Pub Date : 2025-03-05DOI: 10.1016/j.dam.2025.02.039
E.M.M. Coelho , H. Coelho , L. Faria , M.P. Ferreira , S. Klein
<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, the size of the largest clique <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is always less than or equal to the chromatic number <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>. The oriented coloring of an oriented graph <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> assigns colors to the vertices of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>, such that the arcs connecting vertices in different color classes always have the same direction and the smallest number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> of colors in an oriented coloring is the oriented chromatic number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>. Oriented colorings have fundamental implications for homomorphisms of oriented graphs and significant applications in distributed processing and task scheduling. In 2004, Klostermeyer and MacGillivray defined the concept of an “analogue of clique” for oriented coloring in which a subgraph <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> is an absolute oriented clique if the oriented distance between a pair of vertices of <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> in <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> is at most 2. The authors defined the absolute oriented clique number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> as the number of vertices <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> in a maximum absolute oriented clique <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></move
{"title":"On the absolute and relative oriented clique problems’ time complexity","authors":"E.M.M. Coelho , H. Coelho , L. Faria , M.P. Ferreira , S. Klein","doi":"10.1016/j.dam.2025.02.039","DOIUrl":"10.1016/j.dam.2025.02.039","url":null,"abstract":"<div><div>Given a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, the size of the largest clique <span><math><mrow><mi>ω</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is always less than or equal to the chromatic number <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>G</mi></math></span>. The oriented coloring of an oriented graph <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> assigns colors to the vertices of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>, such that the arcs connecting vertices in different color classes always have the same direction and the smallest number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> of colors in an oriented coloring is the oriented chromatic number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>. Oriented colorings have fundamental implications for homomorphisms of oriented graphs and significant applications in distributed processing and task scheduling. In 2004, Klostermeyer and MacGillivray defined the concept of an “analogue of clique” for oriented coloring in which a subgraph <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> is an absolute oriented clique if the oriented distance between a pair of vertices of <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> in <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover></math></span> is at most 2. The authors defined the absolute oriented clique number of <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> as the number of vertices <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover><mo>)</mo></mrow></mrow></math></span> in a maximum absolute oriented clique <span><math><mover><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi><mi>o</mi></mrow></msub></mrow><mo>⃗</mo></move","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 53-65"},"PeriodicalIF":1.0,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552968","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}
Pub Date : 2025-03-04DOI: 10.1016/j.dam.2025.02.034
Litao Guo , Wantao Ning
We define a new class of graphs called enhanced folded hypercube-like networks . We also investigate the reliability of this class of graphs in terms of the (edge) connectivity and super (edge) connectivity.
{"title":"Connectivity and super connectivity of enhanced folded hypercube-like networks","authors":"Litao Guo , Wantao Ning","doi":"10.1016/j.dam.2025.02.034","DOIUrl":"10.1016/j.dam.2025.02.034","url":null,"abstract":"<div><div>We define a new class of graphs called enhanced folded hypercube-like networks <span><math><mrow><mi>E</mi><mi>F</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>. We also investigate the reliability of this class of graphs in terms of the (edge) connectivity and super (edge) connectivity.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 14-19"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534673","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}
Pub Date : 2025-03-04DOI: 10.1016/j.dam.2025.02.033
Shaofei Du, Tianlei Zhou
It was shown by Kutnar and Šparl in 2009 that every connected vertex-transitive graph of order , where is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except for the Petersen graph by replacing each vertex by a triangle.
{"title":"Hamilton cycles in vertex-transitive graphs of order 6p","authors":"Shaofei Du, Tianlei Zhou","doi":"10.1016/j.dam.2025.02.033","DOIUrl":"10.1016/j.dam.2025.02.033","url":null,"abstract":"<div><div>It was shown by Kutnar and Šparl in 2009 that every connected vertex-transitive graph of order <span><math><mrow><mn>6</mn><mi>p</mi></mrow></math></span>, where <span><math><mi>p</mi></math></span> is a prime, contains a Hamilton path. In this paper, it will be shown that every such graph contains a Hamilton cycle, except for the Petersen graph by replacing each vertex by a triangle.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 165-175"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552118","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}
Pub Date : 2025-03-04DOI: 10.1016/j.dam.2025.02.026
Suchismita Mishra
In this paper, we discuss the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a partitionable (bull, diamond)-free graph can be found in polynomial time. Additionally, we show that the cop number of a (bull, diamond)-free graph containing a triangle is at most two less than its diameter. Furthermore, the cop number of a connected (, bull, diamond)-free graph with a triangle, is at most , for any natural number . We also discuss a couple of applications of the structural theorem of the (bull, diamond)-free graphs in the conclusions.
