European Journal of Combinatorics最新文献

{"title":"Kempe changes in degenerate graphs","authors":"Marthe Bonamy,&nbsp;Vincent Delecroix,&nbsp;Clément Legrand–Duchesne","doi":"10.1016/j.ejc.2023.103802","DOIUrl":"10.1016/j.ejc.2023.103802","url":null,"abstract":"<div><p>We consider Kempe changes on the <span><math><mi>k</mi></math></span>-colorings of a graph on <span><math><mi>n</mi></math></span> vertices. If the graph is <span><math><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>-degenerate, then all its <span><math><mi>k</mi></math></span>-colorings are equivalent up to Kempe changes. However, the sequence between two <span><math><mi>k</mi></math></span>-colorings that arises from the proof may have length exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most <span><math><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></math></span>. Namely, any two <span><math><mi>k</mi></math></span>-colorings are equivalent up to <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As one of the main results, we derive that given an <span><math><mi>n</mi></math></span><span>-vertex graph with maximum degree </span><span><math><mi>Δ</mi></math></span>, the <span><math><mi>Δ</mi></math></span>-colorings are all equivalent up to <span><math><mrow><msub><mrow><mi>O</mi></mrow><mrow><mi>Δ</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> Kempe changes, unless <span><math><mrow><mi>Δ</mi><mo>=</mo><mn>3</mn></mrow></math></span> and some connected component is a 3-prism, that is <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>□</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>, in which case there exist some non-equivalent 3-colorings.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103802"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138536325","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}

{"title":"Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation","authors":"Fabricio Benevides ,&nbsp;Jean-Claude Bermond ,&nbsp;Hicham Lesfari ,&nbsp;Nicolas Nisse","doi":"10.1016/j.ejc.2023.103801","DOIUrl":"10.1016/j.ejc.2023.103801","url":null,"abstract":"<div><p>Let <span><math><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></math></span><span> be any non negative integer and let </span><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 any undirected graph in which a subset <span><math><mrow><mi>D</mi><mo>⊆</mo><mi>V</mi></mrow></math></span> of vertices are initially <em>infected</em>. We consider the process in which, at every step, each non-infected vertex with at least <span><math><mi>r</mi></math></span> infected neighbours becomes infected and an infected vertex never becomes non-infected. The problem consists in determining the minimum size <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of an initially infected vertices set <span><math><mi>D</mi></math></span> that eventually infects the whole graph <span><math><mi>G</mi></math></span>. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math></span><span> for any connected graph </span><span><math><mi>G</mi></math></span>. The case when <span><math><mi>G</mi></math></span> is the <span><math><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></math></span> grid, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, and <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> is well known and appears in many puzzle books, in particular due to the elegant proof that shows that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>n</mi></mrow></math></span> for all <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. We study the cases of square grids, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, and tori, <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub></math></span>, when <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></mrow></mrow></math></span>. We show that <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>⌈</mo><mfrac><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>⌉</mo></mrow></mrow></math></span> for every <span><math><mi>n</mi></math></span> even and that <span><math><mrow><mrow><mo>⌈</mo><mfrac><mrow><ms","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103801"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894821","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}

{"title":"Walks avoiding a quadrant and the reflection principle","authors":"Mireille Bousquet-Mélou,&nbsp;Michael Wallner","doi":"10.1016/j.ejc.2023.103803","DOIUrl":"10.1016/j.ejc.2023.103803","url":null,"abstract":"<div><p>We continue the enumeration of plane lattice walks with small steps avoiding the negative quadrant, initiated by the first author in 2016. We solve in detail a new case, namely the king model where all eight nearest neighbour steps are allowed. The associated generating function is proved to be the sum of a simple, explicit D-finite series (related to the number of walks confined to the first quadrant), and an algebraic one. This was already the case for the two models solved by the first author in 2016. The principle of the approach is also the same, but challenging theoretical and computational difficulties arise as we now handle algebraic series of larger degree.</p><p>We expect a similar algebraicity phenomenon to hold for the seven <em>Weyl</em> step sets, which are those for which walks confined to the first quadrant can be counted using the reflection principle. With this paper, this is now proved for three of them. For the remaining four, we predict the D-finite part of the solution, and in three of the four cases, give evidence for the algebraicity of the remaining part.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103803"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119309","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}

