We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges and sharing a common end, it has a Hamiltonian cycle using and avoiding unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length , and each edge belongs to cycles of every length . For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).
{"title":"Note on Hamiltonicity of Basis Graphs of Even Delta-Matroids","authors":"Donggyu Kim, Sang-il Oum","doi":"10.1002/jgt.23237","DOIUrl":"https://doi.org/10.1002/jgt.23237","url":null,"abstract":"<p>We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two 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> sharing a common end, it has a Hamiltonian cycle using <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>e</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and avoiding <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>3</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, and each edge belongs to cycles of every length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>ℓ</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"446-453"},"PeriodicalIF":0.9,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23237","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256304","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>Given a graph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> and a collection <span></span><math> <semantics> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> </semantics></math> of subsets of <span></span><math> <semantics> <mrow> <mrow> <msup> <mi>R</mi> <mi>d</mi> </msup> </mrow> </mrow> </semantics></math> indexed by the subsets of vertices of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math>, a constrained drawing of <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> </semantics></math> is a drawing where each edge is drawn inside some set from <span></span><math> <semantics> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> </semantics></math>, in such a way that nonadjacent edges are drawn in sets with disjoint indices. In this paper we prove a Ramsey-type result for such drawings. Furthermore, we show how the results can be used to obtain Helly-type theorems. More precisely, we prove the following. For each <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>b</mi> </mrow> </mrow> </semantics></math>, there is <span></span><math> <semantics> <mrow> <mrow> <mi>N</mi> <mo>=</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup>
{"title":"A Sharper Ramsey Theorem for Constrained Drawings","authors":"Pavel Paták","doi":"10.1002/jgt.23226","DOIUrl":"https://doi.org/10.1002/jgt.23226","url":null,"abstract":"<p>Given a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and a collection <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of subsets of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 \u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> indexed by the subsets of vertices of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, a constrained drawing of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a drawing where each edge is drawn inside some set from <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, in such a way that nonadjacent edges are drawn in sets with disjoint indices. In this paper we prove a Ramsey-type result for such drawings. Furthermore, we show how the results can be used to obtain Helly-type theorems. More precisely, we prove the following. For each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>, there is <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>N</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>O</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <msup>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 4","pages":"401-411"},"PeriodicalIF":0.9,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144256493","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}
Ján Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera
A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for 3-edge-colourable cubic graphs, but remains widely open for graphs that are not 3-edge-colourable. The aim of this paper is to verify the validity of Berge's conjecture for cubic graphs that are in a certain sense close to 3-edge-colourable graphs. We measure the closeness by looking at the colouring defect, which is defined as the minimum number of edges left uncovered by any collection of three perfect matchings. While 3-edge-colourable graphs have defect 0, every bridgeless cubic graph with no 3-edge-colouring has defect at least 3. In 2015, Steffen proved that the Berge conjecture holds for cyclically 4-edge-connected cubic graphs with colouring defect 3 or 4. Our aim is to improve Steffen's result in two ways. We show that all bridgeless cubic graphs with defect 3 satisfy Berge's conjecture irrespectively of their cyclic connectivity. If, additionally, the graph in question is cyclically 4-edge-connected, then four perfect matchings suffice, unless the graph is the Petersen graph. The result is best possible as there exists an infinite family of cubic graphs with cyclic connectivity 3 which have defect 3 but cannot be covered with four perfect matchings.
{"title":"Berge's Conjecture for Cubic Graphs With Small Colouring Defect","authors":"Ján Karabáš, Edita Máčajová, Roman Nedela, Martin Škoviera","doi":"10.1002/jgt.23231","DOIUrl":"https://doi.org/10.1002/jgt.23231","url":null,"abstract":"<div>\u0000 \u0000 <p>A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for 3-edge-colourable cubic graphs, but remains widely open for graphs that are not 3-edge-colourable. The aim of this paper is to verify the validity of Berge's conjecture for cubic graphs that are in a certain sense close to 3-edge-colourable graphs. We measure the closeness by looking at the colouring defect, which is defined as the minimum number of edges left uncovered by any collection of three perfect matchings. While 3-edge-colourable graphs have defect 0, every bridgeless cubic graph with no 3-edge-colouring has defect at least 3. In 2015, Steffen proved that the Berge conjecture holds for cyclically 4-edge-connected cubic graphs with colouring defect 3 or 4. Our aim is to improve Steffen's result in two ways. We show that all bridgeless cubic graphs with defect 3 satisfy Berge's conjecture irrespectively of their cyclic connectivity. If, additionally, the graph in question is cyclically 4-edge-connected, then four perfect matchings suffice, unless the graph is the Petersen graph. The result is best possible as there exists an infinite family of cubic graphs with cyclic connectivity 3 which have defect 3 but cannot be covered with four perfect matchings.</p></div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"387-396"},"PeriodicalIF":0.9,"publicationDate":"2025-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944819","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}
Given a finite, simple, connected graph with , we consider the associated graph Laplacian matrix with eigenvalues . One can also consider the same graph equipped with positive edge weights
给定一个有限简单连通图G = (V)，E)与∣V∣= n,我们考虑特征值为0的关联图拉普拉斯矩阵L = D−Aλ 1 <；λ 2≤λ n。也可以考虑具有正边权w的图：E→R >；0归一化到∑e∈ewe =∣e∣和相关的加权拉普拉斯矩阵Lw . 我们说G是保形刚性的如果恒边权使第二个特征值λ 2 (w) （L w / w）最小化λ n （w '）lw '除以所有w‘，也就是说，对于所有的w， w ’，λ 2 (w)≤λ 2(1)≤λ n(1)≤λ n （w '）。保形刚度在G中需要大量的结构。每一个边传递图都是保角刚性的。我们证明了每一个距离正则图，从而每一个强正则图，都是保角刚性的。某些特殊的图嵌入可以用来表征保形刚度。
In this article, we investigate the problem of finding in a given weighted hypergraph a subhypergraph with the maximum possible density. Using the notion of a support matrix we prove that the density of an optimal subhypergraph is equal to for an optimal support matrix . Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We study the density decomposition of a hypergraph and show that it is a significant refinement of the Dulmage–Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the density decomposition for a given weighted hypergraph.