{"title":"Structure of (bull, diamond)-free graphs and its applications","authors":"Suchismita Mishra","doi":"10.1016/j.dam.2025.02.026","DOIUrl":"10.1016/j.dam.2025.02.026","url":null,"abstract":"<div><div>In this paper, we discuss the complete structure of the (bull, diamond)-free graphs. As an application of that, we give the characterization of the partitionable (bull, diamond)-free graphs. Moreover, we show that such a partition for a partitionable (bull, diamond)-free graph can be found in polynomial time. Additionally, we show that the cop number of a (bull, diamond)-free graph containing a triangle is at most two less than its diameter. Furthermore, the cop number of a connected (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, bull, diamond)-free graph with a triangle, is at most <span><math><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span>, for any natural number <span><math><mrow><mi>n</mi><mo>></mo><mn>3</mn></mrow></math></span>. We also discuss a couple of applications of the structural theorem of the (bull, diamond)-free graphs in the conclusions.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 176-183"},"PeriodicalIF":1.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552119","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}
Pub Date : 2025-03-03DOI: 10.1016/j.dam.2025.02.041
Priyamvada
An odd coloring of a graph is an assignment of colors to the vertices of such that is a proper vertex coloring and for every non-isolated vertex , there is a color that occurs an odd number of times within its open neighborhood. The minimum number of colors required by any odd coloring of is called the odd chromatic number of and is denoted by . In this paper, we give tight upper bounds on the odd chromatic number of various standard graph products and operations, including the lexicographic product, corona product, edge corona product and Mycielskian of a graph. Moreover, we give tight lower bounds on the odd chromatic number of corona product and edge corona product of graphs.
{"title":"Tight bounds on odd chromatic number of some standard graph products","authors":"Priyamvada","doi":"10.1016/j.dam.2025.02.041","DOIUrl":"10.1016/j.dam.2025.02.041","url":null,"abstract":"<div><div>An <em>odd coloring</em> of a graph <span><math><mi>G</mi></math></span> is an assignment <span><math><mrow><mi>f</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><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> of colors to the vertices of <span><math><mi>G</mi></math></span> such that <span><math><mi>f</mi></math></span> is a proper vertex coloring and for every non-isolated vertex <span><math><mi>v</mi></math></span>, there is a color that occurs an odd number of times within its open neighborhood. The minimum number of colors required by any odd coloring of <span><math><mi>G</mi></math></span> is called the <em>odd chromatic number</em> of <span><math><mi>G</mi></math></span> and is denoted by <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>o</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, we give tight upper bounds on the odd chromatic number of various standard graph products and operations, including the lexicographic product, corona product, edge corona product and Mycielskian of a graph. Moreover, we give tight lower bounds on the odd chromatic number of corona product and edge corona product of graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 1-13"},"PeriodicalIF":1.0,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143534446","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}
Pub Date : 2025-03-03DOI: 10.1016/j.dam.2025.02.019
Xiaodong Chen, Tianhao Li, Jiayuan Zhang
A dominating set of a graph is secure if for each vertex contains a neighbor of such that is a dominating set of . The minimum cardinality of a secure dominating set in is the secure domination number of and denoted by A graph is -free if it does not contain two independent edges as an induced subgraph. Let denote the independence number of Several results gave the upper bound of by a function of In this note, we shows that for every -free graph moreover, we give an example to show the bound in our result is best possible.
{"title":"A note on secure domination number in 2K2-free graphs","authors":"Xiaodong Chen, Tianhao Li, Jiayuan Zhang","doi":"10.1016/j.dam.2025.02.019","DOIUrl":"10.1016/j.dam.2025.02.019","url":null,"abstract":"<div><div>A dominating set <span><math><mi>D</mi></math></span> of a graph <span><math><mi>G</mi></math></span> is secure if for each vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>D</mi><mo>,</mo></mrow></math></span>\u0000 <span><math><mi>D</mi></math></span> contains a neighbor <span><math><mi>u</mi></math></span> of <span><math><mi>v</mi></math></span> such that <span><math><mrow><mrow><mo>(</mo><mi>D</mi><mo>−</mo><mrow><mo>{</mo><mi>u</mi><mo>}</mo></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>{</mo><mi>v</mi><mo>}</mo></mrow></mrow></math></span> is a dominating set of <span><math><mi>G</mi></math></span>. The minimum cardinality of a secure dominating set in <span><math><mi>G</mi></math></span> is the secure domination number of <span><math><mi>G</mi></math></span> and denoted by <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> A graph is <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free if it does not contain two independent edges as an induced subgraph. Let <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote the independence number of <span><math><mrow><mi>G</mi><mo>.</mo></mrow></math></span> Several results gave the upper bound of <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> by a function of <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> In this note, we shows that <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span> for every <span><math><mrow><mn>2</mn><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>-free graph <span><math><mrow><mi>G</mi><mo>;</mo></mrow></math></span> moreover, we give an example to show the bound in our result is best possible.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 162-164"},"PeriodicalIF":1.0,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552120","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}
Pub Date : 2025-02-28DOI: 10.1016/j.dam.2025.02.038
Athira Divakaran , Tanja Dravec , Tijo James , Sandi Klavžar , Latha S Nair
The Maker–Breaker domination game (MBD game) is a two-player game played on a graph by Dominator and Staller. They alternately select unplayed vertices of . The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.
