Pub Date : 2025-12-18DOI: 10.1016/j.dam.2025.12.016
Sasmita Rout , Gautam Kumar Das
Let be a simple undirected graph with no isolated vertex. A set is a dominating set if each vertex is either in or is adjacent to a vertex . A set is called a semi-total dominating set if is a dominating set, and for every vertex , there exists another vertex such that the distance between and in is at most 2. Given a graph , the semi-total domination problem finds a semi-total dominating set of minimum size. This problem is known to be NP-complete for general graphs and remains NP-complete for some special graph classes, such as planar, split, and chordal bipartite graphs. In this paper, we demonstrate that the problem is also NP-complete for unit disk graphs and propose a factor approximation algorithm. The algorithm’s running time is , where is the number of vertices in the given unit disk graph. In addition, we show that the minimum semi-total domination problem in a graph with maximum degree admits a factor approximation algorithm, which is an improvement over the best-known result .
{"title":"Semi-total domination in unit disk graphs and general graphs","authors":"Sasmita Rout , Gautam Kumar Das","doi":"10.1016/j.dam.2025.12.016","DOIUrl":"10.1016/j.dam.2025.12.016","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> be a simple undirected graph with no isolated vertex. A set <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> is a dominating set if each vertex <span><math><mrow><mi>u</mi><mo>∈</mo><mi>V</mi></mrow></math></span> is either in <span><math><mi>D</mi></math></span> or is adjacent to a vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>D</mi></mrow></math></span>. A set <span><math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mn>2</mn></mrow></msub><mo>⊆</mo><mi>V</mi></mrow></math></span> is called a semi-total dominating set if <span><math><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></math></span>\u0000 <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mn>2</mn></mrow></msub></math></span> is a dominating set, and <span><math><mrow><mo>(</mo><mi>i</mi><mi>i</mi><mo>)</mo></mrow></math></span> for every vertex <span><math><mrow><mi>u</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mn>2</mn></mrow></msub></mrow></math></span>, there exists another vertex <span><math><mrow><mi>v</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>t</mi><mn>2</mn></mrow></msub></mrow></math></span> such that the distance between <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span> in <span><math><mi>G</mi></math></span> is at most 2. Given a graph <span><math><mi>G</mi></math></span>, the semi-total domination problem finds a semi-total dominating set of minimum size. This problem is known to be NP-complete for general graphs and remains NP-complete for some special graph classes, such as planar, split, and chordal bipartite graphs. In this paper, we demonstrate that the problem is also NP-complete for unit disk graphs and propose a <span><math><mrow><mn>6</mn><mo>-</mo></mrow></math></span>factor approximation algorithm. The algorithm’s running time is <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>log</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>n</mi></math></span> is the number of vertices in the given unit disk graph. In addition, we show that the minimum semi-total domination problem in a graph with maximum degree <span><math><mi>Δ</mi></math></span> admits a <span><math><mrow><mn>2</mn><mo>+</mo><mo>ln</mo><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>-</mo></mrow></math></span>factor approximation algorithm, which is an improvement over the best-known result <span><math><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo>ln</mo><mrow><mo>(</mo><mi>Δ</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 358-367"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790230","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 : 2025-12-18DOI: 10.1016/j.dam.2025.12.036
Xiaodan Chen, Xiaoning Yang
The path cover number of a graph is the minimum integer such that contains vertex-disjoint paths that cover all of its vertices. In this paper, we first establish an upper bound on the path cover number for regular graphs. Then we apply this bound to help to derive sufficient conditions for a -tough graph to be Hamiltonian with integer , in terms of the edge number of the graph, which improve some known results in the literature. Another key tool we used to derive these sufficient conditions is the (complete) toughness closure lemma due to Hoàng and Robin (2024) and Shan and Tanyel (2025).
{"title":"An upper bound on path cover number of regular graphs and its application to Hamiltonian cycle in tough graphs","authors":"Xiaodan Chen, Xiaoning Yang","doi":"10.1016/j.dam.2025.12.036","DOIUrl":"10.1016/j.dam.2025.12.036","url":null,"abstract":"<div><div>The path cover number of a graph <span><math><mi>G</mi></math></span> is the minimum integer <span><math><mi>β</mi></math></span> such that <span><math><mi>G</mi></math></span> contains <span><math><mi>β</mi></math></span> vertex-disjoint paths that cover all of its vertices. In this paper, we first establish an upper bound on the path cover number for regular graphs. Then we apply this bound to help to derive sufficient conditions for a <span><math><mi>t</mi></math></span>-tough graph to be Hamiltonian with integer <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, in terms of the edge number of the graph, which improve some known results in the literature. Another key tool we used to derive these sufficient conditions is the (complete) toughness closure lemma due to Hoàng and Robin (2024) and Shan and Tanyel (2025).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 113-121"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792205","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 : 2025-12-18DOI: 10.1016/j.dam.2025.12.034
Feihu Liu
Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is palindromic, using Stanley’s reciprocity theorem. Furthermore, we introduce hypergraph polytopes and establish that every simple, connected, unimodular hypergraph polytope is an integer polytope. Additionally, for simple connected uniform hypergraph polytopes, we demonstrate that the numerator polynomial of their Ehrhart series is palindromic.
