Pub Date : 2026-04-15Epub Date: 2025-12-15DOI: 10.1016/j.dam.2025.12.017
Xuelian Mao, Zhenyu Ni, Ming-Zhu Chen
For a set of graphs , a graph is called -free if it does not contain any graph in as a subgraph. Let SPEX denote the graphs with the maximum spectral radius among all -free graphs of order . In this paper, for any non-bipartite graph , we give some characterizations for the graphs in SPEX for sufficiently large , where , , and there exists at least one not equal to 3. As an application, we completely characterize the graphs in SPEX and SPEX for sufficiently large , where is a friendship graph on vertices consisting of triangles which intersect in exactly one common vertex.
对于一组图F，如果图G不包含F中的任何图作为子图，则称为F自由图。设SPEX（n，F）表示在所有n阶的F-free图中具有最大谱半径的图。在本文中，对于任意非二部图H，我们给出了对于足够大的n，其中，r≥1,k≥1,k≥2，且存在至少一个ki不等于3的SPEX（n,{∈i=1∑Pki,H}）中的图的一些刻画。作为一个应用，对于足够大的n，我们完全刻画了SPEX（n,{∈i=1 l Pki,Kk+1}）和SPEX（n,{∈i=1 l Pki,Fk}）中的图，其中Fk是由k个恰好相交于一个公共顶点的k个三角形组成的2k+1个顶点的友谊图。
{"title":"Spectral extrema of graphs: Forbidden linear forests and non-bipartite graphs","authors":"Xuelian Mao, Zhenyu Ni, Ming-Zhu Chen","doi":"10.1016/j.dam.2025.12.017","DOIUrl":"10.1016/j.dam.2025.12.017","url":null,"abstract":"<div><div>For a set of graphs <span><math><mi>F</mi></math></span>, a graph <span><math><mi>G</mi></math></span> is called <span><math><mi>F</mi></math></span>-free if it does not contain any graph in <span><math><mi>F</mi></math></span> as a subgraph. Let SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></math></span> denote the graphs with the maximum spectral radius among all <span><math><mi>F</mi></math></span>-free graphs of order <span><math><mi>n</mi></math></span>. In this paper, for any non-bipartite graph <span><math><mi>H</mi></math></span>, we give some characterizations for the graphs in SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><mi>H</mi><mo>}</mo></mrow><mo>)</mo></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>, where <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>≥</mo><mn>2</mn></mrow></math></span>, and there exists at least one <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> not equal to 3. As an application, we completely characterize the graphs in SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></math></span> and SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a friendship graph on <span><math><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices consisting of <span><math><mi>k</mi></math></span> triangles which intersect in exactly one common vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 8-14"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748178","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 : 2026-04-15Epub Date: 2025-12-29DOI: 10.1016/j.dam.2025.12.043
Yong-De Feng, Yawen Chen, Baoyindureng Wu
An odd graph is a graph for which every vertex satisfies . An odd spanning tree of is a spanning tree such that for all . It is known that for any complete graph of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in . By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in is given by (where is even).
{"title":"The number of odd spanning trees in the complete graphs","authors":"Yong-De Feng, Yawen Chen, Baoyindureng Wu","doi":"10.1016/j.dam.2025.12.043","DOIUrl":"10.1016/j.dam.2025.12.043","url":null,"abstract":"<div><div>An odd graph is a graph <span><math><mi>G</mi></math></span> for which 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> satisfies <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. An odd spanning tree <span><math><mi>T</mi></math></span> of <span><math><mi>G</mi></math></span> is a spanning tree such that <span><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>T</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>≡</mo><mn>1</mn><mspace></mspace><mrow><mo>(</mo><mo>mod</mo><mspace></mspace><mn>2</mn><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>. It is known that for any complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of even order has an odd spanning tree. In this paper, we establish the exact number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. By employing the classical Prüfer sequence and constructing the corresponding generating function, we prove that the number of labeled odd spanning trees in <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is given by <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced><msup><mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span>\u0000 <!--> <!-->(where <span><math><mi>n</mi></math></span> is even).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 295-299"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885019","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 : 2026-04-15Epub Date: 2026-01-13DOI: 10.1016/j.dam.2025.12.067
Harmender Gahlawat
Cops and Robber is a game played on graphs where a set of cops aim to capture the position of a single robber. The main parameter of interest in this game is the cop number, which is the minimum number of cops that are sufficient to guarantee the capture of the robber.
In a directed graph , the push operation on a vertex reverses the orientation of all arcs incident to . We consider a variation of the classical Cops and Robber on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an out-neighbor in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if is an orientation of a 3-degenerate graph, or of a graph with maximum degree 4, then one cop with push ability has a winning strategy. Moreover, we establish that if can be made to be a directed acyclic graph, then one cop with push ability has a winning strategy.