{"title":"Maker–Breaker domination game critical graphs","authors":"Athira Divakaran , Tanja Dravec , Tijo James , Sandi Klavžar , Latha S Nair","doi":"10.1016/j.dam.2025.02.038","DOIUrl":"10.1016/j.dam.2025.02.038","url":null,"abstract":"<div><div>The Maker–Breaker domination game (MBD game) is a two-player game played on a graph <span><math><mi>G</mi></math></span> by Dominator and Staller. They alternately select unplayed vertices of <span><math><mi>G</mi></math></span>. The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 126-134"},"PeriodicalIF":1.0,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-28DOI: 10.1016/j.dam.2025.02.028
Mark Dukes, Anton Sohn
In this paper we consider a general system of activities that can, but do not have to, occur. This system is governed by a set containing two types of constraints: precedence and response. A precedence constraint dictates that an activity can only occur if it has been preceded by some other specified activity. Response constraints are similarly defined. An execution of the system is a listing of activities in the order they occur and which satisfies all constraints. These listings are known as traces. Such systems naturally arise in areas of theoretical computer science and decision science. An outcome of the freedom with which activities can occur is that there are many different possible executions, and gaining a combinatorial insight into these is a non-trivial problem.
We characterize all of the ways in which such a system can be executed. Our approach uses order theory to provide a classification in terms of the linear extensions of posets constructed from the constraint sets. This characterization is essential in calculating the stakeholder utility metrics that have been developed by the first author that allow for quantitative comparisons of such systems/processes. It also allows for a better understanding of the theoretical backbone to these processes.
{"title":"Characterizing traces of processes defined by precedence and response constraints: An order theory approach","authors":"Mark Dukes, Anton Sohn","doi":"10.1016/j.dam.2025.02.028","DOIUrl":"10.1016/j.dam.2025.02.028","url":null,"abstract":"<div><div>In this paper we consider a general system of activities that can, but do not have to, occur. This system is governed by a set containing two types of constraints: precedence and response. A precedence constraint dictates that an activity can only occur if it has been preceded by some other specified activity. Response constraints are similarly defined. An execution of the system is a listing of activities in the order they occur and which satisfies all constraints. These listings are known as <em>traces</em>. Such systems naturally arise in areas of theoretical computer science and decision science. An outcome of the freedom with which activities can occur is that there are many different possible executions, and gaining a combinatorial insight into these is a non-trivial problem.</div><div>We characterize all of the ways in which such a system can be executed. Our approach uses order theory to provide a classification in terms of the linear extensions of posets constructed from the constraint sets. This characterization is essential in calculating the stakeholder utility metrics that have been developed by the first author that allow for quantitative comparisons of such systems/processes. It also allows for a better understanding of the theoretical backbone to these processes.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 112-125"},"PeriodicalIF":1.0,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-28DOI: 10.1016/j.dam.2025.02.040
Zemin Jin
It is well known that a maximum matching in a given graph can be found in polynomial time. The maximum rainbow matching problem is to find a rainbow matching of maximum size in an edge-colored graph. This problem is equivalent to the multiple choice matching problem which is -Complete. Moreover, it is surprising that the rainbow matching problem is even -Complete for paths. So far, there is few efficient algorithm for rainbow matchings. The only positive result is to reduce it to the maximum independent sets in -free graphs, which can be approximated by a polynomial algorithm with approximation ratio for every . In this paper, we give a heuristic polynomial algorithm to find a large rainbow matching in an edge-colored . For any given integer , we can find either a rainbow , or a with at most colors for some . It is interesting that our result is useful for the existence of a monochromatic against a rainbow matching in . We give applications of the algorithm and, based on it, we generalize the previous results about the rainbow Ramsey number for matchings.
