Jan Dreier, Jakub Gajarsky, Yiting Jiang, Patrice Ossona de Mendez, Jean-Florent Raymond
In this paper, we prove that a graph $G$ with no $K_{s,s}$-subgraph and twin-width $d$ has $r$-admissibility and $r$-coloring numbers bounded from above by an exponential function of $r$ and that we can construct graphs achieving such a dependency in $r$.
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Recently, several vertex identifying notions were introduced (identifying coloring, lid-coloring, ...), these notions were inspired by identifying codes. All of them, as well as original identifying code, are based on separating two vertices according to some conditions on their closed neighborhood. Therefore, twins can not be identified. So most of known results focus on twin-free graph. Here, we show how twins can modify optimal value of vertex-identifying parameters for identifying coloring and locally identifying coloring. \textbf{Keyword :} Identifying coloring, locally identifying colori…
Perfect colorings of the infinite square grid: coverings and twin colors
本文研究无限方格图的完美着色与覆盖问题，证明所有此类着色要么对应图自同构群作用下的轨道划分，要么具有合并后仍保持完美着色的孪生颜色，并对孪生颜色情形进行了分类。
Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced vertex coloring $c : V (G) \rightarrow \mathbb{Z}_k$ defined by $c(v) = \sum_{e\in E_v} s(e) \mbox{ in } \mathbb{Z}_k,$ (where the indicated sum is computed in $\mathbb{Z}_k$ and $E_v$ denotes the set of all edges incident to $v$) results in a proper vertex coloring of $G$, then we refer to such a coloring as an improper twin $k$-edge coloring. The minimum $k$ fo…
Twin-width is a recently introduced graph parameter. In this article, we compute twin-width of various finite graphs. In particular, we prove that the twin-widths of finite graphs with 4 and 5 vertices are less than equal to 1 and 2, respectively. We show that the constructions of dual graph and line graph do not preserve twin-width. Also, we give upper bounds for the twin-width of King's graph and Rook's graph.
The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have seen first small explicit upper bounds of 183 by Jacob and Pilipczuk [arXiv, January 2022], 583 by Bonnet et al. [arXiv, February 2022], and of 37 by Bekos et al. [arXiv, April 2022]. We prove that the twin-width of planar graphs is at most 9. Furthermore, if a planar graph is also bipartite, then its twin-width is at most 6.
Twin-width of graphs with tree-structured decompositions
本文研究树结构分解下图的孪生宽度与其分量宽度的关系，证明孪生宽度至多为强树宽的两倍，并针对双连通、三连通及拟4连通分量分别给出线性或二次上界，同时分析了树分解粘合度的影响。
Group twin coloring of graphs
本文研究图的群孪生色指数，提出猜想 $\chi'_g(G)\leq \Delta(G)+3$，证明对树成立，并推广到一般图得到上界 $2(\Delta(G)+{\rm col}(G))-5$，改进了已知结果。
Coloring Hardness on Low Twin-Width Graphs
研究显示，尽管某些NP难问题在双宽度为4的图类上仍难解，但Min Coloring在双宽度为3的图类上为NP难，揭示了该类图在着色问题上的新特性。
On Ahn-Hendrey-Kim-Oum question for twin-width of graphs with 6 vertices
本文研究了Ahn-Hendrey-Kim-Oum提出的关于是否存在twin-width至少为n/2的n顶点图的开放问题，证明了不存在具有6个顶点的此类图，即任何6顶点图的twin-width均不超过2。
Bounded Twin-Width Tournaments are $\dchi$-Bounded
该研究证明有界双宽锦标赛是dχ-有界的，通过提出混合参数传递定理，结合有序图相关定理与障碍，解决了特定学者的猜想。
Twin-width of Planar Graphs; a Short Proof
本文针对平面图孪生宽度的上界问题，给出了一个简短且自包含的证明，证明平面图的孪生宽度至多为 11，从而为理解平面图及超平面图的孪生宽度上界提供了新的简化视角。
We prove that every triangle-free graph of tree-width t has chromatic number at most ceil((t + 3)/2), and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t.
