Gary R. W. Greaves, Jeven Syatriadi, Charissa I. Utomo
We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.
{"title":"Chromatic Polynomials of Signed Graphs and Dominating-Vertex Deletion Formulae","authors":"Gary R. W. Greaves, Jeven Syatriadi, Charissa I. Utomo","doi":"10.1002/jgt.23236","DOIUrl":"https://doi.org/10.1002/jgt.23236","url":null,"abstract":"<div>\u0000 \u0000 <p>We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"101-110"},"PeriodicalIF":0.9,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624857","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}
Jiangdong Ai, Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou
<p>An oriented graph <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> </semantics></math> is <i>converse invariant</i> if, for any tournament <span></span><math> <semantics> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </semantics></math>, the number of copies of <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> </semantics></math> in <span></span><math> <semantics> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </semantics></math> is equal to that of its converse <span></span><math> <semantics> <mrow> <mrow> <mo>−</mo> <mi>D</mi> </mrow> </mrow> </semantics></math>. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> </semantics></math> with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if <span></span><math> <semantics> <mrow> <mrow> <mi>D</mi> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mo>−</mo> <mi>D</mi> </mrow> </mrow> </semantics></math> have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, w
{"title":"Number of Subgraphs and Their Converses in Tournaments and New Digraph Polynomials","authors":"Jiangdong Ai, Gregory Gutin, Hui Lei, Anders Yeo, Yacong Zhou","doi":"10.1002/jgt.23257","DOIUrl":"https://doi.org/10.1002/jgt.23257","url":null,"abstract":"<p>An oriented graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is <i>converse invariant</i> if, for any tournament <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, the number of copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is equal to that of its converse <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>−</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>−</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, w","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 2","pages":"127-131"},"PeriodicalIF":1.0,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23257","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810990","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}
<p>An <i>inversion</i> of a tournament <span></span><math> <semantics> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </semantics></math> is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let <span></span><math> <semantics> <mrow> <mrow> <msub> <mtext>inv</mtext> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> be the minimum length of a sequence of inversions using sets of size at most <span></span><math> <semantics> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </semantics></math> that result in the transitive tournament. Let <span></span><math> <semantics> <mrow> <mrow> <msub> <mtext>inv</mtext> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> be the maximum of <span></span><math> <semantics> <mrow> <mrow> <msub> <mtext>inv</mtext> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </mrow> </semantics></math> taken over <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math>-vertex tournaments. It is well k
一个锦标赛T的反转是通过反转所有边的方向，在一些顶点集合中有两个端点。设inv k (T)为最小长度使用大小不超过k的集合的反转序列，从而导致传递比武。设inv k (n)是的最大值invk (T)代入N顶点锦标赛。众所周知，inv2 (n) =（1 + 0 (1)）) n∕4，最近由Alon等人证明Inv (n)是对象的集合(n) = n (1)+ 0 (1) 一个锦标赛T的反转是通过反转所有边的方向，在一些顶点集合中有两个端点。设inv k (T)为最小长度使用大小不超过k的集合的反转序列，从而导致传递比武。设inv k (n)是的最大值invk (T)代入N顶点锦标赛。众所周知，inv2 (n) =（1 + 0 (1)）) n∕4，最近由Alon等人证明。 其中，inv (n)是对的(n) = n (1)+ 0 (1))。在这两种极端情况下（k = 2和n），随机比赛是极端目标。证明了invk (n)不是当k≥k 0时，通过随机比赛得到，并推测Inv 3 (n)只能通过（准）随机比赛获得。 进一步证明了（1 + 0）(1); (3)N)∕N 2∈[1,12,0。
{"title":"On Tournament Inversion","authors":"Raphael Yuster","doi":"10.1002/jgt.23251","DOIUrl":"https://doi.org/10.1002/jgt.23251","url":null,"abstract":"<p>An <i>inversion</i> of a tournament <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>inv</mtext>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>T</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be the minimum length of a sequence of inversions using sets of size at most <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> that result in the transitive tournament. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>inv</mtext>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be the maximum of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>inv</mtext>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>T</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> taken over <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-vertex tournaments. It is well k","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"82-91"},"PeriodicalIF":0.9,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23251","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624602","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}
We study extensions of Turán Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey-Turán type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey-Turán density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt, and Murley.
{"title":"Weighted Turán Theorems With Applications to Ramsey-Turán Type of Problems","authors":"József Balogh, Domagoj Bradač, Bernard Lidický","doi":"10.1002/jgt.23244","DOIUrl":"https://doi.org/10.1002/jgt.23244","url":null,"abstract":"<p>We study extensions of Turán Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey-Turán type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey-Turán density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt, and Murley.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"59-71"},"PeriodicalIF":0.9,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23244","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624865","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}
Tutte proved that 4-connected planar graphs are Hamiltonian. It is unknown if there is an analogous result on 1-planar graphs. In this paper, we characterize 4-connected 1-planar chordal graphs and show that all such graphs are Hamiltonian-connected. A crucial tool used in our proof is a characteristic of 1-planar 4-trees.