《Cops and robbers》是一款基于图表的游戏，其中一组警察的目标是抓住一名抢劫犯的位置。在这个游戏中最重要的参数是警察数量，即能够保证抓住抢劫犯的最少警察数量。在有向图G - l - l中，顶点v上的推操作反转了所有与v相关的弧线的方向。我们考虑有向图上经典cop和抢劫者的一种变体，其中每个cop可以移动到其当前顶点的外邻居或推图的某个顶点，而抢劫者可以移动到其外邻居。[Das et al.， CALDAM， 2023]引入了这种变体，并建立了如果G / l是亚立方图的一个方向，那么具有推送能力的一方具有获胜策略。我们扩展了这些结果来证明如果G - l是一个3-简并图的一个方向，或者是一个最大度为4的图的一个方向，那么一个具有推能力的cop有一个获胜策略。此外，我们还证明了如果G / l可以构成一个有向无环图，那么一个具有推能力的cop有一个获胜策略。
{"title":"Pushing Cops and Robber on graphs of maximum degree four","authors":"Harmender Gahlawat","doi":"10.1016/j.dam.2025.12.067","DOIUrl":"10.1016/j.dam.2025.12.067","url":null,"abstract":"<div><div><span>Cops and Robber</span> is a game played on graphs where a set of <em>cops</em> aim to <em>capture</em> the position of a single <em>robber</em>. The main parameter of interest in this game is the <em>cop number</em>, which is the minimum number of cops that are sufficient to guarantee the capture of the robber.</div><div>In a directed graph <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span>, the <em>push</em> operation on a vertex <span><math><mi>v</mi></math></span> reverses the orientation of all arcs incident to <span><math><mi>v</mi></math></span>. We consider a variation of the classical <span>Cops and Robber</span> on oriented graphs, where in its turn, each cop can either move to an out-neighbor of its current vertex or push some vertex of the graph, whereas, the robber can move to an out-neighbor in its turn. [Das et al., CALDAM, 2023] introduced this variant and established that if <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> is an orientation of a subcubic graph, then one cop with push ability has a winning strategy. We extend these results to establish that if <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> is an orientation of a 3-degenerate graph, or of a graph with maximum degree 4, then one cop with push ability has a winning strategy. Moreover, we establish that if <span><math><mover><mrow><mi>G</mi></mrow><mo>⃗</mo></mover></math></span> can be made to be a directed acyclic graph, then one cop with push ability has a winning strategy.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 327-338"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977708","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 : 2026-04-15Epub Date: 2025-12-22DOI: 10.1016/j.dam.2025.12.028
Jaroslav Garvardt , Christian Komusiewicz, Nils Morawietz
In the NP-hard Weighted Cluster Deletion problem, the input is an undirected graph and an edge-weight function , and the task is to partition the vertex set into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that Weighted Cluster Deletion is NP-hard on some graph classes where Cluster Deletion, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value of the largest edge weight assigned by on the problem complexity for such graph classes. Our main results are that Weighted Cluster Deletion is fixed-parameter tractable with respect to on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for Weighted Cluster Deletion. For example, we show that Weighted Cluster Deletion is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.
{"title":"When can Cluster Deletion with bounded weights be solved efficiently?","authors":"Jaroslav Garvardt , Christian Komusiewicz, Nils Morawietz","doi":"10.1016/j.dam.2025.12.028","DOIUrl":"10.1016/j.dam.2025.12.028","url":null,"abstract":"<div><div>In the NP-hard <span>Weighted Cluster Deletion</span> problem, the input is an undirected graph <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> and an edge-weight function <span><math><mrow><mi>ω</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>N</mi></mrow></math></span>, and the task is to partition the vertex set <span><math><mi>V</mi></math></span> into cliques so that the total weight of edges in the cliques is maximized. Recently, it has been shown that <span>Weighted Cluster Deletion</span> is NP-hard on some graph classes where <span>Cluster Deletion</span>, the special case where every edge has unit weight, can be solved in polynomial time. We study the influence of the value <span><math><mi>t</mi></math></span> of the largest edge weight assigned by <span><math><mi>ω</mi></math></span> on the problem complexity for such graph classes. Our main results are that <span>Weighted Cluster Deletion</span> is fixed-parameter tractable with respect to <span><math><mi>t</mi></math></span> on graph classes whose graphs consist of well-separated clusters that are connected by a sparse periphery. Concrete examples for such classes are split graphs and graphs that are close to cluster graphs. We complement our results by strengthening previous hardness results for <span>Weighted Cluster Deletion</span>. For example, we show that <span>Weighted Cluster Deletion</span> is NP-hard on restricted subclasses of cographs even when every edge has weight 1 or 2.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 169-183"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842056","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 : 2026-04-15Epub Date: 2025-12-22DOI: 10.1016/j.dam.2025.12.026
Stefan Kratsch , Van Bang Le
A stable cutset in a graph is a set such that vertices of are pairwise non-adjacent and such that is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is -complete to determine whether a given graph has any stable cutset. Recently, Rauch et al. [FCT 2023 & JCSS 2025] gave a number of fixed-parameter tractable (FPT) algorithms, running in time , for Stable Cutset under a variety of parameters such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to -free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size.