We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in \mathbb{N}$, every graph $G\in \mathcal{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgrap…
Planar graph with twin-width seven
本文构造了一个双宽等于七的平面图，解决了平面图双宽的具体构造问题。
For any small positive real $\varepsilon$ and integer $t > \frac{1}{\varepsilon}$, we build a graph with a vertex deletion set of size $t$ to a tree, and twin-width greater than $2^{(1-\varepsilon) t}$. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least $2-\varepsilon$. Except for the one in oriented twin-width, these lower bounds are essentially tight.
For building up twin-graphic lattices towards topological cryptograph, we define four kinds of new odd-magic-type colorings: odd-edge graceful-difference total coloring, odd-edge edge-difference total coloring, odd-edge edge-magic total coloring, and odd-edge felicitous-difference total coloring in this article. Our RANDOMLY-LEAF-ADDING algorithms are based on adding randomly leaves to graphs for producing continuously graphs admitting our new odd-magic-type colorings. We use complex graphs to make caterpillar-graphic lattices and complementary graphic lattices, such that each graph in these n…
Twin-width is a new parameter informally measuring how diverse are the neighbourhoods of the graph vertices, and it extends also to other binary relational structures, e.g. to digraphs and posets. It was introduced just very recently, in 2020 by Bonnet, Kim, Thomasse and Watrigant. One of the core results of these authors is that FO model checking on graph classes of bounded twin-width is in FPT. With that result, they also claimed that posets of bounded width have bounded twin-width, thus capturing prior result on FO model checking of posets of bounded width in FPT. However, their translation…
Quasirandomness and Uniform Twin-Width
本文研究有限群的拟随机度与均匀孪生宽度的关系，证明前者给出后者的多项式下界、最小忠实复表示次数给出线性上界，非阿贝尔有限单群中二者多项式等价，还构造了特定有限表现群并确定三个Thompson群的均匀孪生宽度。
On the twin-width of near-regular graphs
本文研究近正则图的孪生宽度，发现高孪生宽度的有界度极值图不对称，并为循环图、d-退化图及强正则图建立界，确定了约翰逊图和循环拉丁方图的孪生宽度。
The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the algorithmic point of view. In particular, classes of efficiently bounded twin-width include proper interval graphs, and (as digraphs) posets of width k. Inspired by an existing generalization of interval graphs into so-called k-thin graphs, we define a new class of proper k-mixed-thin graphs which largely generalizes proper interval graphs. We prove that prope…
The graph invariant twin-width was recently introduced by Bonnet, Kim, Thomassé, and Watrigan. Problems expressible in first-order logic, which includes many prominent NP-hard problems, are tractable on graphs of bounded twin-width if a certificate for the twin-width bound is provided as an input. Computing such a certificate, however, is an intrinsic problem, for which no nontrivial algorithm is known. In this paper, we propose the first practical approach for computing the twin-width of graphs together with the corresponding certificate. We propose efficient SAT-encodings that rely on a char…
Bonnet, Kim, Thomassé, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p…
We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $χ$-bounded.
We prove that it is always possible to color online nonrepetitively any (partial) $k$-tree (that is, graphs with tree-width at most $k$) with $4^k$ colors. This implies that it is always possible to color online nonrepetitively cycles, trees and series-parallel graphs with $16$ colors. Our results generalize the respective (offline) nonrepetitive coloring results.
We prove that for every $t\in \mathbb{N}$ there is a constant $γ_t$ such that every graph with twin-width at most $t$ and clique number $ω$ has chromatic number bounded by $2^{γ_t \log^{4t+3} ω}$. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially $χ$-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially $χ$-bounded.