{"title":"Proof of a conjecture on graph polytope","authors":"Feihu Liu","doi":"10.1016/j.dam.2025.12.034","DOIUrl":"10.1016/j.dam.2025.12.034","url":null,"abstract":"<div><div>Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is palindromic, using Stanley’s reciprocity theorem. Furthermore, we introduce hypergraph polytopes and establish that every simple, connected, unimodular hypergraph polytope is an integer polytope. Additionally, for simple connected uniform hypergraph polytopes, we demonstrate that the numerator polynomial of their Ehrhart series is palindromic.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 122-129"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792199","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 : 2025-12-18DOI: 10.1016/j.dam.2025.12.025
Carl Feghali , Hoang-Oanh Le , Van Bang Le
This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. (2024). A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection-colorable if it admits a (proper) vertex -coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph and an integer , whether is strongly conflict-free vertex-connection -colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption , deciding, for a given graph , whether is strongly conflict-free vertex-connection 3-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal -coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.
{"title":"The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability","authors":"Carl Feghali , Hoang-Oanh Le , Van Bang Le","doi":"10.1016/j.dam.2025.12.025","DOIUrl":"10.1016/j.dam.2025.12.025","url":null,"abstract":"<div><div>This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. (2024). A path in a vertex-colored graph is called <em>conflict-free</em> if there is a color that appears exactly once on its vertices. A connected graph is said to be <em>strongly conflict-free vertex-connection</em> <span><math><mi>k</mi></math></span><em>-colorable</em> if it admits a (proper) vertex <span><math><mi>k</mi></math></span>-coloring such that any two distinct vertices are connected by a conflict-free <em>shortest</em> path. Among others, we show that deciding, for a given graph <span><math><mi>G</mi></math></span> and an integer <span><math><mi>k</mi></math></span>, whether <span><math><mi>G</mi></math></span> is strongly conflict-free vertex-connection <span><math><mi>k</mi></math></span>-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption <span><math><mrow><mi>NP</mi><mo>⁄</mo><mo>⊆</mo><mi>coNP/poly</mi></mrow></math></span>, deciding, for a given graph <span><math><mi>G</mi></math></span>, whether <span><math><mi>G</mi></math></span> is strongly conflict-free vertex-connection 3-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal <span><math><mi>k</mi></math></span>-<span>coloring</span> problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 85-93"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792201","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 : 2025-12-18DOI: 10.1016/j.dam.2025.12.032
Yu-Yue Zhang, Jian-Hua Yin
<div><div>The generalized Turán number <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of copies of a complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> in any <span><math><mi>H</mi></math></span>-free graph on <span><math><mi>n</mi></math></span> vertices. Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denote the star on <span><math><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices, and let <span><math><mrow><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math></span> denote the disjoint union of <span><math><mi>k</mi></math></span> copies of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. Gan et al. and Chase determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all integers <span><math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Recently, Liu and Yin further investigated the problem of determining the generalized Turán number of star forests. They determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>5</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Moreover, they also determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>4</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1<
广义的Turán数ex（n,Ks，H）被定义为在任意n个顶点上的无H图中完全图k的最大拷贝数。设sz表示在1个顶点上的星，kz表示sz的k个拷贝的不相交并。Gan et al.和Chase确定了所有整数S≥3、r≥1和n≥1的ex（n,Ks，S）。最近，Liu和Yin进一步研究了确定广义Turán星林数的问题。当s≥4、r≥1、n≥1时确定ex(n,Ks,2S)，当s≥5、r≥1、n≥1时确定ex（n,Ks,3S）。此外，他们还确定了s≥6，r≥1和n≥1时的ex（n,Ks,4S）。然而，确定2≤k≤4和3≤s≤k+1的ex（n,Ks, Ks）的问题似乎是困难和具有挑战性的。本文研究了上述问题，并确定了当s=3时ex（n,Ks,2S）和当3≤s≤4时ex（n,Ks,3S）。此外，我们还确定了3≤s≤5时ex（n,Ks,4S）。