We complement these findings by giving the first results on the existence of polynomial kernelizations for Stable Cutset, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that , we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given answering whether there is a stable cutset of size at most . On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.
{"title":"On polynomial kernelization for Stable Cutset","authors":"Stefan Kratsch , Van Bang Le","doi":"10.1016/j.dam.2025.12.026","DOIUrl":"10.1016/j.dam.2025.12.026","url":null,"abstract":"<div><div>A <em>stable cutset</em> in a graph <span><math><mi>G</mi></math></span> is a set <span><math><mrow><mi>S</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that vertices of <span><math><mi>S</mi></math></span> are pairwise non-adjacent and such that <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> is disconnected, i.e., it is both stable (or independent) set and a cutset (or separator). Unlike general cutsets, it is <span><math><mi>NP</mi></math></span>-complete to determine whether a given graph <span><math><mi>G</mi></math></span> has any stable cutset. Recently, Rauch et al. [FCT 2023 & JCSS 2025] gave a number of fixed-parameter tractable (FPT) algorithms, running in time <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><mi>⋅</mi><msup><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow><mrow><mi>c</mi></mrow></msup></mrow></math></span>, for <span>Stable Cutset</span> under a variety of parameters <span><math><mi>k</mi></math></span> such as the size of a (given) dominating set, the size of an odd cycle transversal, or the deletion distance to <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graphs. Earlier works imply FPT algorithms relative to clique-width and relative to solution size.</div><div>We complement these findings by giving the first results on the existence of polynomial kernelizations for <span>Stable Cutset</span>, i.e., efficient preprocessing algorithms that return an equivalent instance of size polynomial in the parameter value. Under the standard assumption that <span><math><mi>NP ⊈ coNP/poly</mi></math></span>, we show that no polynomial kernelization is possible relative to the deletion distance to a single path, generalizing deletion distance to various graph classes, nor by the size of a (given) dominating set. We also show that under the same assumption no polynomial kernelization is possible relative to solution size, i.e., given <span><math><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span> answering whether there is a stable cutset of size at most <span><math><mi>k</mi></math></span>. On the positive side, we show polynomial kernelizations for parameterization by modulators to a single clique, to a cluster or a co-cluster graph, and by twin cover.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 184-202"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842073","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 : 2026-04-15Epub 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":"2026-04-15","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 : 2026-04-15Epub Date: 2025-12-19DOI: 10.1016/j.dam.2025.12.029
Vadim Lozin
We study properties of graph classes that are closed under taking subclasses, such as boundedness of graph parameters or polynomial-time solvability of algorithmic problems. In the universe of minor-closed classes of graphs, any such property can be described by a set of minimal classes that do not possess the property, because the minor relation is a well-quasi-order. This, however, is not the case for the subgraph relation, implying that in the universe of monotone classes, which extends the family of minor-closed classes, the existence of minimal classes is not guaranteed. To overcome this difficulty, we employ the notion of boundary classes. Together with minimal classes they play a critical role for classes defined by finitely many forbidden subgraphs. In the present paper, we identify several levels in the hierarchy of monotone classes and describe respective critical classes. In particular, we show that a finitely-defined monotone class has bounded chromatic number, degeneracy, functionality and admits an implicit representation if and only if excludes a forest. We also show that has bounded tree-, clique- and twin-width and admits polynomial-time solutions for a variety of algorithmic problems if and only if excludes a tripod, i.e. a subcubic forest every connected component of which has at most one cubic vertex. The last result, however, does not apply to the Hamiltonian cycle problem. Towards identifying critical classes for this problem we determine complexity of the Hamiltonian cycle problem in some monotone classes.