Accessibility and Twin-width
研究有限生成群中有限孪生宽度与可达性的关系，通过结合威尔克斯构造及博内等人的结果，证明存在有限生成群$\Gamma$，其有有限一致孪生宽度但不可达，相应凯莱图也如此。
Twin-width of sparse random graphs
该研究证明了稀疏随机图的双宽度上限，并展示了其在实际随机图中的应用。
The essential annihilating-ideal graph $\mathcal{EG}(R)$ of a commutative unital ring $R$ is a simple graph whose vertices are non-zero ideals of $R$ with non-zero annihilator and there exists an edge between two distinct vertices $I,J$ if and only if $Ann(IJ)$ has a non-zero intersection with any non-zero ideal of $R$. In this paper, we show that $\mathcal{EG}(R)$ is weakly perfect, if $R$ is Noetherian and an explicit formula for the clique number of $\mathcal{EG}(R)$ is given. Moreover, the structures of all rings whose essential annihilating-ideal graphs have chromatic number $2$ are fully…
Twin-width is a recently introduced graph parameter with applications in algorithmics, combinatorics, and finite model theory. For graphs of bounded degree, finiteness of twin-width is preserved by quasi-isometry. Thus, through Cayley graphs, it defines a group invariant. We prove that groups which are abelian, hyperbolic, ordered, solvable, or with polynomial growth, have finite twin-width. Twin-width can be characterised by excluding patterns in the self-action by product of the group elements. Based on this characterisation, we propose a strengthening called uniform twin-width, which is sta…
The Twin-Width of Graphs of Bounded VC-Dimension
本文研究有界VC维图类的孪生宽度，通过修改会议图证明分裂图、二部图和余二部图可具有线性孪生宽度，但利用邻域划分收缩工具证明有界VC维图的孪生宽度为次线性。
We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Cli…
Bounds on the Twin-Width of Product Graphs
本文探讨了积图的twin-width是否受原图twin-width及最大度的函数界定，证明了对多种图积成立，模积不成立，并给出了紧的界限示例及改进。
We continue developing the theory around the twin-width of totally ordered binary structures, initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable algorithm for first-order model checking on structures given with an $O(1)$-sequence (certificate of boun…
Bounded twin-width graphs are polynomially $χ$-bounded
本文证明有界 twin-width 图满足多项式 $χ$-有界性，并通过区分延迟扩展与右扩展、重构分解树论证，推广了已有拟多项式和 clique-width 结果。
Counting equitable $k$-colorings in graphs of bounded clique-width
研究有界团宽图中公平 $k$ 着色的计数问题，给出多项式时间算法并证明团宽依赖最优，同时改进线性团宽算法，并证明 $P_t$-free 图中公平列表 $3$ 着色可次指数时间计数。
Odd coloring graphs with linear neighborhood complexity
该研究证明了具有线性邻域复杂度的图类具有有界不良奇色数，并展示了圆图、双宽度有界类等情形下的结论。
Weisfeiler-Leman on graphs of small twin-width
本文证明双宽1的图可由3维Weisfeiler-Leman算法解决同构问题，而双宽4的图则无法用固定维度的算法解决，同时证明双宽2的稳定图具有有界秩宽。
Twin-width of subdivisions of multigraphs
本文构造有限多重图集合$F_d$，刻画了多重图经充分细分后所得图的twin-width界，并据此证明了$7\times7$网格的twin-width为$4$。
Twin-width of random graphs
本文研究 Erdős-Rényi 随机图的双宽度，发现临界概率约 0.4 处存在行为转变，并给出稠密、均匀及稀疏情形下双宽度的渐近估计与集中性结果。
Twin-width of graphs on surfaces
本文证明了可嵌入欧拉亏格$g$曲面的图的孪生宽度上界为$18\sqrt{47g}+O(1)$，并给出二次时间算法及更强的乘积结构定理。
Sparse Graphs of Twin-width 2 Have Bounded Tree-width
本文证明孪生宽度至多为2且不含大完全二部子图的稀疏图具有多项式有界树宽，并据此给出稀疏图类上判定孪生宽度是否超过2的多项式时间算法，同时说明该结果无法推广到孪生宽度3。
Even-hole-free graphs are a graph class of much interest. Foley et al. [Graphs Comb. 36(1): 125-138 (2020)] have recently studied $(4K_1, C_4, C_6)$-free graphs, which form a subclass of even-hole-free graphs. Specifically, Foley et al. have shown an algorithm for coloring these graphs via bounded clique-width if they contain a $C_7$. In this note, we give a simpler and much faster algorithm via a more restrictive graph parameter, neighborhood diversity.