{"title":"A heuristic algorithm for rainbow matchings and its application in rainbow Ramsey number for matchings","authors":"Zemin Jin","doi":"10.1016/j.dam.2025.02.040","DOIUrl":"10.1016/j.dam.2025.02.040","url":null,"abstract":"<div><div>It is well known that a maximum matching in a given graph can be found in polynomial time. The maximum rainbow matching problem is to find a rainbow matching of maximum size in an edge-colored graph. This problem is equivalent to the multiple choice matching problem which is <span><math><mrow><mi>N</mi><mi>P</mi></mrow></math></span>-Complete. Moreover, it is surprising that the rainbow matching problem is even <span><math><mrow><mi>A</mi><mi>P</mi><mi>X</mi></mrow></math></span>-Complete for paths. So far, there is few efficient algorithm for rainbow matchings. The only positive result is to reduce it to the maximum independent sets in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span>-free graphs, which can be approximated by a polynomial algorithm with approximation ratio <span><math><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>−</mo><mi>ϵ</mi></mrow></math></span> for every <span><math><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></math></span>. In this paper, we give a heuristic polynomial algorithm to find a large rainbow matching in an edge-colored <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. For any given integer <span><math><mi>k</mi></math></span>, we can find either a rainbow <span><math><mrow><mi>k</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math></span>, or a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn><mi>i</mi></mrow></msub></math></span> with at most <span><math><mrow><mi>k</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></math></span> colors for some <span><math><mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi><mo>−</mo><mn>2</mn></mrow></math></span>. It is interesting that our result is useful for the existence of a monochromatic <span><math><mi>G</mi></math></span> against a rainbow matching in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We give applications of the algorithm and, based on it, we generalize the previous results about the rainbow Ramsey number for matchings.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 153-161"},"PeriodicalIF":1.0,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512581","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}
We study a new version of the Traveling Salesperson Problem, called Vector TSP, where the traveler is subject to discrete acceleration constraints, as defined in the paper-and-pencil game Racetrack (also known as Vector Racer). In this model, the degrees of freedom at a certain point in time depends on the current velocity, and the speed is not limited.
The paper introduces this problem and initiates its study, discussing also the main differences with existing versions of TSP. Not surprisingly, the problem turns out to be NP-hard. A key feature of Vector TSP is that it deals with acceleration in a discrete, combinatorial way, making the problem more amenable to algorithmic investigation. The problem involves two layers of trajectory planning: (1) the order in which cities are visited, and (2) the physical trajectory realizing such a visit, both interacting with each other. This interaction is formalized as an interactive protocol between a high-level tour algorithm and a trajectory oracle, the former calling the latter repeatedly. We present an exact implementation of the trajectory oracle, adapting the A* algorithm for paths over multiple checkpoints whose ordering is given (this algorithm being possibly of independent interest). To motivate the problem further, we perform experiments showing that the naive approach consisting of solving the instance as an Euclidean TSP first, then optimizing the trajectory of the resulting tour, is typically suboptimal and outperformed by simple (but dedicated) heuristics.
{"title":"Vector TSP: A Traveling Salesperson Problem with Racetrack-like acceleration constraints","authors":"Arnaud Casteigts , Mathieu Raffinot , Mikhail Raskin , Jason Schoeters","doi":"10.1016/j.dam.2025.02.021","DOIUrl":"10.1016/j.dam.2025.02.021","url":null,"abstract":"<div><div>We study a new version of the Traveling Salesperson Problem, called <span>Vector TSP</span>, where the traveler is subject to discrete acceleration constraints, as defined in the paper-and-pencil game Racetrack (also known as Vector Racer). In this model, the degrees of freedom at a certain point in time depends on the current velocity, and the speed is not limited.</div><div>The paper introduces this problem and initiates its study, discussing also the main differences with existing versions of TSP. Not surprisingly, the problem turns out to be NP-hard. A key feature of <span>Vector TSP</span> is that it deals with acceleration in a discrete, combinatorial way, making the problem more amenable to algorithmic investigation. The problem involves two layers of trajectory planning: (1) the order in which cities are visited, and (2) the physical trajectory realizing such a visit, both interacting with each other. This interaction is formalized as an interactive protocol between a high-level tour algorithm and a trajectory oracle, the former calling the latter repeatedly. We present an exact implementation of the trajectory oracle, adapting the A* algorithm for paths over multiple checkpoints whose ordering is <em>given</em> (this algorithm being possibly of independent interest). To motivate the problem further, we perform experiments showing that the naive approach consisting of solving the instance as an <span>Euclidean TSP</span> first, then optimizing the trajectory of the resulting tour, is typically suboptimal and outperformed by simple (but dedicated) heuristics.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"368 ","pages":"Pages 135-152"},"PeriodicalIF":1.0,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143512586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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