Pub Date : 2025-01-22DOI: 10.1016/j.dam.2025.01.010
Sarah E. Anderson, Brenda K. Kroschel
Zero forcing is a process that passes information along a network. In particular, a vertex that has information may pass that information to one of its neighbors only if that neighbor is its only neighbor that does not have that information. If the information is given to an initial set of vertices that eventually passes the information to the entire network, then that initial set is called a zero forcing set. The zero forcing number of a graph is the size of a minimum zero forcing set of that graph. In this paper, bounds on the zero forcing number of generalized hierarchical products, which are a generalization of the Cartesian product, are provided. In particular, the zero forcing number is determined exactly for certain hierarchical products by constructing zero forcing sets in the product that utilize knowledge about forcing chains for each factor of the product.
{"title":"Zero forcing of generalized hierarchical products","authors":"Sarah E. Anderson, Brenda K. Kroschel","doi":"10.1016/j.dam.2025.01.010","DOIUrl":"10.1016/j.dam.2025.01.010","url":null,"abstract":"<div><div>Zero forcing is a process that passes information along a network. In particular, a vertex that has information may pass that information to one of its neighbors only if that neighbor is its only neighbor that does not have that information. If the information is given to an initial set of vertices that eventually passes the information to the entire network, then that initial set is called a zero forcing set. The zero forcing number of a graph is the size of a minimum zero forcing set of that graph. In this paper, bounds on the zero forcing number of generalized hierarchical products, which are a generalization of the Cartesian product, are provided. In particular, the zero forcing number is determined exactly for certain hierarchical products by constructing zero forcing sets in the product that utilize knowledge about forcing chains for each factor of the product.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 120-126"},"PeriodicalIF":1.0,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160892","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 : 2025-01-21DOI: 10.1016/j.dam.2024.12.032
Florent Foucaud , Clara Marcille , Zin Mar Myint , R.B. Sandeep , Sagnik Sen , S. Taruni
A monitoring edge-geodetic set, or simply an MEG-set, of a graph is a vertex subset such that given any edge of , lies on every shortest - path of , for some . The monitoring edge-geodetic number of , denoted by , is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem.
In this article, we compare with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs that have as their minimum MEG-set, which settles an open problem due to Foucaud et al. (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of . We examine the change in with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.
{"title":"Bounds and extremal graphs for monitoring edge-geodetic sets in graphs","authors":"Florent Foucaud , Clara Marcille , Zin Mar Myint , R.B. Sandeep , Sagnik Sen , S. Taruni","doi":"10.1016/j.dam.2024.12.032","DOIUrl":"10.1016/j.dam.2024.12.032","url":null,"abstract":"<div><div>A monitoring edge-geodetic set, or simply an MEG-set, of a graph <span><math><mi>G</mi></math></span> is a vertex subset <span><math><mrow><mi>M</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that given any edge <span><math><mi>e</mi></math></span> of <span><math><mi>G</mi></math></span>, <span><math><mi>e</mi></math></span> lies on every shortest <span><math><mi>u</mi></math></span>-<span><math><mi>v</mi></math></span> path of <span><math><mi>G</mi></math></span>, for some <span><math><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>M</mi></mrow></math></span>. The monitoring edge-geodetic number of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>m</mi><mi>e</mi><mi>g</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the minimum cardinality of such an MEG-set. This notion provides a graph theoretic model of the network monitoring problem.</div><div>In this article, we compare <span><math><mrow><mi>m</mi><mi>e</mi><mi>g</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with some other graph theoretic parameters stemming from the network monitoring problem and provide examples of graphs having prescribed values for each of these parameters. We also characterize graphs <span><math><mi>G</mi></math></span> that have <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> as their minimum MEG-set, which settles an open problem due to Foucaud et al. (CALDAM 2023), and prove that some classes of graphs fall within this characterization. We also provide a general upper bound for <span><math><mrow><mi>m</mi><mi>e</mi><mi>g</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for sparse graphs in terms of their girth, and later refine the upper bound using the chromatic number of <span><math><mi>G</mi></math></span>. We examine the change in <span><math><mrow><mi>m</mi><mi>e</mi><mi>g</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> with respect to two fundamental graph operations: clique-sum and subdivisions. In both cases, we provide a lower and an upper bound of the possible amount of changes and provide (almost) tight examples.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 106-119"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161747","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 : 2025-01-21DOI: 10.1016/j.dam.2025.01.021
Jingchao Lai, Weigen Yan, Xing Feng
Let be a connected graph with vertices and edges, and let and be the line graph and middle graph of , respectively. Let be the number of perfect matchings of . Dong, Yan and Zhang (Discrete Applied Mathematics, 161 (2013) 794-801.) proved that if is even, then , and characterized the extremal graphs with equality. In this paper, we obtain a similar result for the middle graph of , that is, if is even, then , and characterize the extremal graphs such that . As applications, we enumerate perfect matchings of the middle graphs of the Cartesian products , and , where and are the path and cycle with vertices.