A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts red edges between two vertices representing non-homogeneous subsets, the twin-width is the minimum integer $d$ such that a contraction sequence keeps red degree at most $d$. By changing the condition imposed on the trigraphs (i.e., graphs with some edges being red) and possibly slightly tweaking the notion of contractions, we show how to characterize the well-establi…
We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs which have unbounded twin-width. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -- under common complexity-theoretic assumptions -- only if the class has bounded twin-width. For hereditary classes of ordered graphs, we show that bounded twin-width is equival…
Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs
该研究探讨图的无冲突着色问题，分析团宽度与最少着色数的关系，针对多种图类给出紧的上下界与判定复杂度，解决了区间图开邻域无冲突着色的公开问题。
We show that determining if an $n$-vertex graph has twin-width at most 4 is NP-complete, and requires time $2^{Ω(n/\log n)}$ unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that $n$-vertex graphs subdivided at least $2 \log n$ times have twin-width at most 4. We also show how to encode trigraphs $H$ (2-edge colored graphs involved in the definition of twin-width) into graphs $G$, in the sense that every $d$-sequence (sequence of vertex contractions witnessing that the twin-width is at most $d$) of $G$ inevitably creates $H$ as an induced subtrigraph, wh…
Twin-width and permutations
本文证明二元关系结构类具有有界孪生宽度当且仅当它是真排列类的一阶传导，并由此推出有界孪生宽度类中n顶点图的两两不同构数量至多为$2^{O(n)}$。
Twin-width one
本文研究 twin-width 至多为 1 的图的结构，证明其为排列图，并基于递归分解给出线性时间算法以生成 1-收缩序列或判定 twin-width 大于 1，同时刻画了距离遗传图并给出线性时间最优序列算法。
Twin-width of Planar Graphs is at most 8, and some Related Bounds
本文改进平面图孪生宽度上界至 8，并给出线性时间收缩序列，同时界定二部平面图、1-平面图和地图图。
First Order Logic and Twin-Width in Tournaments and Dense Oriented Graphs
本文刻画了竞赛图中易处理的一阶模型检测，证明其固定参数易处理性与有界twin-width等价，并提出了一个多项式时间近似算法，结果扩展至特定稠密有向图。
We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for $k$-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected $k$-Dominating Set and Total $k$-Dominating Set (albeit with a worse upper bound on the twin-width). The $k$-Independent Set problem admits…
Counting List Colorings of Unlabeled Graphs
本文研究无标号图的列表着色计数问题，定义了无标号列表着色函数，并证明在k足够大时，该函数与经典色多项式相等。
Twin-width is a newly introduced graph width parameter that aims at generalizing a wide range of "nicely structured" graph classes. In this work, we focus on obtaining good bounds on twin-width $\text{tww}(G)$ for graphs $G$ from a number of classic graph classes. We prove the following: - $\text{tww}(G) \leq 3\cdot 2^{\text{tw}(G)-1}$, where $\text{tw}(G)$ is the treewidth of $G$, - $\text{tww}(G) \leq \max(4\text{bw}(G),\frac{9}{2}\text{bw}(G)-3)$ for a planar graph $G$ with $\text{bw}(G) \geq 2$, where $\text{bw}(G)$ is the branchwidth of $G$, - $\text{tww}(G) \leq 183$ for a planar graph $…
The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$. We show that if a graph admits a $d$-contraction sequence, then it also has a linear-arity tree of $f(d)$-contractions, for some function $f$. First this permits to show that every bounded twin-width class is small, i.e., has at most $n…
Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)
本研究提出降带宽概念，证明真小调闭类有界降带宽，强化孪生宽度结果，并给出平面图等具体界，区分两者。
A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with $q$ colors. For fixed $q \ge 2$, we give an $\mathcal{O}^{\star}(q^{tw})$-time algorithm when the input graph is given together with one of its tree decompositions of width $tw$. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unb…
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper …
First-Order Logic and Twin-Width for Some Geometric Graphs
本文证明了某些几何图类中一阶逻辑可计算性与双宽度参数的等价性，包括非退化轴对齐单位线段图和圆弧图，但不适用于1.5D地形的可视图。
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first as…
We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures $\mathcal D$ of subclasses of $\mathcal C$, FO model checking is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs …
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