{"title":"Eigenvalue Approach to Dense Clusters in Hypergraphs","authors":"Yuly Billig","doi":"10.1002/jgt.23218","DOIUrl":"https://doi.org/10.1002/jgt.23218","url":null,"abstract":"<p>In this article, we investigate the problem of finding in a given weighted hypergraph a subhypergraph with the maximum possible density. Using the notion of a support matrix we prove that the density of an optimal subhypergraph is equal to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mo>∥</mo>\u0000 \u0000 <msup>\u0000 <mi>A</mi>\u0000 \u0000 <mi>T</mi>\u0000 </msup>\u0000 \u0000 <mi>A</mi>\u0000 \u0000 <mo>∥</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> for an optimal support matrix <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We study the density decomposition of a hypergraph and show that it is a significant refinement of the Dulmage–Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the density decomposition for a given weighted hypergraph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"353-365"},"PeriodicalIF":0.9,"publicationDate":"2025-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23218","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944639","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 define the average -TSP distance of a graph as the average length of a shortest closed walk visiting vertices, that is, the expected length of the solution for a random TSP instance with uniformly random chosen vertices. We prove relations with the average -Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for given the order of the graph.
{"title":"The Average Solution of a TSP Instance in a Graph","authors":"Stijn Cambie","doi":"10.1002/jgt.23232","DOIUrl":"https://doi.org/10.1002/jgt.23232","url":null,"abstract":"<div>\u0000 \u0000 <p>We define the average <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-TSP distance <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>μ</mi>\u0000 \u0000 <mrow>\u0000 <mtext>tsp</mtext>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of a graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> as the average length of a shortest closed walk visiting <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> vertices, that is, the expected length of the solution for a random TSP instance with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> uniformly random chosen vertices. We prove relations with the average <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>μ</mi>\u0000 \u0000 <mrow>\u0000 <mtext>tsp</mtext>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> given the order of the graph.</p>\u0000 </div>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"332-338"},"PeriodicalIF":0.9,"publicationDate":"2025-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945031","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}
An independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let
有限无向图G中大小为k的独立集是k的集合图的顶点，其中没有两个被一条边连接。设x k (G)为图G中大小为k的独立集合，设α(G) = max {k≥0 :x k (G)≠0}。1987年，阿拉维，马尔德，施文克，Erdős询问独立集合序列x 0 (G)， x1 (G)，... ,x α (G)(G)是单峰的（序列先上升后下降）。这个问题仍然悬而未决。2006年，Levit和Mandrescu证明了树的独立集合序列的最后三分之一是递减的。我们展示了前46个。 随机树中8%的独立集合序列随着顶点数量趋于无穷而呈（指数）高概率增长。所以，关于Alavi， Malde， Schwenk和Erdős的问题有“五分之四正确”的高概率。我们还展示了Erdős-Rényi随机图的独立集合序列的单模性，当单个顶点的期望程度很大时（随着图中顶点的数量趋于无穷，除了模态附近的一个小区域外，具有[指数]高概率）。对于随机正则图，给出了较弱的结果。大小为k的独立集合随k变化的结构在概率、统计物理、组合学和计算机科学中都很有趣。
{"title":"Independent Sets of Random Trees and Sparse Random Graphs","authors":"Steven Heilman","doi":"10.1002/jgt.23225","DOIUrl":"https://doi.org/10.1002/jgt.23225","url":null,"abstract":"<p>An independent set of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in a finite undirected graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> is a set of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> vertices of the graph, no two of which are connected by an edge. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>x</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> be the number of independent sets of size <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in the graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> and let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>α</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>max</mi>\u0000 \u0000 <mrow>\u0000 <mo>{</mo>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"294-309"},"PeriodicalIF":0.9,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944734","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}
Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition of adhesion less than that efficiently distinguishes every two distinct -profiles, and which has the further property that every separable -block is equal to the unique part of in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than . For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.
{"title":"Refining Tree-Decompositions so That They Display the k-Blocks","authors":"Sandra Albrechtsen","doi":"10.1002/jgt.23230","DOIUrl":"https://doi.org/10.1002/jgt.23230","url":null,"abstract":"<p>Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> of adhesion less than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> that efficiently distinguishes every two distinct <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-profiles, and which has the further property that every separable <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>-block is equal to the unique part of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>T</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>V</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math> in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </mrow>\u0000 </semantics></math>. For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 3","pages":"310-314"},"PeriodicalIF":0.9,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143944735","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号