{"title":"On the number of perfect matchings of middle graphs","authors":"Jingchao Lai, Weigen Yan, Xing Feng","doi":"10.1016/j.dam.2025.01.021","DOIUrl":"10.1016/j.dam.2025.01.021","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>m</mi></math></span> edges, and let <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the line graph and middle graph of <span><math><mi>G</mi></math></span>, respectively. Let <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of perfect matchings of <span><math><mi>G</mi></math></span>. Dong, Yan and Zhang (Discrete Applied Mathematics, 161 (2013) 794-801.) proved that if <span><math><mi>m</mi></math></span> is even, then <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>L</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></math></span>, and characterized the extremal graphs <span><math><mi>G</mi></math></span> with equality. In this paper, we obtain a similar result for the middle graph of <span><math><mi>G</mi></math></span>, that is, if <span><math><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></math></span> is even, then <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span>, and characterize the extremal graphs <span><math><mi>G</mi></math></span> such that <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mrow><mn>3</mn></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span>. As applications, we enumerate perfect matchings of the middle graphs of the Cartesian products <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span>, where <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are the path and cycle with <span><math><mi>n</mi></math></span> vertices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 86-91"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161748","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 : 2025-01-21DOI: 10.1016/j.dam.2025.01.020
Jianxin Wei , Yujun Yang
In 2021, Ö. Eǧecioǧlu, V. Iršič introduced the concept of Fibonacci-run graph as an induced subgraph of Hypercube. They conjectured that the diameter of is given by . In this paper, we introduce the concept of distance-barriers between vertices in and provide a lower bound for the diameter of via this concept. By constructing different types of distance-barriers, we show that the conjecture fails to hold for all and for some between 91 and 229. The lower bounds obtained turn out to be better than the result given in the conjecture.
{"title":"On a conjecture of Eǧecioǧlu and Iršič","authors":"Jianxin Wei , Yujun Yang","doi":"10.1016/j.dam.2025.01.020","DOIUrl":"10.1016/j.dam.2025.01.020","url":null,"abstract":"<div><div>In 2021, Ö. Eǧecioǧlu, V. Iršič introduced the concept of Fibonacci-run graph <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as an induced subgraph of Hypercube. They conjectured that the diameter of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given by <span><math><mrow><mi>n</mi><mo>−</mo><mrow><mo>⌊</mo><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>⌋</mo></mrow></mrow></math></span>. In this paper, we introduce the concept of distance-barriers between vertices in <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and provide a lower bound for the diameter of <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> via this concept. By constructing different types of distance-barriers, we show that the conjecture fails to hold for all <span><math><mrow><mi>n</mi><mo>≥</mo><mn>230</mn></mrow></math></span> and for some <span><math><mi>n</mi></math></span> between 91 and 229. The lower bounds obtained turn out to be better than the result given in the conjecture.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 92-105"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161749","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 : 2025-01-21DOI: 10.1016/j.dam.2025.01.018
Cai-Xia Wang, Yu Yang, Shou-Jun Xu
<div><div>Given a graph <span><math><mi>G</mi></math></span> with vertex set <span><math><mi>V</mi></math></span>, a secure dominating set of <span><math><mi>G</mi></math></span> is a set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> with the property that for each vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><mi>S</mi></mrow></math></span>, there is a vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></math></span> adjacent to <span><math><mi>u</mi></math></span> such that <span><math><mrow><mrow><mo>(</mo><mi>S</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 secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph.</div><div>In this paper, firstly, we show that the MSDS problem is APX-hard in <span><math><mi>d</mi></math></span>-box graphs for any fixed integer <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. Secondly, we obtain a PTAS for the MSDS problem which is <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation (resp. <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation) and runs in <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> (resp. <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span>) time for unit cube graphs and unit ball graphs (resp. unit square graphs and unit disk graphs). Thirdly, we give a dynamic programming algorithm for the MSDS problem in unit-height box graphs (resp. unit-height rectangle graphs), where the centers of all boxes (resp. rectangles) are inside a column of length <span><math><mi>l</mi></math></span> and width <span><math><mi>w</mi></math></span> for some integers <span><math><mrow><mi>l</mi><mo>,</mo><mi>w</mi><mo>≥</mo><mn>1</mn></mrow></math></span> (resp. a strip of width <span><math><mi>w</mi></math></span> for some integer <span><math><mrow><mi>w</mi><mo>≥</mo><mn>1</mn></mrow></math></span>). Finally, with the help of the dynamic programming algorithm, we improve the PTAS to <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation (resp. <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation) and <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mr