{"title":"A note on the generalized Turán number of star forests","authors":"Yu-Yue Zhang, Jian-Hua Yin","doi":"10.1016/j.dam.2025.12.032","DOIUrl":"10.1016/j.dam.2025.12.032","url":null,"abstract":"<div><div>The generalized Turán number <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of copies of a complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> in any <span><math><mi>H</mi></math></span>-free graph on <span><math><mi>n</mi></math></span> vertices. Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denote the star on <span><math><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices, and let <span><math><mrow><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math></span> denote the disjoint union of <span><math><mi>k</mi></math></span> copies of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span>. Gan et al. and Chase determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for all integers <span><math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Recently, Liu and Yin further investigated the problem of determining the generalized Turán number of star forests. They determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>4</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>5</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Moreover, they also determined <span><math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>4</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>s</mi><mo>≥</mo><mn>6</mn></mrow></math></span>, <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>1<","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 94-112"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792204","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 : 2025-12-18DOI: 10.1016/j.dam.2025.12.007
Yanbo Zhang , Yaojun Chen , Yunqing Zhang
Given two graphs and , the Ramsey number is the smallest positive integer such that every graph on vertices contains as a subgraph or its complement contains as a subgraph. Let denote a tree on vertices, and let denote a wheel on vertices. Chen et al. (2004) asked for a characterization of those trees satisfying for and . Subsequently, Hafidh and Baskoro (2021) conjectured that if is a tree with maximum degree at most , then for and . More recently, Britz et al. (2025) verified this conjecture partially for and large .
In this note, we resolve the conjecture of Hafidh and Baskoro for all large .
{"title":"On tree–wheel Ramsey numbers","authors":"Yanbo Zhang , Yaojun Chen , Yunqing Zhang","doi":"10.1016/j.dam.2025.12.007","DOIUrl":"10.1016/j.dam.2025.12.007","url":null,"abstract":"<div><div>Given two graphs <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, the Ramsey number <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the smallest positive integer <span><math><mi>r</mi></math></span> such that every graph on <span><math><mi>r</mi></math></span> vertices contains <span><math><mi>G</mi></math></span> as a subgraph or its complement contains <span><math><mi>H</mi></math></span> as a subgraph. Let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote a tree on <span><math><mi>n</mi></math></span> vertices, and let <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> denote a wheel on <span><math><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices. Chen et al. (2004) asked for a characterization of those trees <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfying <span><math><mrow><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> for <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>. Subsequently, Hafidh and Baskoro (2021) conjectured that if <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a tree with maximum degree at most <span><math><mrow><mi>n</mi><mo>−</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>2</mn></mrow></math></span>, then <span><math><mrow><mi>R</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> for <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></math></span>. More recently, Britz et al. (2025) verified this conjecture partially for <span><math><mrow><mi>m</mi><mo>=</mo><mn>4</mn></mrow></math></span> and large <span><math><mi>n</mi></math></span>.</div><div>In this note, we resolve the conjecture of Hafidh and Baskoro for all large <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 80-84"},"PeriodicalIF":1.0,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792203","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 : 2025-12-17DOI: 10.1016/j.dam.2025.11.036
Davoud Foroutannia , Davoud Mahmoudinia
In this paper, we examine coalitional semi-cooperative games in strategic form, where players within each coalition adhere to the equitable preference relation. Equitable preference is a refinement of the Pareto preference relation. It adds to the reflexivity, strict monotonicity, and transitivity of the Pareto preference order the requirements of impartiality and the Pigou–Dalton principle of transfers. We introduce the concept of -equitable, in which is a complete set of coalitions. By generating a finite subset of Pareto optimal solutions, this concept can help decision-makers select the most suitable solution, especially when there is a large number of Pareto equilibria in some real situations.
A scalar equilibrium achieved by maximizing an appropriate utility function over the acceptable joint actions may not be -equitable in many instances. Another goal of this paper is to generate the -equitable equilibria using utility functions. To achieve this, we require the aggregate utility functions to be Schur-convex, in addition to assuming strict increases, to ensure that each scalar equilibrium is a -equitable equilibrium.
The greedy scalar equilibrium is one of the well-known scalar equilibria that assigns actions to players that yield the largest individual payoffs jointly possible. Our final concept is -equitable greedy scalar equilibrium, which incorporates the notions of equitability and greed to select the optimal set of coalitions for a semi-cooperative game.
