP，在直径至少为3的图上，互可视性和对偶互可视性问题在n1/3−ε因子范围内是不可近似的，而对于任意常数ε >； 0，外互可视性和全互可视性问题在n1/2−ε因子范围内是不可近似的。最后，我们研究了互可见数与一般位置数的关系，其中X的三个不同的顶点u， v， w不属于G的任何最短路径。
{"title":"On the approximability of graph visibility problems","authors":"Davide Bilò, Alessia Di Fonso, Gabriele Di Stefano, Stefano Leucci","doi":"10.1016/j.tcs.2026.115766","DOIUrl":"10.1016/j.tcs.2026.115766","url":null,"abstract":"<div><div>Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The <em>mutual-visibility</em> problem in a graph <em>G</em> of <em>n</em> vertices asks to find the largest set of vertices <em>X</em>⊆<em>V</em>(<em>G</em>), also called <em>μ</em>-set, such that for any two vertices <em>u, v</em> ∈ <em>X</em>, there is a shortest <em>u, v</em>-path <em>P</em> where all internal vertices of <em>P</em> are not in <em>X</em>. This means that <em>u</em> and <em>v</em> are visible w.r.t. <em>X</em>. Variations of this problem are known as <em>total, outer</em>, and <em>dual</em> mutual-visibility problems, depending on the visibility property of vertices inside and/or outside <em>X</em>. The mutual-visibility problem and all its variants are known to be <span><math><mi>NP</mi></math></span>-complete on graphs of diameter 4.</div><div>We design a polynomial-time algorithm that finds a <em>μ</em>-set of size <span><math><mrow><mstyle><mi>Ω</mi></mstyle><mrow><mo>(</mo><msqrt><mrow><mi>n</mi><mo>/</mo><mi>D</mi></mrow></msqrt><mo>)</mo></mrow></mrow></math></span>, where <em>D</em> is the average distance in <em>G</em>, we show inapproximability results for all visibility problems on graphs of diameter 2, and we strengthen the inapproximability ratios for graphs of diameter 3 or larger. More precisely, assuming <span><math><mrow><mi>P</mi><mo>≠</mo><mi>NP</mi></mrow></math></span>, the mutual-visibility and dual mutual-visibility problems are not approximable within a factor of <span><math><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>−</mo><mrow><mi>ε</mi></mrow></mrow></msup></math></span> on graphs of diameter at least 3, while the outer and total mutual-visibility problems are not approximable within a factor of <span><math><msup><mi>n</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mrow><mi>ε</mi></mrow></mrow></msup></math></span>, for any constant ε > 0. Finally, we study the relationship between the mutual-visibility number and the general position number, in which no three distinct vertices <em>u, v, w</em> of <em>X</em> belong to any shortest path of <em>G</em>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115766"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039654","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}
Pub Date : 2026-01-17DOI: 10.1016/j.tcs.2026.115756
Benjamin Hellouin De Menibus , Victor H. Lutfalla , Pascal Vanier
We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending the results of [1] on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).
{"title":"Decision problems on geometric tilings","authors":"Benjamin Hellouin De Menibus , Victor H. Lutfalla , Pascal Vanier","doi":"10.1016/j.tcs.2026.115756","DOIUrl":"10.1016/j.tcs.2026.115756","url":null,"abstract":"<div><div>We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending the results of [1] on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115756"},"PeriodicalIF":1.0,"publicationDate":"2026-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039624","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}
Pub Date : 2026-01-15DOI: 10.1016/j.tcs.2026.115764
Tota Yamada, Yonghwan Kim, Yoshiaki Katayama
A Minus Dominating (MD) Function of a graph is a function that assigns a value from to each vertex i ∈ V such that the sum of the values of vertex i and all its neighboring vertices is positive (i.e., equal to or greater than 1). An MD function is minimal if decreasing the value of any vertex by 1 violates the conditions of the MD function.
As an extension of the MD function, we introduce the k-Minimal Minus Dominating (MMD) Function , which is a minimal MD function with additional condition such that no other MD function can be obtained by increasing the values of some vertices by a total of k while decreasing the values of some vertices by at least in total. According to the definition, any minimal MD function corresponds to a 0-MMD function.
In this paper, we propose a silent self-stabilizing algorithm to solve the 1-Minimal Minus Domination Problemon an arbitrary graph. This algorithm employs a composition technique, known as loop composition, which repeatedly applies several self-stabilizing algorithms in order. The algorithm converges within rounds, where D is the diameter and Δ is the maximum degree of the graph. Each vertex requires bits of memory.