{"title":"The algorithm and complexity of secure domination in 3-dimensional box graphs","authors":"Cai-Xia Wang, Yu Yang, Shou-Jun Xu","doi":"10.1016/j.dam.2025.01.018","DOIUrl":"10.1016/j.dam.2025.01.018","url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi></math></span> with vertex set <span><math><mi>V</mi></math></span>, a secure dominating set of <span><math><mi>G</mi></math></span> is a set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> with the property that for each vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>∖</mo><mi>S</mi></mrow></math></span>, there is a vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>S</mi></mrow></math></span> adjacent to <span><math><mi>u</mi></math></span> such that <span><math><mrow><mrow><mo>(</mo><mi>S</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 secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph.</div><div>In this paper, firstly, we show that the MSDS problem is APX-hard in <span><math><mi>d</mi></math></span>-box graphs for any fixed integer <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span>. Secondly, we obtain a PTAS for the MSDS problem which is <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation (resp. <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation) and runs in <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span> (resp. <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></msup></math></span>) time for unit cube graphs and unit ball graphs (resp. unit square graphs and unit disk graphs). Thirdly, we give a dynamic programming algorithm for the MSDS problem in unit-height box graphs (resp. unit-height rectangle graphs), where the centers of all boxes (resp. rectangles) are inside a column of length <span><math><mi>l</mi></math></span> and width <span><math><mi>w</mi></math></span> for some integers <span><math><mrow><mi>l</mi><mo>,</mo><mi>w</mi><mo>≥</mo><mn>1</mn></mrow></math></span> (resp. a strip of width <span><math><mi>w</mi></math></span> for some integer <span><math><mrow><mi>w</mi><mo>≥</mo><mn>1</mn></mrow></math></span>). Finally, with the help of the dynamic programming algorithm, we improve the PTAS to <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation (resp. <span><math><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>k</mi></mrow></mfrac><mo>)</mo></mrow></math></span>-approximation) and <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mrow><mo>(</mo><msup><mr","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 63-74"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161744","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 : 2025-01-21DOI: 10.1016/j.dam.2025.01.017
Simon Brezovnik , Zhongyuan Che , Niko Tratnik , Petra Žigert Pleteršek
We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if is a plane elementary bipartite graph other than , then the resonance graph of is a daisy cube if and only if the Fries number of equals the number of finite faces of . Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph is a daisy cube if and only if is weakly elementary bipartite such that each of its elementary component other than holds the property that the Fries number of equals the number of finite faces of . Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.
{"title":"Resonance graphs of plane bipartite graphs as daisy cubes","authors":"Simon Brezovnik , Zhongyuan Che , Niko Tratnik , Petra Žigert Pleteršek","doi":"10.1016/j.dam.2025.01.017","DOIUrl":"10.1016/j.dam.2025.01.017","url":null,"abstract":"<div><div>We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if <span><math><mi>G</mi></math></span> is a plane elementary bipartite graph other than <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, then the resonance graph of <span><math><mi>G</mi></math></span> is a daisy cube if and only if the Fries number of <span><math><mi>G</mi></math></span> equals the number of finite faces of <span><math><mi>G</mi></math></span>. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph <span><math><mi>G</mi></math></span> is a daisy cube if and only if <span><math><mi>G</mi></math></span> is weakly elementary bipartite such that each of its elementary component <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> other than <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> holds the property that the Fries number of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> equals the number of finite faces of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 75-85"},"PeriodicalIF":1.0,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161745","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}
Given two graphs and , a graph is -free if it contains no induced subgraph isomorphic to nor . A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond.
In this paper, we show that there are finitely many -vertex-critical -free graphs for . To prove these results, we use induction on and perform a careful structural analysis via the Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for we characterize all -vertex-critical -free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the -colorability of -free graphs for where the certificate is either a -coloring or a -vertex-critical induced subgraph.