{"title":"Group coloring and group connectivity with non-isomorphic groups of the same order","authors":"Rikke Langhede,&nbsp;Carsten Thomassen","doi":"10.1016/j.ejc.2023.103816","DOIUrl":"10.1016/j.ejc.2023.103816","url":null,"abstract":"<div><p>For every natural number <span><math><mi>k</mi></math></span>, there exists a planar graph which is <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span>-colorable, but not <span><math><mi>Γ</mi></math></span>-colorable for any other Abelian group <span><math><mi>Γ</mi></math></span> of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>. Its dual graph is <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span>-connected, but not <span><math><mi>Γ</mi></math></span>-connected for any other Abelian group <span><math><mi>Γ</mi></math></span> of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103816"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669823001336/pdfft?md5=4d2c19bd40b56019f9f9869e2dedb235&pid=1-s2.0-S0195669823001336-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135348627","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}

{"title":"Extremal graphs without long paths and large cliques","authors":"Gyula O.H. Katona ,&nbsp;Chuanqi Xiao","doi":"10.1016/j.ejc.2023.103807","DOIUrl":"10.1016/j.ejc.2023.103807","url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of graphs. A graph is called <span><math><mi>F</mi></math></span>-free if it does not contain any member of <span><math><mi>F</mi></math></span> as a subgraph. The Turán number of <span><math><mi>F</mi></math></span> is the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graph and is denoted by <span><math><mrow><mo>ex</mo><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></math></span>. The same maximum under the additional condition that the graphs are connected is <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>conn</mi></mrow></msub><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></math></span>. Let <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the path on <span><math><mi>k</mi></math></span> vertices, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be the clique on <span><math><mi>m</mi></math></span> vertices. We determine <span><math><mrow><mo>ex</mo><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>k</mi><mo>&gt;</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mo>ex</mo></mrow><mrow><mi>conn</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> if <span><math><mrow><mi>k</mi><mo>&gt;</mo><mi>m</mi></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103807"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669823001245/pdfft?md5=1271cd195bffe5dd20cfa6c9c7c1cd05&pid=1-s2.0-S0195669823001245-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388839","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}

{"title":"Finding strong components using depth-first search","authors":"Robert E. Tarjan ,&nbsp;Uri Zwick","doi":"10.1016/j.ejc.2023.103815","DOIUrl":"10.1016/j.ejc.2023.103815","url":null,"abstract":"<div><p>We survey three algorithms that use depth-first search to find the strong components of a directed graph in linear time: (1) Tarjan’s algorithm; (2) a cycle-finding algorithm; and (3) a bidirectional search algorithm.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103815"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135963016","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}

{"title":"Variations on a tree","authors":"Pascale Kuntz ,&nbsp;Bruno Pinaud","doi":"10.1016/j.ejc.2023.103808","DOIUrl":"10.1016/j.ejc.2023.103808","url":null,"abstract":"<div><p>The family tree is like an inherited “object” that has been passed down through many generations, with many and varied definitions which distort the tree both as a combinatorial object and in its visual representations. Moreover, whether used by amateur genealogists or academic researchers, it is always contextualized by both validated exogenous knowledge and by implicit knowledge. In this paper, we explore introducing certain contextual information that is associated with a locally defined dissimilarity between individuals of the same generation. We propose a new heuristic based on a radial representation of a node-link model which seeks to preserve local proximities in the layout. This heuristic is applied in an original form, which is that of Pierre Rosenstiehl’s “scientific family tree”.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103808"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934597","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}

{"title":"Meanders: A personal perspective to the memory of Pierre Rosenstiehl","authors":"Alexander K. Zvonkin","doi":"10.1016/j.ejc.2023.103817","DOIUrl":"10.1016/j.ejc.2023.103817","url":null,"abstract":"<div><p></p><blockquote><p> <!-->J’errais dans un méandre<!--> <!-->; <!--> <!-->J’avais trop de partis, <!--> <!-->Trop compliqués, à prendre... <!--> <!-->(Edmond Rostand, <!--> <!-->Cyrano de Bergerac)</p></blockquote><span> Meander is a self-avoiding closed curve on a plane which intersects<span><span> a straight line in a given set of points. Meander is a very simple object. In the elementary school, we may ask children to draw a few meanders and to admire their strange beauty. In the middle school, we may ask children to perform an exhaustive search of the meanders with a small number of intersections with the line. Then, gradually, we start to perceive an incredible profoundness of the subject, whose relations go from enumeration to quantum field theory and </span>string theory. Pierre Rosenstiehl was one of the pioneers in the study of the algorithmic aspects of meanders, and he also was a passionate connoisseur of labyrinths, of which the meanders are a particular case.</span></span></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103817"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135389045","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}