{"title":"Graph problems and monotone classes","authors":"Vadim Lozin","doi":"10.1016/j.dam.2025.12.029","DOIUrl":"10.1016/j.dam.2025.12.029","url":null,"abstract":"<div><div>We study properties of graph classes that are closed under taking subclasses, such as boundedness of graph parameters or polynomial-time solvability of algorithmic problems. In the universe of minor-closed classes of graphs, any such property can be described by a set of minimal classes that do not possess the property, because the minor relation is a well-quasi-order. This, however, is not the case for the subgraph relation, implying that in the universe of monotone classes, which extends the family of minor-closed classes, the existence of minimal classes is not guaranteed. To overcome this difficulty, we employ the notion of boundary classes. Together with minimal classes they play a critical role for classes defined by finitely many forbidden subgraphs. In the present paper, we identify several levels in the hierarchy of monotone classes and describe respective critical classes. In particular, we show that a finitely-defined monotone class <span><math><mi>X</mi></math></span> has bounded chromatic number, degeneracy, functionality and admits an implicit representation if and only if <span><math><mi>X</mi></math></span> excludes a forest. We also show that <span><math><mi>X</mi></math></span> has bounded tree-, clique- and twin-width and admits polynomial-time solutions for a variety of algorithmic problems if and only if <span><math><mi>X</mi></math></span> excludes a tripod, i.e. a subcubic forest every connected component of which has at most one cubic vertex. The last result, however, does not apply to the Hamiltonian cycle problem. Towards identifying critical classes for this problem we determine complexity of the Hamiltonian cycle problem in some monotone classes.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 152-164"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792197","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}
Csáji, Jungers, and Blondel prove that while a PageRank optimization problem with edge selection constraints is NP-hard, it can be solved optimally in polynomial time for the unconstrained case. This theoretical result is accompanied by several observations, which we leverage to develop valid inequalities in polynomial time for this class of NP-hard problems. We show that these observations can be exploited to derive stronger inequalities than the standard valid inequality available in the literature. These valid inequalities provide a theoretical basis for reducing the optimality gap of the constrained PageRank problem without changing NP-hardness.
{"title":"A note on valid inequalities for PageRank optimization with edge selection constraints","authors":"Shang-Ru Yang, Yung-Han Liao, Chih-Ching Chien, Hao-Hsiang Wu","doi":"10.1016/j.dam.2025.12.023","DOIUrl":"10.1016/j.dam.2025.12.023","url":null,"abstract":"<div><div>Csáji, Jungers, and Blondel prove that while a PageRank optimization problem with edge selection constraints is NP-hard, it can be solved optimally in polynomial time for the unconstrained case. This theoretical result is accompanied by several observations, which we leverage to develop valid inequalities in polynomial time for this class of NP-hard problems. We show that these observations can be exploited to derive stronger inequalities than the standard valid inequality available in the literature. These valid inequalities provide a theoretical basis for reducing the optimality gap of the constrained PageRank problem without changing NP-hardness.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 44-51"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792198","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 : 2026-04-15Epub 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":"2026-04-15","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}
In this work, assuming that preferences are asymmetric, complete and transitive, we aim to prove the equivalence, in bilateral conflicts, between the concepts of Maximin stability and Generalized Metarationality within the graph model for conflict resolution (GMCR). This result is surprising, as these concepts are apparently distinct, due to the movements of the opponent of the focal DM, in the Generalized Metarationality, be limited by the use of a policy. To achieve our goal, we define a Maximin tree to build maximin policies from the choices made by the focal DM opponent in this tree. We show that two kinds of maximin policies are necessary: one for conflict analysis with odd horizon and another for even horizon conflict analysis. This result, in addition to improving our understanding of these concepts, facilitates the determination of states that satisfy generalized metarationality, since maximin states can be determined by means of a backward induction procedure.
{"title":"On the relation between Maximinh stability and Generalized Metarationality in bilateral conflicts","authors":"Alecio Soares Silva , Giannini Italino Alves Vieira , Leandro Chaves Rêgo","doi":"10.1016/j.dam.2025.12.041","DOIUrl":"10.1016/j.dam.2025.12.041","url":null,"abstract":"<div><div>In this work, assuming that preferences are asymmetric, complete and transitive, we aim to prove the equivalence, in bilateral conflicts, between the concepts of Maximin<span><math><msub><mrow></mrow><mrow><mi>h</mi></mrow></msub></math></span> stability and Generalized Metarationality within the graph model for conflict resolution (GMCR). This result is surprising, as these concepts are apparently distinct, due to the movements of the opponent of the focal DM, in the Generalized Metarationality, be limited by the use of a policy. To achieve our goal, we define a Maximin<span><math><msub><mrow></mrow><mrow><mi>h</mi></mrow></msub></math></span> tree to build maximin policies from the choices made by the focal DM opponent in this tree. We show that two kinds of maximin policies are necessary: one for conflict analysis with odd horizon and another for even horizon conflict analysis. This result, in addition to improving our understanding of these concepts, facilitates the determination of states that satisfy generalized metarationality, since maximin states can be determined by means of a backward induction procedure.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 280-294"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842077","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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