{"title":"Equitable greedy scalar equilibria for coalitional semi-cooperative games","authors":"Davoud Foroutannia , Davoud Mahmoudinia","doi":"10.1016/j.dam.2025.11.036","DOIUrl":"10.1016/j.dam.2025.11.036","url":null,"abstract":"<div><div>In this paper, we examine coalitional semi-cooperative games in strategic form, where players within each coalition adhere to the equitable preference relation. Equitable preference is a refinement of the Pareto preference relation. It adds to the reflexivity, strict monotonicity, and transitivity of the Pareto preference order the requirements of impartiality and the Pigou–Dalton principle of transfers. We introduce the concept of <span><math><mi>C</mi></math></span>-equitable, in which <span><math><mi>C</mi></math></span> is a complete set of coalitions. By generating a finite subset of Pareto optimal solutions, this concept can help decision-makers select the most suitable solution, especially when there is a large number of Pareto equilibria in some real situations.</div><div>A scalar equilibrium achieved by maximizing an appropriate utility function over the acceptable joint actions may not be <span><math><mi>C</mi></math></span>-equitable in many instances. Another goal of this paper is to generate the <span><math><mi>C</mi></math></span>-equitable equilibria using utility functions. To achieve this, we require the aggregate utility functions to be Schur-convex, in addition to assuming strict increases, to ensure that each scalar equilibrium is a <span><math><mi>C</mi></math></span>-equitable equilibrium.</div><div>The greedy scalar equilibrium is one of the well-known scalar equilibria that assigns actions to players that yield the largest individual payoffs jointly possible. Our final concept is <span><math><mi>C</mi></math></span>-equitable greedy scalar equilibrium, which incorporates the notions of equitability and greed to select the optimal set of coalitions for a semi-cooperative game.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 52-64"},"PeriodicalIF":1.0,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792200","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 : 2025-12-17DOI: 10.1016/j.dam.2025.12.020
Wenjuan Wei, Min Chen
An -dynamic coloring of a graph is a proper vertex coloring such that for every vertex , at least distinct colors appear in its neighborhood . Given a list assignment , if there exists an -dynamic coloring of such that for every vertex , then we say that is -dynamically -colorable. If is -dynamically -colorable for every list assignment with for every vertex , then is called to be -dynamically -choosable. It is known that every planar graph is 3-dynamically 10-choosable and it has been conjectured that they are 3-dynamically 8-choosable. In this paper, we sure prove that every planar graph is 3-dynamically 9-choosable.
{"title":"List 3-dynamic colorings of planar graphs","authors":"Wenjuan Wei, Min Chen","doi":"10.1016/j.dam.2025.12.020","DOIUrl":"10.1016/j.dam.2025.12.020","url":null,"abstract":"<div><div>An <span><math><mi>r</mi></math></span>-dynamic coloring of a graph <span><math><mi>G</mi></math></span> is a proper vertex coloring such that for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, at least <span><math><mrow><mo>min</mo><mrow><mo>{</mo><mi>r</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span> distinct colors appear in its neighborhood <span><math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. Given a list assignment <span><math><mrow><mi>L</mi><mo>=</mo><mrow><mo>{</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∣</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>, if there exists an <span><math><mi>r</mi></math></span>-dynamic coloring <span><math><mi>π</mi></math></span> of <span><math><mi>G</mi></math></span> such that <span><math><mrow><mi>π</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, then we say that <span><math><mi>G</mi></math></span> is <span><math><mi>r</mi></math></span>-dynamically <span><math><mi>L</mi></math></span>-colorable. If <span><math><mi>G</mi></math></span> is <span><math><mi>r</mi></math></span>-dynamically <span><math><mi>L</mi></math></span>-colorable for every list assignment <span><math><mi>L</mi></math></span> with <span><math><mrow><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mi>k</mi></mrow></math></span> for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, then <span><math><mi>G</mi></math></span> is called to be <span><math><mi>r</mi></math></span>-dynamically <span><math><mi>k</mi></math></span>-choosable. It is known that every planar graph is 3-dynamically 10-choosable and it has been conjectured that they are 3-dynamically 8-choosable. In this paper, we sure prove that every planar graph is 3-dynamically 9-choosable.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"381 ","pages":"Pages 349-357"},"PeriodicalIF":1.0,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790229","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 : 2025-12-17DOI: 10.1016/j.dam.2025.12.030
Colin Cooper, Tomasz Radzik
Given a graph , and an ordering of its vertices, a permutation forest is a spanning forest of whose components are obtained as follows. For each vertex , connect to its first neighbour in that appears after in the ordering . If we regard this edge as directed forward, from to , then each vertex has at most one forward edge, and the components of are arborescences. The roots of the components formed by this process are those vertices of with no forward edge in the ordering .