{"title":"Critical (P5,dart)-free graphs","authors":"Wen Xia , Jorik Jooken , Jan Goedgebeur , Shenwei Huang","doi":"10.1016/j.dam.2025.01.011","DOIUrl":"10.1016/j.dam.2025.01.011","url":null,"abstract":"<div><div>Given two graphs <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, a graph is <span><math><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span>-free if it contains no induced subgraph isomorphic to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> nor <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond.</div><div>In this paper, we show that there are finitely many <span><math><mi>k</mi></math></span>-vertex-critical <span><math><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mi>d</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo>)</mo></mrow></math></span>-free graphs for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. To prove these results, we use induction on <span><math><mi>k</mi></math></span> and perform a careful structural analysis via the Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for <span><math><mrow><mi>k</mi><mo>∈</mo><mrow><mo>{</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></mrow></math></span> we characterize all <span><math><mi>k</mi></math></span>-vertex-critical <span><math><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mi>d</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo>)</mo></mrow></math></span>-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the <span><math><mi>k</mi></math></span>-colorability of <span><math><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mi>d</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo>)</mo></mrow></math></span>-free graphs for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span> where the certificate is either a <span><math><mi>k</mi></math></span>-coloring or a <span><math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-vertex-critical induced subgraph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 44-52"},"PeriodicalIF":1.0,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160888","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 : 2025-01-18DOI: 10.1016/j.dam.2025.01.013
Songnian Xu , Dein Wong , Yan Wang , Fenglei Tian
<div><div>In 2016, the authors of Wang (2016) have determined the automorphisms of the zero-divisor graph <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, where the graph is defined in such a way: the vertices are all nonzero zero-divisors of <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and there is a directed edge from <span><math><mi>A</mi></math></span> to <span><math><mi>B</mi></math></span> if and only if <span><math><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Regretfully, the graph <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is over square matrices, directed, finite and with some loops. Thus we are motivated to improve the definition such that the new defined graph can be infinite, undirected, without loops and it can be over rectangular matrices. Let <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the set of all <span><math><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></math></span> rectangular matrices over the field of complex (or real) number. By <span><math><msup><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>T</mi></mrow></msup></math></span> we mean the conjugate transpose of <span><math><mrow><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> be the graph whose vertex set consists of all nonzero matrices in <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> with rank less than <span><math><mi>n</mi></math></span>, and two vertices <span><math><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></math></span> are adjacent if and only if <span><math><mrow><mi>A</mi><msup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span> (if <span><math><mi>F</mi></math></span> is the real field, that is <span><math><mrow><mi>A</mi><msup><mrow><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span>). Note that <span><math><
{"title":"Automorphism group of a graph related to zero-divisor graphs","authors":"Songnian Xu , Dein Wong , Yan Wang , Fenglei Tian","doi":"10.1016/j.dam.2025.01.013","DOIUrl":"10.1016/j.dam.2025.01.013","url":null,"abstract":"<div><div>In 2016, the authors of Wang (2016) have determined the automorphisms of the zero-divisor graph <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>, where the graph is defined in such a way: the vertices are all nonzero zero-divisors of <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and there is a directed edge from <span><math><mi>A</mi></math></span> to <span><math><mi>B</mi></math></span> if and only if <span><math><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mn>0</mn></mrow></math></span>. Regretfully, the graph <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is over square matrices, directed, finite and with some loops. Thus we are motivated to improve the definition such that the new defined graph can be infinite, undirected, without loops and it can be over rectangular matrices. Let <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the set of all <span><math><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></math></span> rectangular matrices over the field of complex (or real) number. By <span><math><msup><mrow><mover><mrow><mi>A</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>T</mi></mrow></msup></math></span> we mean the conjugate transpose of <span><math><mrow><mi>A</mi><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> be the graph whose vertex set consists of all nonzero matrices in <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> with rank less than <span><math><mi>n</mi></math></span>, and two vertices <span><math><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></math></span> are adjacent if and only if <span><math><mrow><mi>A</mi><msup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span> (if <span><math><mi>F</mi></math></span> is the real field, that is <span><math><mrow><mi>A</mi><msup><mrow><mi>B</mi></mrow><mrow><mi>T</mi></mrow></msup><mo>=</mo><mn>0</mn></mrow></math></span>). Note that <span><math><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 35-43"},"PeriodicalIF":1.0,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143161758","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 : 2025-01-18DOI: 10.1016/j.dam.2025.01.019