{"title":"4-Connected 1-Planar Chordal Graphs Are Hamiltonian-Connected","authors":"Licheng Zhang, Yuanqiu Huang, Shengxiang Lv, Fengming Dong","doi":"10.1002/jgt.23250","DOIUrl":"https://doi.org/10.1002/jgt.23250","url":null,"abstract":"<div>\u0000 \u0000 <p>Tutte proved that 4-connected planar graphs are Hamiltonian. It is unknown if there is an analogous result on 1-planar graphs. In this paper, we characterize 4-connected 1-planar chordal graphs and show that all such graphs are Hamiltonian-connected. A crucial tool used in our proof is a characteristic of 1-planar 4-trees.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"72-81"},"PeriodicalIF":0.9,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624866","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 article, we extend the duality relation between face colorings and integer flows of graphs on orientable surfaces in Tutte's flow theory to both orientable and nonorientable surfaces and study Bouchet's 6-flow conjecture from point of embeddings of graphs on surfaces. Consequently, we verify Bouchet's conjecture for a family of embedded graphs, which have a crosscap-contractible circuit.
{"title":"Signed Graphs, Nonorientable Surfaces, and Integer Flows","authors":"You Lu, Rong Luo, Cun-Quan Zhang, Zhang Zhang","doi":"10.1002/jgt.23249","DOIUrl":"https://doi.org/10.1002/jgt.23249","url":null,"abstract":"<div>\u0000 \u0000 <p>In this article, we extend the duality relation between face colorings and integer flows of graphs on orientable surfaces in Tutte's flow theory to both orientable and nonorientable surfaces and study Bouchet's 6-flow conjecture from point of embeddings of graphs on surfaces. Consequently, we verify Bouchet's conjecture for a family of embedded graphs, which have a crosscap-contractible circuit.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"48-58"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624818","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}
<div> <p>A <i>separating system</i> of a graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is a family <span></span><math> <semantics> <mrow> <mrow> <mi>S</mi> </mrow> </mrow> </semantics></math> of subgraphs of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> for which the following holds: for all distinct edges <span></span><math> <semantics> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> </semantics></math> of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>, there exists an element in <span></span><math> <semantics> <mrow> <mrow> <mi>S</mi> </mrow> </mrow> </semantics></math> that contains <span></span><math> <semantics> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> </semantics></math> but not <span></span><math> <semantics> <mrow> <mrow> <mi>f</mi> </mrow> </mrow> </semantics></math>. Recently, it has been shown that every graph of order <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math> admits a separating system consisting of <span></span><math> <semantics> <mrow> <mrow> <mn>19</mn> <mi>n</mi> </mrow> </mrow> </semantics></math> paths, improving the previous almost linear bound of <span></span><math> <semantics> <mrow> <mrow> <mi>O</
{"title":"Separating the Edges of a Graph by Cycles and by Subdivisions of \u0000 \u0000 \u0000 \u0000 \u0000 K\u0000 4","authors":"Fábio Botler, Tássio Naia","doi":"10.1002/jgt.23248","DOIUrl":"https://doi.org/10.1002/jgt.23248","url":null,"abstract":"<div>\u0000 \u0000 <p>A <i>separating system</i> of a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a family <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of subgraphs of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for which the following holds: for all distinct edges <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, there exists an element in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> that contains <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> but not <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Recently, it has been shown that every graph of order <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> admits a separating system consisting of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>19</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> paths, improving the previous almost linear bound of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>O</","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"41-47"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624596","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}
Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Paul Wollan
We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small “width”, which is measured in terms of its adhesion and the path-likeness of its torsos.
{"title":"A Grid Theorem for Strong Immersions of Walls","authors":"Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Paul Wollan","doi":"10.1002/jgt.23245","DOIUrl":"https://doi.org/10.1002/jgt.23245","url":null,"abstract":"<p>We show that a graph contains a large wall as a strong immersion minor if and only if the graph does not admit a tree-cut decomposition of small “width”, which is measured in terms of its adhesion and the path-likeness of its torsos.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"110 1","pages":"23-32"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23245","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144624421","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}
<div> <p>The recognition of Pfaffian bipartite graphs in polynomial time has been obtained, but this fact is still unknown for Pfaffian nonbipartite graphs. For a path <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </mrow> </semantics></math> and a cycle <span></span><math> <semantics> <mrow> <mrow> <msub> <mi>C</mi> <mi>n</mi> </msub> </mrow> </mrow> </semantics></math>, the Pfaffian graphs of Cartesian products <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </mrow> </semantics></math> were characterized by Lu and Zhang in 2014. Recently, Li and Wang characterized the Pfaffian graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> <mo>×</mo> <msub> <mi>P</mi> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mrow> </semantics></math> for <span></span><math> <semantics> <mrow>
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