This paper shows that the permutation forests of the random graphs have a threshold for the emergence of a linear size component around . In contrast to the w.h.p. emergence of a unique giant in , the permutation forest process has the unusual property that, with positive probability, a number of linear size components occur within the threshold window.
{"title":"A random forest process with a variable number of giant components in the threshold window","authors":"Colin Cooper, Tomasz Radzik","doi":"10.1016/j.dam.2025.12.030","DOIUrl":"10.1016/j.dam.2025.12.030","url":null,"abstract":"<div><div>Given a graph <span><math><mi>G</mi></math></span>, and an ordering <span><math><mi>π</mi></math></span> of its vertices, a permutation forest <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>π</mi><mo>)</mo></mrow></mrow></math></span> is a spanning forest of <span><math><mi>G</mi></math></span> whose components are obtained as follows. For each vertex <span><math><mi>v</mi></math></span>, connect <span><math><mi>v</mi></math></span> to its first neighbour <span><math><mi>w</mi></math></span> in <span><math><mi>G</mi></math></span> that appears after <span><math><mi>v</mi></math></span> in the ordering <span><math><mi>π</mi></math></span>. If we regard this edge <span><math><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></math></span> as directed forward, from <span><math><mi>v</mi></math></span> to <span><math><mi>w</mi></math></span>, then each vertex has at most one forward edge, and the components of <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>π</mi><mo>)</mo></mrow></mrow></math></span> are arborescences. The roots of the components formed by this process are those vertices of <span><math><mi>G</mi></math></span> with no forward edge in the ordering <span><math><mi>π</mi></math></span>.</div><div>This paper shows that the permutation forests of the random graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow></msub></math></span> have a threshold for the emergence of a linear size component around <span><math><mrow><mi>p</mi><mo>=</mo><msqrt><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msqrt></mrow></math></span>. In contrast to the w.h.p. emergence of a unique giant in <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow></msub></math></span>, the permutation forest process has the unusual property that, with positive probability, a number of linear size components occur within the threshold window.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 65-79"},"PeriodicalIF":1.0,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792202","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 : 2025-12-16DOI: 10.1016/j.dam.2025.12.024
Gennaro Cordasco, Luisa Gargano, Adele A. Rescigno
This paper explores a recently introduced graph problem called ()-Broadcast Domination. In this problem, a set of “broadcasting” towers transmit a signal of initial strength . This signal’s strength decreases linearly along the edges. The objective is to find the smallest set of broadcasting towers such that every vertex in the graph receives a cumulative signal strength of at least . The ()-Broadcast Domination problem, which generalizes the concept of distance domination, has been recently introduced and studied in a few specific class of graphs, like grids and lattices. However, the ()-Broadcast Domination problem has not been studied for general graphs. We present the first study of the complexity of this problem in general graphs, including a general approximation algorithm and optimal polynomial-time algorithms for cographs. We also provide an algorithm parameterized by the neighborhood diversity of the input graph and an algorithm parameterized by modular-width and the solution size.
{"title":"(t,r)-Broadcast Domination in graphs","authors":"Gennaro Cordasco, Luisa Gargano, Adele A. Rescigno","doi":"10.1016/j.dam.2025.12.024","DOIUrl":"10.1016/j.dam.2025.12.024","url":null,"abstract":"<div><div>This paper explores a recently introduced graph problem called (<span><math><mrow><mi>t</mi><mo>,</mo><mi>r</mi></mrow></math></span>)-<span>Broadcast Domination</span>. In this problem, a set of “broadcasting” towers transmit a signal of initial strength <span><math><mi>t</mi></math></span>. This signal’s strength decreases linearly along the edges. The objective is to find the smallest set of broadcasting towers such that every vertex in the graph receives a cumulative signal strength of at least <span><math><mi>r</mi></math></span>. The (<span><math><mrow><mi>t</mi><mo>,</mo><mi>r</mi></mrow></math></span>)-<span>Broadcast Domination</span> problem, which generalizes the concept of distance domination, has been recently introduced and studied in a few specific class of graphs, like grids and lattices. However, the (<span><math><mrow><mi>t</mi><mo>,</mo><mi>r</mi></mrow></math></span>)-<span>Broadcast Domination</span> problem has not been studied for general graphs. We present the first study of the complexity of this problem in general graphs, including a general approximation algorithm and optimal polynomial-time algorithms for cographs. We also provide an algorithm parameterized by the neighborhood diversity of the input graph and an algorithm parameterized by modular-width and the solution size.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"382 ","pages":"Pages 355-367"},"PeriodicalIF":1.0,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790315","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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