Hui Li, Yunshu Gao
<div><div>For a bipartite graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we use <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> to denote the maximal degree of the vertices in the sets <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in <span><math><mi>G</mi></math></span>, respectively. Let the tree <span><math><mrow><mi>T</mi><mrow><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> be an <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>-bipartite graph of order <span><math><mi>n</mi></math></span>. We prove that if <span><math><mrow><mrow><mo>|</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>|</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>, then there is a 2-packing <span><math><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span> of <span><math><mi>T</mi></math></span> in some complete bipartite graph <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> such that for each <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∪</mo><mi>τ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is at most <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. This result is sharp because <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span> cannot be reduced any further. By applying this result, we deduce a corollary for the existence of packing of three bipartite graphs into complete bi
{"title":"Packing two copies of a tree into a bipartite graph with restrained maximum degree","authors":"Hui Li, Yunshu Gao","doi":"10.1016/j.dam.2025.01.019","DOIUrl":"10.1016/j.dam.2025.01.019","url":null,"abstract":"<div><div>For a bipartite graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>, we use <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> to denote the maximal degree of the vertices in the sets <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in <span><math><mi>G</mi></math></span>, respectively. Let the tree <span><math><mrow><mi>T</mi><mrow><mo>(</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> be an <span><math><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></math></span>-bipartite graph of order <span><math><mi>n</mi></math></span>. We prove that if <span><math><mrow><mrow><mo>|</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo>|</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>, then there is a 2-packing <span><math><mrow><mo>(</mo><mi>σ</mi><mo>,</mo><mi>τ</mi><mo>)</mo></mrow></math></span> of <span><math><mi>T</mi></math></span> in some complete bipartite graph <span><math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> such that for each <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∪</mo><mi>τ</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is at most <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mspace></mspace><mrow><mo>(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. This result is sharp because <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span> cannot be reduced any further. By applying this result, we deduce a corollary for the existence of packing of three bipartite graphs into complete bi","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 23-34"},"PeriodicalIF":1.0,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160889","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 : 2025-01-18DOI: 10.1016/j.dam.2025.01.016
Zhipeng Gao , Rongling Lang , Changqing Xi , Jun Yue
Let be a positive integer and let be a graph. A vertex set is an -component dominating set of if every vertex outside has a neighbor in and every component of the subgraph induced by in contains at least vertices. The minimum cardinality of an -component dominating set of is the -component domination number of . Determining the exact values or bounds of domination parameters on graphs is an important, basic, and challenging problem in the graph domination field. The tree and the generalized Petersen graph with are the significant graph classes in graph theory. In this paper, we first give an upper bound of the 3-component domination number of a tree . Then, we study the -component domination numbers on and get the exact values of 3-component domination numbers on and .
{"title":"On 3-component domination numbers in graphs","authors":"Zhipeng Gao , Rongling Lang , Changqing Xi , Jun Yue","doi":"10.1016/j.dam.2025.01.016","DOIUrl":"10.1016/j.dam.2025.01.016","url":null,"abstract":"<div><div>Let <span><math><mi>s</mi></math></span> be a positive integer and let <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> be a graph. A vertex set <span><math><mi>D</mi></math></span> is an <span><math><mi>s</mi></math></span>-component dominating set of <span><math><mi>G</mi></math></span> if every vertex outside <span><math><mi>D</mi></math></span> has a neighbor in <span><math><mi>D</mi></math></span> and every component of the subgraph induced by <span><math><mi>D</mi></math></span> in <span><math><mi>G</mi></math></span> contains at least <span><math><mi>s</mi></math></span> vertices. The minimum cardinality of an <span><math><mi>s</mi></math></span>-component dominating set of <span><math><mi>G</mi></math></span> is the <span><math><mi>s</mi></math></span>-<em>component domination number</em> <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> of <span><math><mi>G</mi></math></span>. Determining the exact values or bounds of domination parameters on graphs is an important, basic, and challenging problem in the graph domination field. The tree <span><math><mi>T</mi></math></span> and the generalized Petersen graph <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></math></span> are the significant graph classes in graph theory. In this paper, we first give an upper bound of the 3-component domination number of a tree <span><math><mi>T</mi></math></span>. Then, we study the <span><math><mi>s</mi></math></span>-component domination numbers on <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> and get the exact values of 3-component domination numbers on <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"366 ","pages":"Pages 53-62"},"PeriodicalIF":1.0,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160550","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}
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