{"title":"The repetition threshold of episturmian sequences","authors":"L’ubomíra Dvořáková,&nbsp;Edita Pelantová","doi":"10.1016/j.ejc.2024.104001","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104001","url":null,"abstract":"<div><p>The repetition threshold of a class <span><math><mi>C</mi></math></span> of infinite <span><math><mi>d</mi></math></span>-ary sequences is the smallest real number <span><math><mi>r</mi></math></span> such that in the class <span><math><mi>C</mi></math></span> there exists a sequence that avoids <span><math><mi>e</mi></math></span>-powers for all <span><math><mrow><mi>e</mi><mo>&gt;</mo><mi>r</mi></mrow></math></span>. This notion was introduced by Dejean in 1972 for the class of all sequences over a <span><math><mi>d</mi></math></span>-letter alphabet. Thanks to the effort of many authors over more than 30 years, the precise value of the repetition threshold in this class is known for every <span><math><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></math></span>. The repetition threshold for the class of Sturmian sequences was determined by Carpi and de Luca in 2000. Sturmian sequences may be equivalently defined in various ways, therefore there exist many generalizations to larger alphabets. Rampersad, Shallit and Vandome in 2020 initiated a study of the repetition threshold for the class of balanced sequences – one of the possible generalizations of Sturmian sequences. Here, we focus on the class of <span><math><mi>d</mi></math></span>-ary episturmian sequences – another generalization of Sturmian sequences introduced by Droubay, Justin and Pirillo in 2001. We show that the repetition threshold of this class is reached by the <span><math><mi>d</mi></math></span>-bonacci sequence and its value equals <span><math><mrow><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, where <span><math><mrow><mi>t</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span> is the unique positive root of the polynomial <span><math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mo>⋯</mo><mo>−</mo><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 104001"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242538","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}

{"title":"Testing the planar straight-line realizability of 2-trees with prescribed edge lengths","authors":"Carlos Alegría,&nbsp;Manuel Borrazzo,&nbsp;Giordano Da Lozzo,&nbsp;Giuseppe Di Battista,&nbsp;Fabrizio Frati,&nbsp;Maurizio Patrignani","doi":"10.1016/j.ejc.2023.103806","DOIUrl":"10.1016/j.ejc.2023.103806","url":null,"abstract":"<div><p>We study a classic problem introduced thirty years ago by Eades and Wormald. Let <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span><span> be a weighted planar graph, where </span><span><math><mrow><mi>λ</mi><mo>:</mo><mi>E</mi><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math></span> is a <em>length function</em>. The <span><span>Fixed Edge-Length Planar Realization</span></span> problem (<span>FEPR</span> for short) asks whether there exists a <em>planar straight-line realization</em> of <span><math><mi>G</mi></math></span>, i.e., a planar straight-line drawing of <span><math><mi>G</mi></math></span> where the Euclidean length of each edge <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow></math></span> is <span><math><mrow><mi>λ</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span>.</p><p>Cabello, Demaine, and Rote showed that the <span>FEPR</span> problem is <span>NP</span>-hard, even when <span><math><mi>λ</mi></math></span> assigns the same value to all the edges and the graph is triconnected. Since the existence of large triconnected minors is crucial to the known <span>NP</span>-hardness proofs, in this paper we investigate the computational complexity of the <span>FEPR</span> problem for weighted 2-trees, which are <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-minor free. We show the <span>NP</span>-hardness of the problem, even when <span><math><mi>λ</mi></math></span> assigns to the edges only up to four distinct lengths. Conversely, we show that the <span>FEPR</span> problem is linear-time solvable when <span><math><mi>λ</mi></math></span> assigns to the edges up to two distinct lengths, or when the input has a prescribed embedding. Furthermore, we consider the <span>FEPR</span> problem for weighted maximal outerplanar graphs and prove it to be linear-time solvable if their dual tree is a path, and cubic-time solvable if their dual tree is a caterpillar. Finally, we prove that the <span>FEPR</span> problem for weighted 2-trees is slice-wise polynomial in the length of the large path.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103806"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135638189","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}