图G=(V，E)（|V|=n）的负支配（MD）函数是这样一个函数，它给每个顶点i ∈ V赋一个{−1,0,1}的值，使得顶点i与其所有相邻顶点的值之和为正（即等于或大于1）。如果将任意顶点的值减少1违反了MD函数的条件，则MD函数是最小的。作为MD函数的扩展，我们引入了k- minimal - Minus (MMD) function(0≤k≤2n−1)，这是一个最小MD函数，它带有附加条件，使得不能通过将某些顶点的值增加k而将某些顶点的值减少至少k+1来获得其他MD函数。根据定义，任何最小MD函数都对应一个0-MMD函数。本文提出了一种无声自稳定算法来解决任意图的1-极小负控制问题。该算法采用了一种复合技术，称为循环复合，它按顺序重复应用几个自稳定算法。算法在O（n（Δ2+D））轮内收敛，其中D为图的直径，Δ为图的最大度。每个顶点需要O（Δ4logn）位内存。
{"title":"k-minimal minus domination and self-stabilization","authors":"Tota Yamada, Yonghwan Kim, Yoshiaki Katayama","doi":"10.1016/j.tcs.2026.115764","DOIUrl":"10.1016/j.tcs.2026.115764","url":null,"abstract":"<div><div>A <em>Minus Dominating (MD) Function</em> of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></math></span><span><math><mrow><mo>(</mo><mo>|</mo><mi>V</mi><mo>|</mo><mo>=</mo><mi>n</mi><mo>)</mo></mrow></math></span> is a function that assigns a value from <span><math><mrow><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span> to each vertex <em>i</em> ∈ <em>V</em> such that the sum of the values of vertex <em>i</em> and all its neighboring vertices is positive (i.e., equal to or greater than 1). An MD function is <em>minimal</em> if decreasing the value of any vertex by 1 violates the conditions of the MD function.</div><div>As an extension of the MD function, we introduce the <em>k</em>-<em>Minimal Minus Dominating (MMD) Function</em> <span><math><mrow><mo>(</mo><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>, which is a minimal MD function with additional condition such that no other MD function can be obtained by increasing the values of some vertices by a total of <em>k</em> while decreasing the values of some vertices by at least <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> in total. According to the definition, any minimal MD function corresponds to a 0-MMD function.</div><div>In this paper, we propose a silent self-stabilizing algorithm to solve the <em>1-Minimal Minus Domination Problem</em>on an arbitrary graph. This algorithm employs a composition technique, known as <em>loop composition</em>, which repeatedly applies several self-stabilizing algorithms in order. The algorithm converges within <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mrow><mo>(</mo><msup><mstyle><mi>Δ</mi></mstyle><mn>2</mn></msup><mo>+</mo><mi>D</mi><mo>)</mo></mrow><mo>)</mo></mrow></math></span> rounds, where <em>D</em> is the diameter and Δ is the maximum degree of the graph. Each vertex requires <span><math><mrow><mi>O</mi><mo>(</mo><msup><mstyle><mi>Δ</mi></mstyle><mn>4</mn></msup><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> bits of memory.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115764"},"PeriodicalIF":1.0,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039650","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}
Pub Date : 2026-01-15DOI: 10.1016/j.tcs.2026.115758
Sourav Mondal , Parikshit Das , Zahid Raza , Anita Pal , Modjtaba Ghorbani
The spectral graph theory investigates the relationships between combinatorial qualities of graphs and algebraic properties of related matrices. The adjacency matrix is currently undergoing significant modification as a result of its well-developed theoretical and application standpoint. The present work deals with one such extension of the adjacency matrix. We propose here the neighborhood Sombor matrix corresponding to the well-known Sombor index. We compute the neighborhood Sombor spectrum of some benchmark graphs. Lower and upper bounds of the spectral radius (ζ1) are derived with identifying extremal graphs. Moreover, extremal trees are characterized in view of spectral radius, where path and star graphs yield minimal and maximal structures, respectively. The role of ζ1 in structure-property modelling is also demonstrated. The isomer-discrimination ability of ζ1 is found to be better than that of some well-known descriptors.
{"title":"Graph spectrum of neighbourhood sombor matrix and structure-Property modelling","authors":"Sourav Mondal , Parikshit Das , Zahid Raza , Anita Pal , Modjtaba Ghorbani","doi":"10.1016/j.tcs.2026.115758","DOIUrl":"10.1016/j.tcs.2026.115758","url":null,"abstract":"<div><div>The spectral graph theory investigates the relationships between combinatorial qualities of graphs and algebraic properties of related matrices. The adjacency matrix is currently undergoing significant modification as a result of its well-developed theoretical and application standpoint. The present work deals with one such extension of the adjacency matrix. We propose here the neighborhood Sombor matrix corresponding to the well-known Sombor index. We compute the neighborhood Sombor spectrum of some benchmark graphs. Lower and upper bounds of the spectral radius (<em>ζ</em><sub>1</sub>) are derived with identifying extremal graphs. Moreover, extremal trees are characterized in view of spectral radius, where path and star graphs yield minimal and maximal structures, respectively. The role of <em>ζ</em><sub>1</sub> in structure-property modelling is also demonstrated. The isomer-discrimination ability of <em>ζ</em><sub>1</sub> is found to be better than that of some well-known descriptors.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1068 ","pages":"Article 115758"},"PeriodicalIF":1.0,"publicationDate":"2026-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081151","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}
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